Recent zbMATH articles in MSC 47https://zbmath.org/atom/cc/472021-01-08T12:24:00+00:00WerkzeugConvergence rate of a new projected-type algorithm solving non-Lipschitz equilibrium problems.https://zbmath.org/1449.471162021-01-08T12:24:00+00:00"Trinh Ngoc Hai"https://zbmath.org/authors/?q=ai:trinh-ngoc-hai.Summary: In this paper, we introduce a new step size strategy for projection-type algorithms for solving strongly pseudomonotone equilibrium problems in a Hilbert space. In contrast to the work by \textit{Pham Ky Anh} and \textit{Nguyen The Vinh} [Numer. Algorithms 81, No. 3, 983--1001 (2019; Zbl 1446.47077)] and by \textit{P. Santos} and \textit{S. Scheimberg} [Comput. Appl. Math. 30, No. 1, 91--107 (2011; Zbl 1242.90265)], our methods do not require the step sizes being square summable. Moreover, at each step of the proposed algorithms, instead of solving a constrained problem, we only have to solve an unconstrained problem and compute a projection onto the feasible set or its intersection with a closed sphere. The strong convergence of the proposed algorithms is proven without any Lipschitz-type condition. Also, we evaluate the convergence rate of these algorithms. Using cutting hyperplanes, we refine the feasible set at the beginning of our algorithms. Thanks to this, we can apply the new algorithms to the equilibrium problems with non-closed and non-convex feasible set. Some numerical experiments and comparisons confirm efficiency of the proposed modification.On a compactness criteria for multidimensional Hardy type operator in \(p\)-convex Banach function spaces.https://zbmath.org/1449.470682021-01-08T12:24:00+00:00"Bandaliev, R. A."https://zbmath.org/authors/?q=ai:bandaliev.rovshan-alifaga-oglySummary: The main goal of this paper is to prove a criteria on compactness of a multidimensional Hardy type operator from weighted Lebesgue spaces into \(p\)-convex weighted Banach function spaces. The analogous problem for the dual operator is considered.The Jacobson radical of certain semicrossed products.https://zbmath.org/1449.471322021-01-08T12:24:00+00:00"Wiart, Jaspar"https://zbmath.org/authors/?q=ai:wiart.jasparSummary: We study the Jacobson radical of the semicrossed product \(A\times_\alpha P\) when \(\mathcal{A}\) is a simple \(C^*\)-algebra and \(P\) is either a semigroup contained in an abelian group or a free semigroup. A full characterization is obtained for a large subset of these semicrossed products and we apply our results to a number of examples.The proof and perturbation for generalized property \( (\omega')\).https://zbmath.org/1449.470302021-01-08T12:24:00+00:00"Zhao, Lingling"https://zbmath.org/authors/?q=ai:zhao.lingling"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: By means of the new spectrum defined in view of the property of consistence in Fredholm and index, we establish for a bounded linear operator \(T\) defined on a Hilbert space the sufficient and necessary conditions under which the generalized property \( (\omega')\) holds. We also study the stability of generalized property \( (\omega')\) under perturbations by finite rank operators and by power finite rank operators commuting with \(T\).Cesàro-like operators.https://zbmath.org/1449.470472021-01-08T12:24:00+00:00"Rhoades, B. E."https://zbmath.org/authors/?q=ai:rhoades.billy-e"Trutt, D."https://zbmath.org/authors/?q=ai:trutt.dSummary: In [\textit{B. K. Ghosh} et al., Proc. Am. Math. Soc. 66, 261--265 (1977; Zbl 0386.47009)], it was shown that the lower triangular generalized Hausdorff matrix \(H_\alpha\) with nonzero entries \(h_{nk}= (n+ \alpha+1)^{-1}\), for \(\alpha \geq 0\), is subnormal on \(\ell^2\) if and only if \(\alpha =0,1,2,\dots\). For \(0<h \leq 1\), the weighted Cesàro operator \(C'_h:\{a_n\} \rightarrow \{b_n\}\) on \(\ell^2\), when \(b_n= (a_0+d_1a_1+ \cdots +d_na_n)/(n+1)d_n\), is subnormal when \(d^2_j= \Gamma (j+1) \Gamma (h)/ \Gamma (j+h)\).
In this paper, we show that, when \(d_j= \Gamma (j+1) \Gamma (h)/ \Gamma (j+h)\), the square of the weights chosen above, then the corresponding operator \(C_h\) is bounded on \(\ell^2\) for \(0<h<3/2\), that \(H_\alpha\) is bounded on \(\ell^2\) for all non-integer \(\alpha <0\), and that \(C_h\) is closely related to \(H_{h-1}\). This relationship leads to our main result that \(C_h\) is only subnormal when \(h=1\), when it corresponds to the original Cesàro operator with \(\alpha =0\) and each \(d_j=1\).Existence of positive solutions of fractional differential equations with integral boundary conditions.https://zbmath.org/1449.340732021-01-08T12:24:00+00:00"He, Xingyue"https://zbmath.org/authors/?q=ai:he.xingyue"Gao, Chenghua"https://zbmath.org/authors/?q=ai:gao.chenghuaSummary: Based on the fixed-point index theory in a cone, by constructing a cone and the properties of the Green function, we give the existence, multiplicity and nonexistence of positive solutions for the following nonlinear boundary value problem
\[\begin{cases}
{}^CD^\alpha u (t) +\lambda f (t,u(t)) = 0, \quad t \in (0,1),\\
u (0) = u'' (0) = 0, \quad u (1) = \mu\int_0^1 u (s)\mathrm{d}s,
\end{cases}\]
with two parameters under different growth conditions, where \(2<\alpha<3\). \(0<\mu<2\) and \(\lambda>0\) are two parameters.Existence of solutions of a positive finite volume scheme for unsteady advection-diffusion equations.https://zbmath.org/1449.652212021-01-08T12:24:00+00:00"Zhang, Yanmei"https://zbmath.org/authors/?q=ai:zhang.yanmei"Lan, Bin"https://zbmath.org/authors/?q=ai:lan.bin"Sheng, Zhiqiang"https://zbmath.org/authors/?q=ai:sheng.zhiqiang"Yuan, Guangwei"https://zbmath.org/authors/?q=ai:yuan.guangweiSummary: A nonlinear positive finite volume scheme is developed in this paper for unsteady advection-diffusion equations on star-shaped polygonal meshes. The scheme has only cell-centered unknowns and preserves local conservation. Moreover, the existence of discrete solution for the nonlinear scheme is proved by using Brouwer fixed-point theorem. Numerical results are presented to show that the scheme has second-order accuracy.Caputo fractional differential inclusions of arbitrary order with nonlocal integro-multipoint boundary conditions.https://zbmath.org/1449.340082021-01-08T12:24:00+00:00"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Garout, Doa'a"https://zbmath.org/authors/?q=ai:garout.doaa"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmedSummary: We study a new class of boundary value problems of Caputo type fractional differential inclusions supplemented with nonlocal integro-multipoint boundary conditions. An existence result for the problem with convex valued (multivalued) map is obtained via nonlinear alternative of Leray-Schauder type, while the existence of solutions for the problem involving nonconvex valued map is established by means of Wegrzyk's fixed point theorem. Our results are well illustrated with examples.The extragradient algorithm with inertial effects extended to equilibrium problems.https://zbmath.org/1449.651132021-01-08T12:24:00+00:00"Ur Rehman, Habib"https://zbmath.org/authors/?q=ai:rehman.habib-ur"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Abubakar, Auwal Bala"https://zbmath.org/authors/?q=ai:bala-abubakar.auwal"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, two algorithms are proposed for a class of pseudomonotone and strongly pseudomonotone equilibrium problems. These algorithms can be viewed as a extension of the paper title, the extragradient algorithm with inertial effects for solving the variational inequality proposed by \textit{Q.-L. Dong} et al. [Optimization 65, No. 12, 2217--2226 (2016; Zbl 1358.90139)]. The weak convergence of the first algorithm is well established based on the standard assumption imposed on the cost bifunction. We provide a strong convergence for the second algorithm without knowing the strongly pseudomonoton and the Lipschitz-type constants of cost bifunction. The practical interpretation of a second algorithm is that the algorithm uses a sequence of step sizes that is converging to zero and non-summable. Numerical examples are used to assist the well-established convergence result, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.On Jensen's multiplicative inequality for positive convex functions of selfadjoint operators in Hilbert spaces.https://zbmath.org/1449.470352021-01-08T12:24:00+00:00"Dragomir, Silvestru Sever"https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: We obtain some multiplicative refinements and reverses of Jensen's inequality for positive convex/concave functions of selfadjoint operators in Hilbert spaces. Natural applications for power and exponential functions are provided.On the positive definite solution of a class of pair of nonlinear matrix equations.https://zbmath.org/1449.150382021-01-08T12:24:00+00:00"Ali, Hasem"https://zbmath.org/authors/?q=ai:ali.hasem"Hossein, Sk M."https://zbmath.org/authors/?q=ai:hossein.sk-monowarSummary: We find some necessary and sufficient conditions for the existence of Hermitian positive definite solution of a pair of nonlinear matrix equations of the form:
\[\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1 \\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}\]
and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.Stability of the solution semigroup for neutral delay differential equations.https://zbmath.org/1449.342452021-01-08T12:24:00+00:00"Fabiano, Richard"https://zbmath.org/authors/?q=ai:fabiano.richard-h"Payne, Catherine"https://zbmath.org/authors/?q=ai:payne.catherineSummary: We derive a new condition for delay-independent stability of systems of linear neutral delay differential equations. The method applies ideas from linear semigroup theory, and involves renorming the underlying Hilbert space to obtain a dissipative inequality on the infinitesimal generator of the solution semigroup. The new stability condition is shown to either improve upon or be independent of existing stability conditions.Almost uniform and strong convergences in ergodic theorems for symmetric spaces.https://zbmath.org/1449.470242021-01-08T12:24:00+00:00"Chilin, V."https://zbmath.org/authors/?q=ai:chilin.vladimir-ivanovich|chilin.vladmir"Litvinov, S."https://zbmath.org/authors/?q=ai:litvinov.s-v|litvinov.semyon-n|litvinov.sergej|litvinov.s-a|litvinov.sergeyA space $X\subset L^0_\nu$ is fully symmetric on $((0,\infty),\nu)$ if $f\in X$, $g\in L^0_\nu$, and the decreasing rearrangement of $f$ dominates that of $g: g^\ast\leq f^\ast$ pointwise (resp., $\int_0^s g^\ast (t)\, dt\leq \int_0^s f^\ast(t)\, dt$ for all $s>0$) implies that $g\in X$ and $\Vert g\Vert_X\leq \Vert f\Vert_X$.
The first main result extends the Dunford-Schwartz pointwise ergodic theorem in characterizing $\mathcal{R}_\mu=\{f\in L^1+L^\infty: \forall\lambda>0,\, \mu(\vert f\vert >\lambda)<\infty\}$ as a space on which pointwise ergodic limits converge uniformly. To be precise, let $(\Omega,\mathcal{A},\mu)$ be a measure space and $X$ a fully symmetric space on $(\Omega,\mathcal{A},\mu)$ such that the constant $1\notin X$. If $T\in DS$ (that is, $T$ is bounded on both $L^1$ and $L^\infty$) and $f\in X$, then the averages $M_n(T)(f)=\frac{1}{n}\sum_{k=0}^{n-1} T^k(f)$ converge a.u. to some $\hat{f}\in X$. In particular, $M_n(T)(f) =\frac{1}{n}\sum_{k=0}^{n-1} T^k (f)\, \to\hat{f}\in \mathcal{R}_\mu$ a.u. when $f\in \mathcal{R}_\mu$.
The proofs involve a reduction to the $L^1$ case and the Hopf maximal theorem ($\int_{M(T)^\ast(f)>0} f\, d\mu>0$), where $M(T)^\ast(f)(x)=\sup \vert M_n(T)(f)(x)\vert $, and the weak type inequalities $$\mu(M(T)^\ast(\vert f\vert)>\lambda)\leq \bigl(2\frac{\Vert f\Vert_p}{\lambda}\bigr)^p,\quad\lambda>0\, .$$
Theorem 3.4 then states that, if $\mu$ is $\sigma$-finite, then $\mathcal{R}_\mu$ is the largest subspace of $L^1+L^\infty$ on which convergence is almost uniform, that is, if $f\in (L^1+L^\infty)\setminus \mathcal{R}_\mu$, then there is a $T\in DS$ such that the sequence $M_n(T)(f)$ fails to converge almost everywhere. In fact, the maximality of $\mathcal{R}_\mu$ for a subspace $X$ is equivalent to constants not belonging to $X$. Orlicz spaces are used to illustrate the condition that constants are not members. Strong convergence of Cesàro means is also discussed in the context of characterizing validity of mean ergodicity for fully symmetric spaces.
Reviewer: Joseph Lakey (Las Cruces)A new system of general nonconvex set-valued variational inequalities.https://zbmath.org/1449.471042021-01-08T12:24:00+00:00"Chen, Rudong"https://zbmath.org/authors/?q=ai:chen.rudong"Jiang, Yaqian"https://zbmath.org/authors/?q=ai:jiang.yaqian"Wu, Chengyu"https://zbmath.org/authors/?q=ai:wu.chengyuSummary: In this article, our main job is to introduce a new kind of general nonconvex set-valued variational inequalities. We first transform the general nonconvex set-valued variational inequalities into the fixed point problems equivalently. By constructing a new perturbed projection algorithm, we prove that the iterative algorithm given in this paper is convergent under certain conditions.Mean value projection method of Fredholm integral operator.https://zbmath.org/1449.470292021-01-08T12:24:00+00:00"Ren, Hanjing"https://zbmath.org/authors/?q=ai:ren.hanjing"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.3"Zhu, Guangtian"https://zbmath.org/authors/?q=ai:zhu.guangtianSummary: In this paper, we introduce a numerical discrete method in which we use the mean of the function on a subinterval as a substitute of the original value on the corresponding subinterval. Then we prove that operators defined by the method are projection operators defined on a normed linear space possessing the property of self-adjointness and are not orthogonal. Besides, we conclude that the projection operators converge pointwise to the identity operator in \({L^p}\). Furthermore, by using the mean-value projection method dealing with the Fredholm integral operator, we gain the rationality of the proposed algorithm.Existence of positive solutions for a class of nonlinear fractional differential equations with boundary values.https://zbmath.org/1449.340712021-01-08T12:24:00+00:00"Cai, Huize"https://zbmath.org/authors/?q=ai:cai.huize"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaolingSummary: In this paper, by using the Schauder fixed point theorem and Krasnoselskii's fixed point theorem, the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation
\[{}^{\mathrm{C}}D_{0^+}^\alpha u(t) = f(t,u(t),u'(t), u''(t)), t \in (0,1),\]
\[u'(0) + u''(0) = 0, u'(1) + u''(1) = 0, u(0) = 0\]
is obtained, where \(2 < \alpha \leq 3,{}^{\mathrm{C}} D_{0^+}^\alpha\) is the Caputo fractional derivative.Periodic solutions for seasonally forced SIRS model with pulse vaccination.https://zbmath.org/1449.341232021-01-08T12:24:00+00:00"Wang, Lin"https://zbmath.org/authors/?q=ai:wang.lin.2|wang.lin.1|wang.lin|wang.lin.3|wang.lin.4"Pang, Yanni"https://zbmath.org/authors/?q=ai:pang.yanni"Li, Wenjin"https://zbmath.org/authors/?q=ai:li.wenjinSummary: Using the coincidence degree theory of Gaines-Mawhin, we prove the existence of periodic solutions for seasonally forced SIRS models with pulse vaccination. The effects of different loss of immunity rates on the infectious disease models are compared with numerical simulations.On a new class of Bernstein type operators based on Beta function.https://zbmath.org/1449.470332021-01-08T12:24:00+00:00"Bhatt, Dhawal J."https://zbmath.org/authors/?q=ai:bhatt.dhawal-j"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayan"Jana, Ranjan Kumar"https://zbmath.org/authors/?q=ai:jana.ranjan-kumarSummary: We develop Bernstein type operators using the Beta function and study their approximation properties. By using Korovkin's theorem, we achieve the uniform convergence of sequences of these operators. We obtain the rate of convergence in terms of modulus of continuity and establish the Voronovskaja type asymptotic result for these operators. At last, the graphical comparison of these newly defined operators with few of the fundamental but significant operators is discussed.\(g\)-\( (h,e)\)-mixed monotone operator and applications.https://zbmath.org/1449.470892021-01-08T12:24:00+00:00"Zheng, Xiaoxia"https://zbmath.org/authors/?q=ai:zheng.xiaoxia"Han, Wei"https://zbmath.org/authors/?q=ai:han.wei"Sang, Yanbing"https://zbmath.org/authors/?q=ai:sang.yanbingSummary: We study the existence and uniqueness of the fixed points by introducing \(g\)-\( (h,e)\)-mixed monotone operator and using the cone theory and the monotone iterative method. For the Sturm-Liouville boundary value problem, the existence and uniqueness of its ordinary solutions are studied by applying the main conclusions. In this paper, some existing results are improved and generalized, and a new method for studying nonlinear equation problems is proposed.Weighted estimates of fractional maximal operator and its commutator on weighted \(\lambda\)-central Morrey spaces.https://zbmath.org/1449.420342021-01-08T12:24:00+00:00"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Yang, Yuhe"https://zbmath.org/authors/?q=ai:yang.yuheSummary: By applying weighted inequalities and real variable methods, the boundedness of the fractional maximal operator with rough kernel is obtained in the weighted \(\lambda\)-central Morrey spaces is obtained. The boundedness of its commutator generated by a \(\lambda\)-central mean oscillation function is also proved.The compactness of commutators of bilinear fractional maximal operators on Multi-Morrey spaces.https://zbmath.org/1449.420312021-01-08T12:24:00+00:00"Guo, Qingdong"https://zbmath.org/authors/?q=ai:guo.qingdong"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangSummary: Let \({\mathcal{M}_\alpha}\) be the bilinear fractional maximal operators and let \(\vec{b} = ({b_1}, {b_2})\) be a collection of locally integrable functions. In this paper, we obtain that the commutators generated by the bilinear fractional maximal operators and the CMO (\(C_c^\infty\) closure under the BMO norm) functions are compact operators from the Morrey spaces to the Multi-Morrey spaces, where the commutators include the fractional maximal linear commutators \({\mathcal{M}_{\alpha, \sum \vec{b}}}\) and fractional maximal iterator commutators \({\mathcal{M}_{\alpha, \prod \vec{b}}}\). The conclusion of this paper is also new when the operators are linear.A modification for the viscosity approximation method for fixed point problems in Hilbert spaces.https://zbmath.org/1449.471262021-01-08T12:24:00+00:00"Liu, Ying"https://zbmath.org/authors/?q=ai:liu.ying.2|liu.ying|liu.ying.1|liu.ying.5|liu.ying.4|liu.ying.6|liu.ying.3"Kong, Hang"https://zbmath.org/authors/?q=ai:kong.hangSummary: In this paper, we present a modification for the viscosity approximation method for fixed point problems of a nonexpansive mapping in Hilbert spaces. The modification removes a control condition of the viscosity approximation method. We establish a strong convergence theorem for the modified algorithm.Boundedness of Marcinkiewicz integral and its commutator on weighted \(\lambda\)-central Morrey spaces.https://zbmath.org/1449.420202021-01-08T12:24:00+00:00"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Chen, Zhuanzhuan"https://zbmath.org/authors/?q=ai:chen.zhuanzhuanSummary: By applying function decompositions and properties of weights, the boundedness of Marcinkiewicz integrals and their commutators is established on the weighted \(\lambda\)-central Morrey spaces with the help of the corresponding boundedness on weighted spaces.Non-global nonlinear Lie triple derivable maps on factor von Nuemann algebras.https://zbmath.org/1449.160852021-01-08T12:24:00+00:00"Su, Yutian"https://zbmath.org/authors/?q=ai:su.yutian"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(M\) be a factor von Neumann algebra with dimension greater than 1 on a Hilbert space \(H\). With the help of algebraic decomposition method, we prove that if a nonlinear map \(\delta:M \to M\) satisfied \(\delta ([[A, B], C]) = [[\delta (A), B], C] + [[A, \delta (B)], C] + [[A, B], \delta (C)]\) for any \(A\), \(B\), \(C \in M\) with \(ABC = 0\), then there existed an additive derivation \(d:M \to M\), such that \(\delta (A) = d (A) + \tau (A)I\) for any \(A \in M\), where \(\tau :M \to \mathbb{C}I\) is a nonlinear map such that \(\tau ([[A, B], C]) = 0\) with \(ABC = 0\) for any \(A\), \(B\), \(C \in M\).New exact solutions for a class of nonlinear fractional evolution equations.https://zbmath.org/1449.471302021-01-08T12:24:00+00:00"Yang, Juan"https://zbmath.org/authors/?q=ai:yang.juan"Zeng, Chunhua"https://zbmath.org/authors/?q=ai:zeng.chunhua"Feng, Qingjiang"https://zbmath.org/authors/?q=ai:feng.qingjiangSummary: Using \(\exp(-\Phi (\xi))\)-expansion method, the new exact solutions of the nonlinear fractional Phi-4 equation, the nonlinear fractional order foam drainage equation and the nonlinear fractional SRLW equation are obtained. The practice proves that this method is simple and convenient, it has very important significance for the research of nonlinear fractional evolution equations.Iterative algorithms for a system of split general variational inequalities in Hilbert spaces.https://zbmath.org/1449.471212021-01-08T12:24:00+00:00"Zhao, Yali"https://zbmath.org/authors/?q=ai:zhao.yali"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.2|liu.xin.4|liu.xin|liu.xin.1|liu.xin.3|liu.xin.5"Han, Dongxue"https://zbmath.org/authors/?q=ai:han.dongxue"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qianSummary: In this paper, a system of split general variational inequalities in Hilbert spaces is introduced and studied. By making use of projection operator, an iterative algorithm for the system of split general variational inequalities is proposed. It is proved that the sequences generated by the iterative algorithm strongly converge to the solution of the system of split general variational inequalities.On the \(F\)-contraction properties of multivalued integral type transformations.https://zbmath.org/1449.470982021-01-08T12:24:00+00:00"Sekman, Derya"https://zbmath.org/authors/?q=ai:sekman.derya"Karakaya, Vatan"https://zbmath.org/authors/?q=ai:karakaya.vatanThe authors consider a class of multivalued integral operators possessing the \(F\)-contraction property [\textit{D. Wardowski}, Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)]. The existence of fixed points is proved.
Reviewer: Anatoly N. Kochubei (Kyïv)Consistent invertibility and the proof of property \( (\omega)\).https://zbmath.org/1449.470152021-01-08T12:24:00+00:00"Yin, Le"https://zbmath.org/authors/?q=ai:yin.le"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: Using the property of consistent invertibility, the necessary condition and sufficient condition for an operator are discussed respectively. On the basis of this work, we give the proofs of an operator and its functional calculus whose property \( (\omega)\) holds.Existence of positive solutions for boundary value problems of second-order systems with nonlinear boundary conditions.https://zbmath.org/1449.340832021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantang"Jia, Kaijun"https://zbmath.org/authors/?q=ai:jia.kaijunSummary: The existence of positive solutions for boundary value problems of second-order singular differential systems with nonlinear boundary conditions
\[\begin{cases}
-u'' = \Lambda G (t)F (u),\, 0 < t < 1, \\
u (0) = 0,\, u' (1) + C (u (1))u (1) = 0
\end{cases}\]
is studied, where \(u = (u_1, u_2, \cdots, u_n)^{\mathrm{T}}\), \(G (t) = \mathrm{diag}[g_1 (t), g_2 (t), \cdots, g_n (t)]\), \(g_i (t) (t = 1, 2, \cdots, n)\), allows singularity at \(t = 0\), \(F (u) = (f^1 (u), f^2 (u), \cdots, f^n (u))^{\mathrm{T}}\), \(C = \mathrm{diag} (c_1, c_2, \cdots, c_n)\), \(\Lambda = \mathrm{diag} (\lambda_1, \lambda_2, \cdots, \lambda_n)\), \(\lambda_i (i = 1, 2, \cdots, n)\) is a positive parameter. Under the condition that the nonlinearity term \(F\) satisfies superlinear, sublinear and asymptotically linear growth respectively, the existence of positive solutions of the problems is obtained by using the fixed-point theorem of cone expansion-compression.Existence of positive solutions for nonhomogeneous boundary value problems of fractional differential equations with sign changing nonlinearities.https://zbmath.org/1449.340772021-01-08T12:24:00+00:00"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.1|li.lin.2|li.lin"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.mei"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Song, Junqiu"https://zbmath.org/authors/?q=ai:song.junqiuSummary: We consider the existence of positive solutions for a class of nonhomogeneous integral boundary value problems of fractional differential equations with sign changing nonlinearities. By using fixed point theorems of cone expansion and cone compression, we establish and prove the existence of positive solutions for the boundary value problem, and give some examples to illustrate the conclusions.Proof of property (\(\omega\)) of bounded linear operators.https://zbmath.org/1449.470122021-01-08T12:24:00+00:00"Guo, Qi"https://zbmath.org/authors/?q=ai:guo.qi"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohong"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: We define a new spectrum of operators based on the single valued extension property of bounded linear operators on a Hilbert space. Using the spectrum, the single valued extension property and Kato property of bounded linear operators, we obtain a new proof for the property (\({\omega_1}\)) and the property (\(\omega\)) of bounded linear operators on a Hilbert space.A general alternative regularization method with line search technique for solving split equilibrium and fixed point problems in Hilbert spaces.https://zbmath.org/1449.651392021-01-08T12:24:00+00:00"Jolaoso, Lateef Olakunle"https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunle"Karahan, Ibrahim"https://zbmath.org/authors/?q=ai:karahan.ibrahimSummary: In this paper, we introduce a new general alternative regularization algorithm for solving split equilibrium and fixed point problems in real Hilbert spaces. The proposed method does not require a prior estimate of the norm of the bounded linear operator nor a fixed stepsize for its convergence. Instead, we employ a line search technique and prove a strong convergence result for the sequence generated by the algorithm. A numerical experiment is given to show that the proposed method converges faster in terms of number of iteration and CPU time of computation than some existing methods in the literature.Sturm-Liouville operators with complex singular coefficients.https://zbmath.org/1449.470812021-01-08T12:24:00+00:00"Goryunov, A. S."https://zbmath.org/authors/?q=ai:goryunov.a-sSummary: We consider on a finite interval the Sturm-Liouville differential expression \(l(y)= -(py')' + qy + i((ry)' + ry')\) with coefficients satisfying the condition: \(q= Q'\), \(1/\sqrt{| p|}\), \(Q/\sqrt{| p|}\), \(r/\sqrt{| p|}\in L_2\), where the derivative of function \(Q\) is understood in the sense of distributions. The corresponding operators are correctly defined as quasi-differential. Conditions for the
minimal operator to be symmetric are obtained and all its self-adjoint, maximal dissipative and maximal accumulative extensions are described in terms of boundary conditions.Variable fractional integral operator and its commutator on the Herz-Hardy space with variable exponent.https://zbmath.org/1449.420242021-01-08T12:24:00+00:00"Yao, Junqing"https://zbmath.org/authors/?q=ai:yao.junqing"Shi, Hui"https://zbmath.org/authors/?q=ai:shi.hui"Zhao, Kai"https://zbmath.org/authors/?q=ai:zhao.kaiSummary: Based on the definitions and basic properties of the function spaces with variable exponents and the variable fractional integral operators, by the atomic decomposition of the Herz-Hardy spaces with variable exponents, using the Hölder and Jensen inequalities, we proved the boundedness of variable fractional integral operators with homogeneous kernel and its commutators on the Herz-Hardy spaces with variable exponents.The existence of solution on Rotenberg model.https://zbmath.org/1449.920112021-01-08T12:24:00+00:00"Wu, Hongxing"https://zbmath.org/authors/?q=ai:wu.hongxing"Zhang, Fen"https://zbmath.org/authors/?q=ai:zhang.fen"Cheng, Guofei"https://zbmath.org/authors/?q=ai:cheng.guofei"Wang, Shenghua"https://zbmath.org/authors/?q=ai:wang.shenghuaSummary: Transport equation is a kind of model to study the macro transport phenomenon caused by the micro effect of the movement of particles in material. Studying such transport equations is of great importance to the development of the basic theory of mathematics. This paper is to research the transport equation of Rotenberg model with cell populations in \({L_1}\) space. The existence of solution of the corresponding transport equation for this model is proved. The paper relies on the theory of linear operators, resolvent operator, and comparison operator methods.Proximal point algorithms involving fixed point iteration for nonexpansive mappings in CAT\((\kappa)\) spaces.https://zbmath.org/1449.471132021-01-08T12:24:00+00:00"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapol"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasit"Suparatulatorn, Raweerote"https://zbmath.org/authors/?q=ai:suparatulatorn.raweerote"Chaipunya, Parin"https://zbmath.org/authors/?q=ai:chaipunya.parinSummary: In this paper, we propose a new modified proximal point algorithm involving fixed point iteration for nonexpansive mappings in CAT(1) spaces. Under some mild conditions, we prove that the sequence generated by our iterative algorithm \(\Delta\)-converges to a common solution of certain convex optimization and fixed point problems.Higher \(\xi \)-Lie derivable maps on triangular algebras at reciprocal elements.https://zbmath.org/1449.160862021-01-08T12:24:00+00:00"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xia"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(\mathcal{U} = {\mathrm{Tri}} (\mathcal{A, M, B})\) be a triangular algebra with identity 1, \({1_\mathcal{A}}\), \({1_\mathcal{B}}\) be the unit of \(\mathcal{A}\) and \(\mathcal{B}\), respectively. For any \(A \in \mathcal{A}, B \in \mathcal{B}\), there are integers \({k_1}, {k_2}\) respectively, making \({k_1}{1_\mathcal{A}}-A, {k_2}{1_\mathcal{B}}-B\) invertible in triangular algebras. \(\{\varphi_n\}_{n \in N}: \mathcal{U} \to \mathcal{U}\) be a sequence of linear maps. In this paper, we prove that, if \(\{\varphi_n\}_{n \in N}\) satisfies \({\varphi_n} ([U, V]_\xi) = \sum\limits_{i + j = n} {\varphi_i} (U){\varphi_j} (V)-\xi{\varphi_i} (V){\varphi_j} (U)\) \((\xi \ne 0, 1)\), for any \(U, V \in \mathcal{U}\) with \(UV = VU = 1\), then \(\{\varphi_n\}_{n \in N}\) is a higher derivation, where \({\varphi_0} = \mathrm{id}_0\) is the identity map, \([U,V]_\xi = UV - \xi VU\).Qualitative properties of solution for hybrid nonlinear fractional differential equations.https://zbmath.org/1449.340292021-01-08T12:24:00+00:00"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-mSummary: In this article we investigate some qualitative properties for a class of hybrid nonlinear fractional differential equations. The existence, uniqueness, monotonicity and positivity of the solution are studied by the method of upper and lower control functions and using Dhage's fixed point theorem. Some examples are introduced to illustrate the applicability of the results.Three new iterative methods for solving inclusion problems and related problems.https://zbmath.org/1449.651362021-01-08T12:24:00+00:00"Gibali, Aviv"https://zbmath.org/authors/?q=ai:gibali.aviv"Thong, Duong Viet"https://zbmath.org/authors/?q=ai:duong-viet-thong."Vinh, Nguyen The"https://zbmath.org/authors/?q=ai:vinh.nguyen-theSummary: In this paper, we study the variational inclusion problem which consists of finding zeros of the sum of a single and multivalued mappings in real Hilbert spaces. Motivated by the viscosity approximation, projection and contraction and inertial forward-backward splitting methods, we introduce two new forward-backward splitting methods for solving this variational inclusion. We present weak and strong convergence theorems for the proposed methods under suitable conditions. Our work generalize and extend some related results in the literature. Several numerical examples illustrate the potential applicability of the methods and comparisons with related methods emphasize it further.Nonlinear Jordan higher derivable maps on triangular algebras by Lie product square zero elements.https://zbmath.org/1449.160802021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.lei"Zhu, Guowei"https://zbmath.org/authors/?q=ai:zhu.guoweiSummary: Let \(\mathcal{U}\) be a 2-torsion free triangular algebra, \(D = \{d_n\}_{n \in \textbf{N}}\) is a nonlinear Jordan higher derivable map on triangular algebra \(\mathcal{U}\) by Lie product square zero elements. In this paper, it is shown that every nonlinear Jordan higher derivable map on triangular algebra \(\mathcal{U}\) by Lie product square zero elements is a higher derivation. As its application, we get that every nonlinear Jordan higher derivable map on a nest algebra or a 2-torsion free block upper triangular matrix algebra \(\mathcal{U}\) by Lie product square zero elements is a higher derivation.Strong new product preserving maps on \(*\)-algebras.https://zbmath.org/1449.470742021-01-08T12:24:00+00:00"Zhang, Fangjuan"https://zbmath.org/authors/?q=ai:zhang.fangjuanSummary: Let \(\mathfrak{R}\) be a unital \(*\)-algebra with a nontrivial symmetric idempotent \(P\) which satisfies: (1) \(A\mathfrak{R}P = \{0\}\) implies \(A = 0\); (2) \(A\mathfrak{R} (I-P) = \{0\}\) implies \(A = 0\). Let \(\phi :\mathfrak{R} \to \mathfrak{R}\) be a surjective map. Then \(\phi\) is strong new product preserving if and only if there exists an element \(Z\in {Z_S} (\mathfrak{R})\) with \({Z^2} = I\) such that \(\phi (X) = ZX\) for all \(X \in \mathfrak{R}\). As an application, characterizations of the strong new product preserving on von Neumann algebras with no central summands of type \({I_1}\) and prime \(*\)-ring are obtained.Some fixed point theorems via measure of noncompactness with applications to differential equations.https://zbmath.org/1449.470952021-01-08T12:24:00+00:00"Banaei, Shahram"https://zbmath.org/authors/?q=ai:banaei.shahram"Mursaleen, Mohammad"https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman"Parvaneh, Vahid"https://zbmath.org/authors/?q=ai:parvaneh.vahidSummary: In this article, we prove some Darbo's type fixed point theorems associated with measure of noncompactness via the concept of operators \(A(f;.)\) and weakly \(JS\)-contractive condition in Banach space. Moreover, as an application of our results, we study the existence of solutions for a system of differential equations. Finally, we present an example to support the effectiveness of our results.A criterion for the existence and uniqueness of solution involving fractional order and impulsive boundary conditions.https://zbmath.org/1449.341022021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"He, Cong"https://zbmath.org/authors/?q=ai:he.cong"Tang, Yuping"https://zbmath.org/authors/?q=ai:tang.yupingSummary: By using the fixed point theorem for the sum of operators, a criterion for the existence and uniqueness of solution involving fractional order and impulsive boundary conditions was obtained. Also, an iterative sequence was constructed to approximate the solution. To illustrate the main results, an example was given in the paper.Existence of solutions for impulsive differential inclusions with upper and lower solutions in the reverse order.https://zbmath.org/1449.340582021-01-08T12:24:00+00:00"Luo, Yan"https://zbmath.org/authors/?q=ai:luo.yan"Xie, Wenzhe"https://zbmath.org/authors/?q=ai:xie.wenzheSummary: In this paper, we discuss the existence of solutions for nonlinear boundary problems of first-order impulsive differential inclusions. In the presence of a lower solution \(\alpha\) and an upper solution \(\beta\) in the reverse order \(\beta \leq \alpha\), we establish existence results by using Martelli's fixed point theorem.On the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations.https://zbmath.org/1449.340152021-01-08T12:24:00+00:00"Cuong, D. X."https://zbmath.org/authors/?q=ai:cuong.d-xSummary: In this paper, by using a Bielecki type norm and the Banach fixed point theorem, we obtain a result on the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations.Property \( (h)\) and perturbations.https://zbmath.org/1449.470172021-01-08T12:24:00+00:00"Wurichaihu"https://zbmath.org/authors/?q=ai:wurichaihu."Alatancang"https://zbmath.org/authors/?q=ai:alatancang.chen|chen.alatancangSummary: In this paper, we introduce and study the property \( (h)\), which extends \(a\)-Weyl's theorem. We consider its stability under commuting finite rank and nilpotent perturbations. We prove that property \( (h)\) on Banach spaces is related to an important property which has a leading role in local spectral theory: the single-valued extension property. From this result we deduce that property \( (h)\) holds for several classes of operators.Boundedness of commutators for the multilinear Calderón-Zygmund operators with kernels of Dini's type.https://zbmath.org/1449.420292021-01-08T12:24:00+00:00"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiang"Hu, Xi"https://zbmath.org/authors/?q=ai:hu.xiSummary: The authors give a sharp maximal estimate for the iterated commutator that is generated by multilinear Calderón-Zygmund operators with kernels of Dini's type and Lipschitz function. Furthermore, in the suitable index case, the commutator is bounded on product Lebesgue spaces.Inequalities for functions of selfadjoint operators on Hilbert spaces: a survey of recent results.https://zbmath.org/1449.470362021-01-08T12:24:00+00:00"Dragomir, Silvestru Sever"https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: The main aim of this survey is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.On the compactness of commutators of bilinear fractional maximal operator.https://zbmath.org/1449.420372021-01-08T12:24:00+00:00"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiang"Guo, Qingdong"https://zbmath.org/authors/?q=ai:guo.qingdongSummary: Let \(\mathcal{M}_\alpha\) be the bilinear fractional maximal operator and let \(\vec{b} = ({b_1}, {b_2})\) be a collection of locally integrable functions. In this paper we mainly study the compactness of commutators of bilinear fractional maximal operators on Lebesgue spaces, of which the commutators include the fractional maximal linear commutator \(\mathcal{M}_{\alpha, \Sigma \vec{b}}\) and the fractional maximal iterated commutator \(\mathcal{M}_{\alpha, \Pi \vec{b}}\). The results are new even in the linear case.Hybrid Bregman projection methods for fixed point and equilibrium problems.https://zbmath.org/1449.471182021-01-08T12:24:00+00:00"Wang, Zi-Ming"https://zbmath.org/authors/?q=ai:wang.zi-ming"Wei, Airong"https://zbmath.org/authors/?q=ai:wei.airong"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poomSummary: The purpose of this article is to investigate a projection algorithm for solving a fixed point problem of a closed multi-valued Bregman quasi-strict pseudocontraction and an equilibrium problem of a bifunction. Strong convergence of the projection algorithm is obtained without any compact assumption in a reflexive Banach space. As applications, monotone variational inequality problems are considered. Finally, a numerical simulation example is presented for demonstrating the feasibility and convergence of the algorithm proposed in main result.\(L_p\)-equivalence between two ordinary impulse differential equations with bounded linear impulse operators in a Banach Space.https://zbmath.org/1449.340482021-01-08T12:24:00+00:00"Kostadinov, G."https://zbmath.org/authors/?q=ai:kostadinov.g-d|kostadinov.georgi"Zahariev, A."https://zbmath.org/authors/?q=ai:zahariev.andrey-ivanovSummary: By means of the fixed point principle of Schauder-Tychonoff and Banach there are found sufficient conditions for the existence of \(L_p\)-equivalence between two ordinary impulse differential equations with bounded linear impulse operators in an arbitrary Banach space.Global attractivity of solutions for a class of multi-term fractional differential equations.https://zbmath.org/1449.341992021-01-08T12:24:00+00:00"Li, Yanfeng"https://zbmath.org/authors/?q=ai:li.yanfeng"Hao, Yanpeng"https://zbmath.org/authors/?q=ai:hao.yanpeng"Wang, Erjing"https://zbmath.org/authors/?q=ai:wang.erjing"Li, Qiaoluan"https://zbmath.org/authors/?q=ai:li.qiaoluanSummary: In this paper, we present results for the global attractivity of solutions of fractional differential equations involving Caputo-Katugampola fractional calculus. By transforming the differential equations into an integral equations, the existence of the solutions is obtained by using the Schauder's fixed point theorem.Cauchy problem for a class of pseudo-differential equations on \(\mathbb{Q}_p^n\).https://zbmath.org/1449.354702021-01-08T12:24:00+00:00"Wu, Bo"https://zbmath.org/authors/?q=ai:wu.bo"Qian, Dandan"https://zbmath.org/authors/?q=ai:qian.dandanSummary: This paper considers a class of pseudo-differential equations on \(\mathbb{Q}_p^n\),
\[ \frac{{\partial^2}u (t,x)}{\partial{t^2}} + 2{a^2}\Delta_p^{\alpha/2}\frac{\partial u (t,x)}{\partial t} + {b^2}\Delta_p^\alpha u (t,x) + {c^2}u (t,x) = q (t,x), \]
where \(t \in [0, T]\) and \({\Delta_p}\) is the Laplacian of \(\mathbb{Q}_p^n\). When nonlinear term and initial value satisfy some conditions, the solutions of pseudo-differential equations are obtained by applying fundamental solutions.Fast self-adaptive regularization iterative algorithm for solving split feasibility problem.https://zbmath.org/1449.902852021-01-08T12:24:00+00:00"Dang, Ya-Zheng"https://zbmath.org/authors/?q=ai:dang.yazheng"Xue, Zhong-Hui"https://zbmath.org/authors/?q=ai:xue.zhonghui"Gao, Yan"https://zbmath.org/authors/?q=ai:gao.yan"Li, Jun-Xiang"https://zbmath.org/authors/?q=ai:li.junxiangSummary: Split feasibility problem (SFP) is to find a point which belongs to one convex set in one space, such that its image under a linear transformation belongs to another convex set in the image space. This paper deals with a unified regularized SFP for the convex case. We first construct a self-adaptive regularization iterative algorithm by using Armijo-like search for the SFP and show it converges at a subliner rate of \(O(1/k)\), where \(k\) represents the number of iterations. More interestingly, inspired by the acceleration technique introduced by \textit{Yu. E. Nesterov} [Sov. Math., Dokl. 27, 372--376 (1983; Zbl 0535.90071); translation from Dokl. Akad. Nauk SSSR 269, 543--547 (1983)], we present a fast Armijo-like regularization iterative algorithm and show it converges at rate of \(O(1/k^2)\). The efficiency of the algorithms is demonstrated by some random data and image debluring problems.The existence of positive periodic solutions for a kind of generalized Liénard equation.https://zbmath.org/1449.341182021-01-08T12:24:00+00:00"Cui, Xiaoxiao"https://zbmath.org/authors/?q=ai:cui.xiaoxiao"Cheng, Zhibo"https://zbmath.org/authors/?q=ai:cheng.zhibo"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, by application of the Manasevich-Mawhin continuation theorem, we prove the existence and asymptotic stability of positive periodic solutions for a class of generalized Liénard equation, where the non-autonomous function satisfies superlinear condition. At last, two examples and numerical solutions (phase portraits and time series portraits) are given to illustrate our conclusions.An accurate numerical method for solving the generalized time-fractional diffusion equation.https://zbmath.org/1449.354492021-01-08T12:24:00+00:00"Syam, Muhammed"https://zbmath.org/authors/?q=ai:syam.muhammed-i"Al-Subaihi, Ibrahim"https://zbmath.org/authors/?q=ai:al-subaihi.ibrahim-aSummary: In this paper, a formulation for the fractional Legendre functions is constructed to solve a class of time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre. Analysis for the presented method is given and numerical results are presented.Measure of noncompactness of operators in Banach spaces.https://zbmath.org/1449.470922021-01-08T12:24:00+00:00"Shen, Qinrui"https://zbmath.org/authors/?q=ai:shen.qinruiSummary: This paper is committed to dealing with the measure of noncompactness of operators in Banach spaces. First, we give a characterization of the Hausdorff measure of noncompactness of operators with respect to the Hausdorff metric. Then, we give a formula of the Hausdorff measure of noncompactness of operators in \({\ell ^p}\) for some \({1 \le p < \infty}\). Finally, several common equivalent measures of noncompactness of operators and related proofs are given.An analytic model for left invertible weighted translation semigroups.https://zbmath.org/1449.470492021-01-08T12:24:00+00:00"Phatak, Geetanjali M."https://zbmath.org/authors/?q=ai:phatak.geetanjali-m"Sholapurkar, V. M."https://zbmath.org/authors/?q=ai:sholapurkar.v-mSummary: \textit{M. R. Embry} and \textit{A. Lambert} [Rocky Mt. J. Math. 7, 333--344 (1977; Zbl 0379.47029)] initiated the study of a semigroup of operators \(\{S_t\}\) indexed by a non-negative real number \(t\) and termed it as weighted translation semigroup. The operators \(S_t\) are defined on \(L^2(\mathbb{R}_+)\) by using a weight function. The operator \(S_t\) can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator \(S_t\) can be modeled as a multiplication by \(z\) on a reproducing kernel Hilbert space \(H\) of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with \(\mathcal{H}\) is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertible, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator \(S_t\).A trace formula for integro-differential operators.https://zbmath.org/1449.470842021-01-08T12:24:00+00:00"Chen, Shirong"https://zbmath.org/authors/?q=ai:chen.shirongSummary: The trace formulae for the integro-differential operator are studied, which have many applications in the inverse problem, the numerical computation of eigenvalues and the theory of integrable system, etc. The trace formula for integro-differential operators with Dirichlet-Robin boundary conditions or Dirichlet boundary conditions are obtained.A class of local nonlinear triple higher derivable maps on triangular algebras.https://zbmath.org/1449.160792021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: Let \(\mathcal{U}\) be a 2-torsion free triangular algebra, \(\Omega = \{x \in \mathcal{U}:{x^2} = 0\}\) and \(D = \{{d_n}\}_{n \in \mathbb{N}}\) be a sequence mapping from \(\mathcal{U}\) into itself (without assumption of additivity). By using the method of algebraic decomposition, we prove that if
\[{d_n} (xyz) = \sum\limits_{i+j+k=n}{d_i} (x){d_j} (y){d_k} (z)\] for all \({n \in \mathbb{N}}\), \(x,y,z \in \mathcal{U}\) with \(xyz \in \Omega\), then \(D\) is a higher derivation.A mathematical basis for the graphene.https://zbmath.org/1449.343072021-01-08T12:24:00+00:00"Conca, Carlos"https://zbmath.org/authors/?q=ai:conca.carlos"Martín, Jorge San"https://zbmath.org/authors/?q=ai:san-martin.jorge-alonso"Solano, Viviana"https://zbmath.org/authors/?q=ai:solano.vivianaSummary: We present a new basis of representation for the graphene honeycomb structure that facilitates the solution of the eigenvalue problem by reducing it to one dimension. We define spaces in these geometrical basis that allow us to solve the Hamiltonian in the edges of the lattice. We conclude that it is enough to analyze a one-dimensional problem in a set of coupled ordinary second-order differential equations to obtain the behavior of the solutions in the whole graphene structure.CFI operators and property \( (\omega)\).https://zbmath.org/1449.470272021-01-08T12:24:00+00:00"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: According to the property of consistence in Fredholm index, a new spectral set is defined. By using the relationship between the spectral set and Browder spectrum, we give the necessary and sufficient conditions for a bounded linear operator on a Hilbert space to satisfy the property \( (\omega)\), and characterize the property \( (\omega)\) of polynomial functions.Application of new strongly convergent iterative methods to split equality problems.https://zbmath.org/1449.471082021-01-08T12:24:00+00:00"Gautam, Pankaj"https://zbmath.org/authors/?q=ai:gautam.pankaj"Dixit, Avinash"https://zbmath.org/authors/?q=ai:dixit.avinash-k"Sahu, D. R."https://zbmath.org/authors/?q=ai:sahu.daya-ram"Som, T."https://zbmath.org/authors/?q=ai:som.tanmoy|som.tonmoy|som.tapas-kSummary: In this paper, we study the generalized problem of split equality variational inclusion problem. For this purpose, we introduced the problem of finding the zero of a nonnegative lower semicontinuous function over the common solution set of a fixed point problem and a monotone inclusion problem. We propose and study the convergence behaviour of different iterative techniques to solve the generalized problem. Furthermore, we study an inertial form of the proposed algorithm and compare the convergence speed. Numerical experiments have been conducted to compare the convergence speed of the proposed algorithm, its inertial form and already existing algorithms to solve the generalized problem.Existence of mild solutions of second order evolution integro-differential equations in the Fréchet spaces.https://zbmath.org/1449.342662021-01-08T12:24:00+00:00"Jawahdou, Adel"https://zbmath.org/authors/?q=ai:jawahdou.adelSummary: In this article, we shall establish sufficient conditions for the existence of mild solutions for second order semilinear integro-differential evolution equations in Fréchet spaces \(C(\mathbb{R}_+ , E)\), where \(E\) is a Banach space. Our approach is based on the concept of a measure of noncompactness and Tychonoff fixed point theorem. For illustration we give an example.A new iterative method for solving pseudomonotone variational inequalities with non-Lipschitz operators.https://zbmath.org/1449.471152021-01-08T12:24:00+00:00"Thong, Duong Viet"https://zbmath.org/authors/?q=ai:duong-viet-thong."Shehu, Yekini"https://zbmath.org/authors/?q=ai:shehu.yekini"Iyiola, Olaniyi S."https://zbmath.org/authors/?q=ai:iyiola.olaniyi-samuelSummary: The purpose of this paper is to study and analyze a new projection-type algorithm for solving pseudomonotone variational inequality problems in real Hilbert spaces. The advantage of the proposed algorithm is the strong convergence proved without assuming Lipschitz continuity of the associated mapping. In addition, the proposed algorithm uses only two projections onto the feasible set in each iteration. The numerical behaviour of the proposed algorithm on a test problem is illustrated and compared with several previously known algorithms.On the graphene Hamiltonian operator.https://zbmath.org/1449.343082021-01-08T12:24:00+00:00"Conca, C."https://zbmath.org/authors/?q=ai:conca.carlos"Orive, R."https://zbmath.org/authors/?q=ai:orive.rafael"Martín, J. San"https://zbmath.org/authors/?q=ai:martin.jo-san|san-martin.jorge-alonso"Solano, V."https://zbmath.org/authors/?q=ai:solano.vivianaSummary: We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by \textit{P. Kuchment} and \textit{O. Post} [Commun. Math. Phys. 275, No. 3, 805--826 (2007; Zbl 1145.81032)] to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.Some results on \(L\)-weakly compact sets and operators.https://zbmath.org/1449.470782021-01-08T12:24:00+00:00"Lhaimer, Driss"https://zbmath.org/authors/?q=ai:lhaimer.driss"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"Moussa, Mohammed"https://zbmath.org/authors/?q=ai:moussa.mohammedSummary: In this paper, we give some new characterizations of \(L\)-weakly compact operators. As consequences, we will give some interesting results. Also, we establish some necessary and sufficient conditions on which a relatively weakly compact set is \(L\)-weakly compact. In particular, we characterize Banach lattices with the positive Schur property.The application of differential characteristic set method to pseudo differential operator and Lax representation.https://zbmath.org/1449.354692021-01-08T12:24:00+00:00"Jia, Yifeng"https://zbmath.org/authors/?q=ai:jia.yifeng"Xiao, Dongliang"https://zbmath.org/authors/?q=ai:xiao.dongliangSummary: Differential characteristic set method is applied to the calculation of pseudo differential operators and Lax representation of nonlinear evolution equations. Firstly, differential characteristic set method and differential division with remainder are used for the calculation of inverse and extraction root of pseudo differential operator, such that the process is simplified since it is unnecessary to solve ordinary differential equation systems and substitute the solutions. Secondly, using differential characteristic set method, the nonlinear partial differential equation systems derived from the generalized Lax equation and Zakharov-Shabat equation, are reduced, and the corresponding nonlinear evolution equation is obtained. The related programs are compiled in Mathematica. A computer-based computer algebra system, and Lax representation of some nonlinear evolution equations can be calculated with the aid of the computer.Existence of positive solutions to a semipositone second-order boundary value problem.https://zbmath.org/1449.340902021-01-08T12:24:00+00:00"Wei, Jinying"https://zbmath.org/authors/?q=ai:wei.jinying"Wang, Suyun"https://zbmath.org/authors/?q=ai:wang.suyun"Li, Yongjun"https://zbmath.org/authors/?q=ai:li.yongjunSummary: We consider the existence of positive solutions to the boundary value problem
\[u'' + c (t)u + \lambda f (t, u) = 0,\, 0 < t < 1,\, u (0) = u (1) = 0,\]
where \(\lambda > 0\), \(c (\cdot) \in C[0, 1]\) satisfies \(-\infty < c (t) < \pi^2\) for \(t \in [0, 1]\), \(f:[0, 1] \times \mathbb{R}^+ \to \mathbb{R}\) is continuous function and satisfies \(f \geq -L\), \(L > 0\) is a constant. By investigating the sign property of the Green function of the associated linear boundary value problem, we show the existence of positive solutions of semipositone problems. The proof of the main result is based on the Krasnosel'skii fixed point theorems in cone.Boundary representations of operator spaces, and compact rectangular matrix convex sets.https://zbmath.org/1449.460472021-01-08T12:24:00+00:00"Fuller, Adam H."https://zbmath.org/authors/?q=ai:fuller.adam-hanley"Hartz, Michael"https://zbmath.org/authors/?q=ai:hartz.michael"Lupini, Martino"https://zbmath.org/authors/?q=ai:lupini.martinoSummary: We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.\(f\)-orthomorphisms and \(f\)-linear operators on the order dual of an \(f\)-algebra revisited.https://zbmath.org/1449.460032021-01-08T12:24:00+00:00"Jaber, Jamel"https://zbmath.org/authors/?q=ai:jaber.jamelSummary: We give a necessary and sufficient condition on an \(f\)-algebra \(A\) for which orthomorphisms, \(f\)-linear operators, and \(f\)-orthomorphisms on the order dual \(A^\sim\) are the same class of operators.Explicit iterative sequences of positive solutions for a class of fractional differential equations on an infinite interval.https://zbmath.org/1449.340942021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Li, Yaohong"https://zbmath.org/authors/?q=ai:li.yaohongSummary: By applying the monotone iterative method, this study develops two explicit monotone iterative sequences for approximating the minimal and maximal positive solutions. At the same time, by applying the Banach fixed-point theory, an explicit iterative sequence and error estimate for approximating the unique positive solution are obtained. Some examples are given to illustrate the application of the results.Functorial properties of \(\mathrm{Ext}_{u}(\cdot, \mathcal{B})\) when \(\mathcal{B}\) is simple with continuous scale.https://zbmath.org/1449.460622021-01-08T12:24:00+00:00"Ng, P. W."https://zbmath.org/authors/?q=ai:ng.ping-wong"Robin, Tracy"https://zbmath.org/authors/?q=ai:robin.tracyTo give a sample of the results proved in this paper we mention the following one. The authors show that \(\mathrm{Ext}_{u}(\mathcal{A}, \mathcal{B})\) is always an abelian group whenever the \(C^*\)-algebra \(\mathcal{A}\) is separable and nuclear and \(\mathcal{B}\) is a simple continuous scale \(C^*\)-algebra. A~similar result is proved in the unital case.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)New exact solutions of \( (2+1)\)-dimensional Boiti-Leon-Manna-Pempinelli equation.https://zbmath.org/1449.353642021-01-08T12:24:00+00:00"Deng, Changrui"https://zbmath.org/authors/?q=ai:deng.changrui"Zhou, Xiaohong"https://zbmath.org/authors/?q=ai:zhou.xiaohongSummary: Nonlinear evolution equations are widely used in real physical model, such as high polymer physics, fluid dynamics, solid state physics, plasma physics and so on. This paper mainly studies the \( (2+1)\)-dimensional Boti-Leon-Manna-Pempinelli equation. Firstly, the Hirota bilinear form of the equation is obtained by Hirota method, and then the extended three-wave test method is used to obtain the periodic soliton solution, periodic double soliton solution and double periodic double soliton solution of the \( (2+1)\)-dimensional Boti-Leon-Manna-Pempinelli equation.A modified contraction method for solving certain class of split monotone variational inclusion problems with application.https://zbmath.org/1449.471102021-01-08T12:24:00+00:00"Izuchukwu, C."https://zbmath.org/authors/?q=ai:izuchukwu.chinedu"Ezeora, Jerry N."https://zbmath.org/authors/?q=ai:ezeora.jerry-n"Martinez-Moreno, J."https://zbmath.org/authors/?q=ai:martinez-moreno.juanSummary: The main purpose of this paper is to propose a new modified contraction method for solving a certain class of split monotone variational inclusion problems in real Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a solution of the aforementioned problem. Our strong convergence result is obtained when the underlying operator is monotone and Lipschitz continuous, and the knowledge of its Lipschitz constant is not required. As application, we solve split linear inverse problems, for which we also considered a special case of this problem, namely, the LASSO problem. We also give some numerical illustrations of the proposed method in comparison with other methods in the literature to further show the applicability and advantage of our results. The results obtained in this paper generalize and improve many recent results in this direction.Iterative approximations for finite families generalized asymptotically quasi pseudo-contractive type nonself-mappings.https://zbmath.org/1449.471292021-01-08T12:24:00+00:00"Zhang, Shuyi"https://zbmath.org/authors/?q=ai:zhang.shuyi"Zhang, Xinyu"https://zbmath.org/authors/?q=ai:zhang.xinyu"Nie, Hui"https://zbmath.org/authors/?q=ai:nie.huiSummary: A new class of finite families generalized asymptotically quasi pseudo-contractive type nonself-mappings of nearly uniformly Lipschitz in real normed linear spaces is introduced and studied. By using a new analytical method, the strong convergence of a modified Reich-Takahashi iterative sequence with mixed errors associated with the finite families of fixed point of generalized asymptotically quasi pseudo-contractive type nonself-mappings with nearly uniformly Lipschitz is established under weaker conditions, which improves and extends the corresponding results of some references.Compact composition operators on model spaces with univalent symbols.https://zbmath.org/1449.470542021-01-08T12:24:00+00:00"Karaki, Muath"https://zbmath.org/authors/?q=ai:karaki.muathSummary: We give a sufficient condition and a necessary condition for the compactness of the composition operators on model spaces \(C_\varphi: K_\theta\to H^2\), where \(\varphi\) is univalent. This is a generalization to a result of Shapiro for the composition operator \(C_\varphi:H^2\to H^2\) [\textit{J. H. Shapiro}, Composition operators and classical function theory. New York: Springer-Verlag (1993; Zbl 0791.30033)].Semidefinite perturbations in the subspace perturbation problem.https://zbmath.org/1449.470312021-01-08T12:24:00+00:00"Seelmann, Albrecht"https://zbmath.org/authors/?q=ai:seelmann.albrechtLet \(A\) be a self-adjoint operator and V be a bounded self-adjoint operator, \(V\ge 0\). Let the spectrum be separated, more precisely, \(\operatorname{Spec}(A)=\sigma\sup\Sigma\), where \(\operatorname{dist}(\sigma, \Sigma)=d>0\). Some new results on separation of the spectrum of the perturbed operator \(A+V\) are obtained in the paper under review.
Reviewer: Michal Zajac (Bratislava)Solution by iteration of split equality problem involving some families of mappings in Banach spaces.https://zbmath.org/1449.471122021-01-08T12:24:00+00:00"Ofoedu, Eric U."https://zbmath.org/authors/?q=ai:ofoedu.eric-u"Araka, Nnamdi N."https://zbmath.org/authors/?q=ai:araka.nnamdi-nSummary: An iterative scheme is proposed for approximation of common element of set of solution of split equality fixed point problems for \(\eta\)-demimetric mappings, set of common fixed points of finite families of relatively quasi-nonexpansive mappings and set of common solutions of systems of equilibrium problems in specified real Banach spaces. Our theorems extend and complement several existing results.\(p\)-convergent operators and the \(p\)-Schur property.https://zbmath.org/1449.460212021-01-08T12:24:00+00:00"Alikhani, M."https://zbmath.org/authors/?q=ai:alikhani.malihe|alikhani.mahdi|alikhani.morteza|alikhani.masoomeh"Fakhar, M."https://zbmath.org/authors/?q=ai:fakhar.majid"Zafarani, J."https://zbmath.org/authors/?q=ai:zafarani.jafarLet \(X,Y\) be Banach spaces. This interesting paper deals with the question of certain subclasses of \({\mathcal{K}}(X,Y)\) being equal to \({\mathcal{K}}(X,Y)\) for all spaces \(Y\) or equivalently for a test space \(Y\).
For \(1 \leq p < \infty\) let \(C_p(X,Y)\) denote the class of operators which map weakly \(p\)-summable sequences to norm null sequences. It is shown that when \(C_p(X,Y) = {\mathcal{K}}(X,Y)\) for \(Y = \ell^{\infty}\), then these spaces are equal for all Banach spaces \(Y\).
For \(1 \leq p <q < \infty\), a similar result holds for the inclusion \(C_p(X,Y) \subset C_q(X,Y)\).
Reviewer: T.S.S.R.K. Rao (Bangalore)Two groups of mixed reverse order laws for generalized inverses of two and three matrix products.https://zbmath.org/1449.150052021-01-08T12:24:00+00:00"Tian, Yongge"https://zbmath.org/authors/?q=ai:tian.yonggeSummary: Generalized inverses of a matrix product can be written as certain matrix expressions that are composed by the given matrices and their generalized inverses, and a challenging task in this respect is to establish various reasonable reverse order laws for generalized inverses of matrix products. In this paper, we present two groups of known and new mixed reverse order laws for the Moore-Penrose inverses of products of two and three matrices through various conventional matrix operations. We also establish four groups of matrix set inclusions that are composed by \(\{1\}\)- and \(\{1,2\}\)-generalized inverses of \(A\), \(B\), \(C\), and their products \(AB\) and \(ABC\).Lower bounds for unbounded operators and semigroups.https://zbmath.org/1449.470232021-01-08T12:24:00+00:00"Batty, Charles J. K."https://zbmath.org/authors/?q=ai:batty.charles-j-k"Geyer, Felix"https://zbmath.org/authors/?q=ai:geyer.felixSummary: Let \(A\) be an unbounded operator on a Banach space \(X\). It is sometimes useful to improve the operator \(A\) by extending it to an operator \(B\) on a larger Banach space \(Y\) with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of \(B\), and also to extend bounded operators related to \(A\), for example a semigroup generated by \(A\). When \(X\) is a Hilbert space, one may also want \(Y\) to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, Müller and Badea, and we give some extensions of their results to unbounded operators and we raise some open questions. A related problem is to improve properties of a \(C_0\)-semigroup satisfying lower bounds by extending it to a \(C_0\)-group on a larger space or by finding left-inverses. Results of this type for Hilbert spaces have been obtained by \textit{J. C. Louis} and \textit{D. Wexler} [J. Differ. Equations 49, 258--269 (1983; Zbl 0477.49022)], and by \textit{H. Zwart} [J. Evol. Equ. 13, No. 2, 335--342 (2013; Zbl 1288.47040)], and we give some additional results.Algebraic properties of Bergman-type Toeplitz operators on the Dirichlet space.https://zbmath.org/1449.470612021-01-08T12:24:00+00:00"Qin, Jie"https://zbmath.org/authors/?q=ai:qin.jie"Liu, Youqi"https://zbmath.org/authors/?q=ai:liu.youqi"Huang, Sui"https://zbmath.org/authors/?q=ai:huang.suiSummary: We study preliminary properties and algebraic properties of Bergman-type Toeplitz operators which are induced by harmonic symbols on the Dirichlet space, including self-adjointness, products, commutativity and invertibility. Moreover, the spectra of the Toeplitz operators are calculated.Closedness of range of symplectic symmetric Hamiltonian operators.https://zbmath.org/1449.470022021-01-08T12:24:00+00:00"Li, Zhengzhang"https://zbmath.org/authors/?q=ai:li.zhengzhang"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangSummary: Using the perturbation theory and the factorization of operator matrices, the authors investigate the closedness of range of symplectic symmetric Hamiltonian operators. Some descriptions on closedness of range are given for the cases such as diagonal dominant and upper dominant, and the results for the general case are actually presented.The common iterative Halpern's type method for fixed points and equilibrium solutions of asymptotically nonexpansive mappings.https://zbmath.org/1449.471272021-01-08T12:24:00+00:00"Wang, Yuanheng"https://zbmath.org/authors/?q=ai:wang.yuanheng.1"Chen, Lingfa"https://zbmath.org/authors/?q=ai:chen.lingfaSummary: This paper studied the approximation of a common element of the set of solutions of equilibrium problems and fixed points of asymptotically nonexpansive mappings in real Hilbert spaces. By using Halpern's type iterative algorithm, a new transformation iterative sequence was constructed. Under suitable weaker conditions, the strong convergence of the sequence was obtained. The results have a wider adaptability and include some recent results as their special cases.Approximating solutions of third-order nonlinear hybrid differential equations via Dhage iteration principle.https://zbmath.org/1449.340422021-01-08T12:24:00+00:00"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://zbmath.org/authors/?q=ai:djoudi.ahceneSummary: We prove the existence and approximation of solutions of the initial value problems of nonlinear third-order hybrid differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.An investigation on the existence and Ulam stability of solution for an impulsive fractional differential equation.https://zbmath.org/1449.342752021-01-08T12:24:00+00:00"Guo, Yuchen"https://zbmath.org/authors/?q=ai:guo.yuchen"Shu, Xiaobao"https://zbmath.org/authors/?q=ai:shu.xiaobaoSummary: In this paper, we investigate the existence and Ulam stability of solution for impulsive Riemann-Liouville fractional neutral function differential equation with infinite delay of order \(1 < \beta < 2\). Firstly, the solution for the equation is proved. By using the fixed point theorem as well as the Hausdorff measure of noncompactness, the existence results are obtained and the Ulam stability of the solution is proved.Relationship between number of zeros of nonlinear term and number of positive solutions of second-order periodic boundary value problems.https://zbmath.org/1449.340892021-01-08T12:24:00+00:00"Wang, Suyun"https://zbmath.org/authors/?q=ai:wang.suyun"Wei, Jinying"https://zbmath.org/authors/?q=ai:wei.jinying"Zhang, Yanhong"https://zbmath.org/authors/?q=ai:zhang.yanhongSummary: By using the fixed point index theorem, we give the existence of multiple positive solutions for periodic boundary value problems of second-order ordinary differential equations: \[\begin{cases}
u''-qu + \lambda f (u) = 0,\, t \in (0, T),\\
u (0) = u (T),\, u' (0) = u' (T),
\end{cases}\]
where \(0 < q < +\infty\), \(f \in C ([0;\infty), [0;\infty))\), \(f (a_i) = 0\), \(f (b_i) = 0\) and \(f (u) > 0\) in \((a_i,b_i)\) with \(a_i,b_i \in (0, \infty)\) and \(a_i < b_i \leq a_{i+1} < b_{i+1}\) for \(i =1,2,\cdots, n\). The results reveal the relationship between the number of zeros of the nonlinear term \(f\) and the number of positive solutions of periodic boundary value problems.A Beurling theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras with unitarily invariant norms.https://zbmath.org/1449.460552021-01-08T12:24:00+00:00"Liu, Wenjing"https://zbmath.org/authors/?q=ai:liu.wenjing"Sager, Lauren"https://zbmath.org/authors/?q=ai:sager.lauren-b-mThe authors deal with completions \(L^\alpha({\mathcal{M}},\tau)\) under \(\alpha\) of a set of elementary operators \({\mathcal{I}}\) in a semifinite von Neumann algebra \({\mathcal{M}}\) with a faithful normal semifinite trace \(\tau\), where \(\alpha\) is a unitarily invariant norm on \({\mathcal{I}}\) satisfying some extra regularity conditions. Many examples of such norms are given. The definition of Arveson's noncommutative Hardy spaces \(H^\infty\) is extended to subspaces \(H^\alpha\) of \(L^\alpha({\mathcal{M}},\tau)\) being the closures of \({\mathcal{A}}\cap L^\alpha({\mathcal{M}}, \tau)\) with respect to the \(\alpha\)-norm, where \({\mathcal{A}}\) is a subdiagonal subalgebra of \({\mathcal{M}}\). The authors prove a Beurling-type theorem for the \(H^\alpha\) spaces, and use it to investigate invariant subspaces of a class of noncommutative Banach function spaces.
Reviewer: Stanisław Goldstein (Łódź)Cyclicity in the harmonic Dirichlet space.https://zbmath.org/1449.460242021-01-08T12:24:00+00:00"Abakumov, Evgueni"https://zbmath.org/authors/?q=ai:abakumov.evgeny-v"El-Fallah, Omar"https://zbmath.org/authors/?q=ai:el-fallah.omar"Kellay, Karim"https://zbmath.org/authors/?q=ai:kellay.karim"Ransford, Thomas"https://zbmath.org/authors/?q=ai:ransford.thomas-jSummary: The harmonic Dirichlet space is the Hilbert space of functions \(f\in L^2(\mathbb{T})\) such that \[ \Vert f\Vert^2_{\mathcal{D}(\mathbb{T})}:= \sum_{n\in\mathbb{Z}}(1+\vert n\vert)\vert\widehat f(n)\vert^2 <\infty. \] We give sufficient conditions for \(f\) to be cyclic in \(\mathcal{D}(\mathbb{T})\), that is, for \(\{\zeta^nf(\zeta):n\ge 0\}\) to span a dense subspace of \(\mathcal{D}(\mathbb{T})\).
For the entire collection see [Zbl 1404.42002].Consistent operator semigroups and their interpolation.https://zbmath.org/1449.470802021-01-08T12:24:00+00:00"ter Elst, A. F. M."https://zbmath.org/authors/?q=ai:ter-elst.antonius-f-m"Rehberg, J."https://zbmath.org/authors/?q=ai:rehberg.joachimSummary: Under a mild regularity condition we prove that the generator of the interpolation of two \(C_0\)-semigroups is the interpolation of the two generators.On self-adjoint realizations of sign-indefinite Laplacians.https://zbmath.org/1449.351912021-01-08T12:24:00+00:00"Pankrashkin, Konstantin"https://zbmath.org/authors/?q=ai:pankrashkin.konstantinLet \(\Omega \subset \mathbb{R}^d\) be a domain divided onto two parts \(\Omega^{\pm}\) having a joint smooth surface. The author considers the spectral problem for the operator \(-\nabla \cdot h \nabla\) in \(L_2(\Omega)\) with the Dirichlet problem on \(\partial \Omega\). The function \(h\) is equal to unity in \(\Omega^{+}\) and \(-\mu <0\) in \(\Omega^{-}\). This case of different signs of \(h\) corresponds to ``metamaterials''. The model case when \(\Omega^{\pm}\) are two rectangles having the joint side is investigated. The obtained results are extended to general case.
Reviewer: Vladimir Mityushev (Kraków)Essential spectrum, quasi-orbits and compactifications: application to the Heisenberg group.https://zbmath.org/1449.470282021-01-08T12:24:00+00:00"Mougel, Jérémy"https://zbmath.org/authors/?q=ai:mougel.jeremyLet \(H\) be the Heisenberg group. Using the natural bijection between \(H\) and \(\mathbb{R}\sp{3}\), there is introduced a compactification \(\bar{H}\) of \(H\) induced by the spherical compactification of \(\mathbb{R}\sp{3}\). Let \(T=-\Delta + V\) be the Schrödinger-type operator on \(L\sp{2}(H)\), where \(V\) is a continuous function on \(\bar{H}\). The main result of the paper gives a representation of the essential spectrum of \(T\) as the union of spectra of some simpler operators. There are also obtained some similar results.
Reviewer: Vladimir S. Pilidi (Rostov-na-Donu)Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations.https://zbmath.org/1449.540462021-01-08T12:24:00+00:00"Alecsa, Cristian Daniel"https://zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [\textit{H. Huang} et al., Positivity 23, No. 1, 21--34 (2019; Zbl 07053419)] and [\textit{H. Liu} and \textit{S. Xu}, Fixed Point Theory Appl. 2013, Paper No. 320, 10 p. (2013; Zbl 1295.54062)]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.Perturbation of outer inverses and the simplest expression.https://zbmath.org/1449.470082021-01-08T12:24:00+00:00"Zhang, Huichun"https://zbmath.org/authors/?q=ai:zhang.huichun"Pan, Weiwei"https://zbmath.org/authors/?q=ai:pan.weiwei"Guo, Zhirong"https://zbmath.org/authors/?q=ai:guo.zhirong"Huang, Qianglian"https://zbmath.org/authors/?q=ai:huang.qianglianSummary: This paper is concerned with the perturbation problem for outer inverses of linear bounded operators in Banach spaces. Some necessary and sufficient conditions for the simplest expression \(B = T^{\{ 2 \}} (I + \delta TT^{\{2 \}})^{-1}\) to be the \(\{2, 3\}\)-inverse, \(\{2, 4\}\)-inverse and \(\{2, 5\}\)-inverse of the perturbed operator \(\overline{T} = T + \delta T\) are obtained.The method of coupled upper and lower solutions for boundary value problems of fractional functional differential equations.https://zbmath.org/1449.342172021-01-08T12:24:00+00:00"Jian, Xingyue"https://zbmath.org/authors/?q=ai:jian.xingyue"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.mei"Luo, Zeyu"https://zbmath.org/authors/?q=ai:luo.zeyuSummary: In this paper, a class of boundary value problems of fractional functional differential equations with time delays is studied. Firstly, the problems studied in this paper are transformed into integral equations. The existence and uniqueness theorems of solutions for boundary value problems are proved by using nonlinear analysis theory. The monotone iterative sequences for solving the solutions of boundary value problems are generated and the error estimates are given. Secondly, by using the generalized monotone iteration technique and the coupled upper and lower solutions method, sufficient conditions for the existence and uniqueness of solutions of boundary value problems are obtained, and the range of solutions is determined. Finally, some examples are given to illustrate the wide applicability of our main results.Eight positive almost periodic solutions to an delay predator-prey system with impulsive and harvesting terms.https://zbmath.org/1449.342912021-01-08T12:24:00+00:00"Lv, Xiaojun"https://zbmath.org/authors/?q=ai:lv.xiaojun"Xie, Haiping"https://zbmath.org/authors/?q=ai:xie.haiping"Zhao, Kaihong"https://zbmath.org/authors/?q=ai:zhao.kaihongSummary: By using Mawhin's continuation theorem of coincidence degree theory and differential inequalities, we study an delay predator-prey system with impulsive and harvesting terms. Finally, we find some sufficient conditions for the existence of eight positive almost periodic solutions for the system under consideration.The existence of solution of set-valued equilibrium and Browder variational inclusion problems.https://zbmath.org/1449.471022021-01-08T12:24:00+00:00"Zhang, Congjun"https://zbmath.org/authors/?q=ai:zhang.congjun"Ju, Guiyin"https://zbmath.org/authors/?q=ai:ju.guiyin"Wang, Yuehu"https://zbmath.org/authors/?q=ai:wang.yuehuSummary: In this paper, we deal with the existence of the solution of set-valued equilibrium problems in the cone by using Ky Fan lemma. Moreover, we study Browder variational inclusion problem in the case of cone and extend the related results in recent literature.Best proximity point for proximal Berinde nonexpansive mappings on starshaped sets.https://zbmath.org/1449.470972021-01-08T12:24:00+00:00"Bunlue, Nuttawut"https://zbmath.org/authors/?q=ai:bunlue.nuttawut"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthepSummary: In this paper, we introduce the new concept of proximal mapping, namely proximal weak contractions and proximal Berinde nonexpansive mappings. We prove the existence of best proximity points for proximal weak contractions in metric spaces, and for proximal Berinde nonexpansive mappings on starshaped sets in Banach spaces. Examples supporting our main results are also given. Our main results extend and generalize some of the well-known best proximity point theorems for proximal nonexpansive mappings in the literature.Some surjectivity results for operators of generalized monotone type via a topological degree.https://zbmath.org/1449.470882021-01-08T12:24:00+00:00"Hong, Suk-Joon"https://zbmath.org/authors/?q=ai:hong.suk-joon"Kim, In-Sook"https://zbmath.org/authors/?q=ai:kim.in-sookSummary: We introduce a topological degree for a class of operators of generalized monotone type in reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we give some surjectivity results for operators of generalized monotone type in reflexive Banach spaces. In the Hilbert space case, this reduces to the celebrated Browder-Minty theorem for monotone operators.Semi-local convergence of a seventh-order method in Banach spaces under \(\omega\)-continuity condition.https://zbmath.org/1449.651112021-01-08T12:24:00+00:00"Gupta, Neha"https://zbmath.org/authors/?q=ai:gupta.neha"Jaiswal, J. P."https://zbmath.org/authors/?q=ai:jaiswal.jai-prakashSummary: The article is about the analysis of semi-local convergence of a seventh-order iterative method used for finding the roots of a nonlinear equation in Banach spaces. In this article, the imposed hypotheses is amiable than the well-known Lipschitz and Hölder continuity conditions. The \(R\)-order convergence of the considered scheme is proved to be at least \(4+3q\). An approximate apriori error bound for this method is also elaborated and the domain of existence and uniqueness of the solution in the convergence theorem. Two numerical illustrations have been worked out to exhibit the virtue of the developed theory.Bounded solutions of singular differential equation with asymptotic conditions.https://zbmath.org/1449.340682021-01-08T12:24:00+00:00"Zhao, Jin"https://zbmath.org/authors/?q=ai:zhao.jinSummary: Applying Schauder's fixed point theorem, the author considers a class of second-order singular differential equations with asymptotic conditions, proves the existence of bounded solutions, and generalizes the conclusion of bounded solutions of the arctic circulation model to general second order singular differential equations.Some spectral properties of a generalized Friedrichs model.https://zbmath.org/1449.810232021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovich"Turdiev, Khalim Khamraevich"https://zbmath.org/authors/?q=ai:turdiev.khalim-khamraevichSummary: We consider self-adjoint generalized Friedrichs model \(h(p)\), \(p \in\mathcal{T}^3\) (\(\mathcal{T}^3\) is the three-dimensional torus), in the case where the parameter functions \(w_1\) and \(w_2\) of this operator has the special forms. These functions has non-degenerate minimum at the several different points. Threshold effects for the considering operator are studied depending on the minimum points of \(w_2\).The Faddeev equation and location of the essential spectrum of a three-particle model operator.https://zbmath.org/1449.810222021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovich"Rakhmonov, Askar Akhmadovich"https://zbmath.org/authors/?q=ai:rakhmonov.askar-akhmadovichSummary: In this paper a model operator \(H\) associated to a system of three-particles on a lattice is considered. The location of the essential spectrum of \(H\) is described by the spectrum of channel operators. The Faddeev type equation for the eigenvectors of \(H\) is obtained.Existence and continuation of solutions of Hilfer fractional differential equations.https://zbmath.org/1449.340132021-01-08T12:24:00+00:00"Bhairat, Sandeep P."https://zbmath.org/authors/?q=ai:bhairat.sandeep-pSummary: In the present paper, we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality.Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup.https://zbmath.org/1449.342082021-01-08T12:24:00+00:00"Xu, Genqi"https://zbmath.org/authors/?q=ai:xu.gen-qiSummary: This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the \({L^p}\) linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system \( (A, B)\). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an \({L^p}\) well-posed system.Viscosity iterative algorithms for a new variational inequality problem and fixed point problem in Hilbert spaces.https://zbmath.org/1449.471032021-01-08T12:24:00+00:00"Cai, Gang"https://zbmath.org/authors/?q=ai:cai.gangSummary: The aim of this paper is to introduce a new viscosity iterative algorithm for finding a common element of the set of solutions of a new variational inequality problem for two inverse-strongly monotone operators and the set of fixed points of a nonexpansive mapping in Hilbert spaces. We give several strong convergence theorems under some suitable assumptions imposed on the parameters by using a modified extra-gradient method. A numerical example is also given to support our main results. The results obtained in this paper extend and improve many recent ones.A new iterative method for the split feasibility problem.https://zbmath.org/1449.471062021-01-08T12:24:00+00:00"Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoli"Jiang, Dan"https://zbmath.org/authors/?q=ai:jiang.danSummary: The split feasibility problem (SFP) has many applications, which can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. In this paper, we introduce a new projection method to solve the SFP and prove its convergence under standard assumptions. Our results improve previously known corresponding methods and results of this area. The preliminary numerical experiments illustrate the advantage of our proposed methods.Positive solutions of integral boundary value problem involving derivative boundary conditions.https://zbmath.org/1449.340972021-01-08T12:24:00+00:00"Zhao, Yang"https://zbmath.org/authors/?q=ai:zhao.yang"Wang, Xiamei"https://zbmath.org/authors/?q=ai:wang.xiamei"Yang, Zhilin"https://zbmath.org/authors/?q=ai:yang.zhilinSummary: This paper deals with the positive solutions for the second-order integral boundary value problem \[\begin{cases}u'' + f (t, u) = 0, \\ u (0) = \int_0^1 u (\tau){\mathrm{d}}\alpha (\tau), \\ u' (1) = \int_0^1 u' (\tau){\mathrm{d}}\beta (\tau),\end{cases}\] where \(f \in C ([0, 1] \times \mathbb{R}^+, \mathbb{R}^+)\). Based on a priori estimates, we use fixed-point index theory to establish the existence and multiplicity of positive solutions for the above problem.An iterative scheme for split variational inclusion and fixed point problem for demicontractive mappings.https://zbmath.org/1449.471202021-01-08T12:24:00+00:00"Zhang, Lijuan"https://zbmath.org/authors/?q=ai:zhang.lijuan"Chen, Junmin"https://zbmath.org/authors/?q=ai:chen.junminSummary: In this paper, we introduce an iterative method for finding a common solution of split variational inclusion problem and fixed point problem for a countable family of demicontractive mappings in Hilbert spaces. Strong convergence of the proposed iterative scheme is obtained under suitable control conditions. We provide some applications and numerical examples.Existence of solutions for a class of periodic boundary value problem of third-order nonlinear ordinary differential equations.https://zbmath.org/1449.340642021-01-08T12:24:00+00:00"Deng, Zhengping"https://zbmath.org/authors/?q=ai:deng.zhengping"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this paper, the existence of solutions for the periodic boundary value problem of the following third-order ordinary differential equation
\[\begin{cases}Lu (t) = f (t,u (t),u' (t),u'' (t)),\; t\in[0,\omega], \\ u^{ (k)} (0) = u^{ (k)} (\omega), k = 0, 1, 2\end{cases}\]
is considered, where \(Lu (t) = u''' (t)+{a_2}u'' (t) + {a_1}u' (t) + {a_0}u (t)\) is a third-order ordinary differential operator, \(f:[0,\omega]\times \mathbb{R}^3\to \mathbb{R}\) is continuous. Applying the Fourier analysis method and Leray-Schauder fixed point theorem, we obtain the existence and uniqueness of the solutions of the equation when the nonlinear term \(f\) satisfies some proper growth conditions.On solvability of inhomogeneous boundary-value problems in Sobolev-Slobodetskiy spaces.https://zbmath.org/1449.471332021-01-08T12:24:00+00:00"Mikhailets, V. A."https://zbmath.org/authors/?q=ai:mikhailets.vladimir-a|mikhailets.volodymyr-a"Skorobohach, T. B."https://zbmath.org/authors/?q=ai:skorobohach.t-bSummary: We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev-Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boundary-value problem. Sufficient conditions for the sequence of the characteristic matrices of a specified boundary-value problems to converge are found.Estimates for some convolution operators with singularities of their kernels on spheres.https://zbmath.org/1449.470692021-01-08T12:24:00+00:00"Gil', Alekseĭ Viktorovich"https://zbmath.org/authors/?q=ai:gil.aleksei-viktorovich"Zadorozhniĭ, Anatoliĭ Ivanovich"https://zbmath.org/authors/?q=ai:zadorozhnii.anatolii-ivanovich"Nogin, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:nogin.vladimir-aleksandrovichSummary: In the framework of Hardy spaces \(H^p\), we study multidimensional convolution operators whose kernels have power-type singularities on a finite union of spheres in \(\mathbb{R}^n\). Necessary and sufficient conditions are obtained for such operators to be bounded from \(H^p\) to \(H^q\), \(0<p\leq q<\infty\), from \(H^p\) to BMO, and from BMO to BMO.No entire inner functions.https://zbmath.org/1449.300892021-01-08T12:24:00+00:00"Cobos, A."https://zbmath.org/authors/?q=ai:cobos.a"Seco, D."https://zbmath.org/authors/?q=ai:seco.daniel|seco.diegoIt is well known that the only entire inner functions in the classical Hardy space \(H^2(\mathbb{D})\) are normalized monomials. The authors discuss the similar problem for other reproducing kernel Hilbert spaces of analytic functions on the unit disk. In particular, they consider the weighted Hardy spaces \(H^2_\omega(\mathbb{D})\) with \(\omega=(\omega_n)_{n\ge0}\), \(\omega_0=1\), and \(\omega_n>0\), \(n=1,2,\ldots\), satisfying \[ \lim_{n\to\infty} \frac{\omega_n}{\omega_{n+1}}=1, \qquad \sup_{n\le k\le 2n}\omega_k\le C\omega_n, \ \ n=1,2,\ldots, \] and the standard inner product \(\langle,\rangle_\omega\) with weights \(\omega_n\). A function \(f\in H^2_\omega(\mathbb{D})\) is called inner, if \(\langle z^n f, f\rangle_\omega=\delta_{n,0}\) for all positive integers \(n\).
A Shapiro-Shields function in \(H^2_\omega(\mathbb{D})\) is a function which is a proper analogue of the finite Blaschke products in \(H^2(\mathbb{D})\). The space \(H^2_\omega(\mathbb{D})\) has no extraneous zeros if all zeros of all Shapiro-Shields functions in the space are regular. The main result of the paper claims that the only entire inner functions for such weighted Hardy spaces are normalized monomials.
Reviewer: Leonid Golinskii (Kharkov)A reductive function algebra.https://zbmath.org/1449.471312021-01-08T12:24:00+00:00"Xu, Anjian"https://zbmath.org/authors/?q=ai:xu.anjian"Zou, Yang"https://zbmath.org/authors/?q=ai:zou.yangSummary: In this paper, we study reducibility of a function algebra which acts on a function space. Let \(X\) be a compact Hausdorff space, \(A\) be a logmodular algebra on \(X\), there is a unique representation measure \(m\) corresponding to a positive linear functional \(\varphi\) on \(A\). \({H^2} (m)\) is the closure of \(A\) in \({L^2} (m)\) which is the Lebesgue space over \(X\) defined by \(m\). It is shown that every function in \({H^2} (m)\) is a quotient of two functions in \({H^\infty} (m)\). \(A\) acts on \({H^2} (m)\) by multiplication. Then every densely defined linear transform \(T\) on \(A\) has compressed spectrum, and if \(B\) is a reductive algebra on \({H^2} (m)\) containing \(A\), then \(B\) is selfadjoint. This generalizes some known results.Existence and multiplicity of solutions for equations of \(p(x)\)-Laplace type in \(\mathbb{R}^N\) without AR-condition.https://zbmath.org/1449.351642021-01-08T12:24:00+00:00"Kim, Jae-Myoung"https://zbmath.org/authors/?q=ai:kim.jaemyoung"Kim, Yun-Ho"https://zbmath.org/authors/?q=ai:kim.yunho"Lee, Jongrak"https://zbmath.org/authors/?q=ai:lee.jun-ikSummary: We are concerned with the following elliptic equations with variable exponents \[ -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{in}\quad\mathbb{R}^N, \] where the function \(\varphi(x,v)\) is of type \(|v|^{p(x)-2}v\) with continuous function \(p:\mathbb{R}^N\to(1,\infty)\), \(V:\mathbb{R}^N\to(0,\infty)\) is a continuous potential function, and \(f:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}\) satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of \(\lambda\)'s for which our problem admits a nontrivial solution with simple assumptions in some sense.A new characterization of generalized composition operators from \({B^\alpha}\) spaces into \({Z^\beta}\) spaces.https://zbmath.org/1449.470522021-01-08T12:24:00+00:00"Chen, Cui"https://zbmath.org/authors/?q=ai:chen.cui"Ji, Jingrong"https://zbmath.org/authors/?q=ai:ji.jingrongSummary: Let \(\varphi \) be analytic self-map of the open unit disk \(D\) in the complex plane \(C\) and \(g\) be an analytic function on \(D\). In this paper, we characterize the boundedness and compactness of the generalized composition operator \(C_\varphi^g: {B^\alpha} \to {Z^\beta}\) in a new way. Moreover, the estimate for the essential norm of the generalized composition operator \(C_\varphi^g: {B^\alpha} \to {Z^\beta}\) is also investigated.The basis property of root vector systems of unbounded Hamiltonian operators via quadratic operator pencils.https://zbmath.org/1449.470792021-01-08T12:24:00+00:00"Qiao, Yanfen"https://zbmath.org/authors/?q=ai:qiao.yanfen"Hou, Guolin"https://zbmath.org/authors/?q=ai:hou.guolin"Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangSummary: In this paper, the Schauder basis property of the root vector systems for a class of unbounded Hamiltonian operators is studied by the induced quadratic operator pencils. First, the relationships between the root vectors of unbounded Hamiltonian operators and the corresponding quadratic operator pencils are established. Next, the eigenvalues distribution and the algebraic index of eigenvalues of unbounded Hamiltonian operators are described with the aid of the spectral properties of quadratic operator pencils. Then, a necessary and sufficient condition is given for the root vector systems of unbounded Hamiltonian operators to be a block Schauder basis of some Hilbert space. Eventually, the obtained results are validated for the bending problems of rectangular plates.Inverse spectral problems of symmetric potential functions.https://zbmath.org/1449.340562021-01-08T12:24:00+00:00"Wang, Wenjing"https://zbmath.org/authors/?q=ai:wang.wenjingSummary: In this paper, we consider the uniqueness problem of inverse Sturm-Liouville (S-L) differential operators defined on the interval \([0, 1]\). By making use of the Weyl function and Marchenko's uniqueness theorem, it is shown that if the potential function \(q (x)\) is multiple, symmetric and is known on some intervals, then the potential function \(q (x)\) on the interval \([0, 1]\) can be uniquely determined in terms of choosing a set of appropriate common eigenvalues.Some properties of the operators corresponding with spectrum determined growth condition.https://zbmath.org/1449.470402021-01-08T12:24:00+00:00"Sun, Lili"https://zbmath.org/authors/?q=ai:sun.lili"Wang, Lijie"https://zbmath.org/authors/?q=ai:wang.lijie"Wang, Hui"https://zbmath.org/authors/?q=ai:wang.hui.1|wang.hui|wang.hui.4|wang.hui.2|wang.hui.5"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.3"Ren, Hanjing"https://zbmath.org/authors/?q=ai:ren.hanjingSummary: Using functional analytic methods and the related properties of the expansion of the \({C_0}\) semigroup on the finite-dimensional subspace, we analyze the operators under spectrum determined growth condition. Then, we prove that the algebraic multiplicity of the eigenvalues on the spectral bound is equal to the geometrical multiplicity when the spectral bound and the growth bound are equal.The point, residual and continuous spectrum of \(3 \times 3\) upper triangular operator matrices.https://zbmath.org/1449.470132021-01-08T12:24:00+00:00"Wu, Xiufeng"https://zbmath.org/authors/?q=ai:wu.xiufeng"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjieSummary: Let
\[M_{D,E,F} = \begin{bmatrix}A&D&E\\ 0&B&E\\ 0&0&C\end{bmatrix}\]
be an upper triangular operator matrix on the Hilbert space \({\mathcal{H}_1} \oplus {\mathcal{H}_2} \oplus {\mathcal{H}_3}\). In this paper, we obtain necessary and sufficient conditions of \({\sigma_*} (M_{D,E,F}) = {\sigma_*} (A) \cup {\sigma_*} (B) \cup {\sigma_*} (C)\) for every \(D \in \mathcal{B} ({\mathcal{H}_2}, {\mathcal{H}_1})\), \(E \in \mathcal{B} ({\mathcal{H}_3}, {\mathcal{H}_1})\) and \(F \in \mathcal{B} ({\mathcal{H}_3}, {\mathcal{H}_2})\), in terms of the spectral properties of the diagonal entries \(A,B\) and \(C\) in \(M_{D,E,F}\), where \({\sigma_*}\) is the point spectrum, the residual spectrum, and the continuous spectrum. Moreover, we construct some examples illustrating our main results.Nonlocal boundary value problem for a Lykov's type system of first-order.https://zbmath.org/1449.353002021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: In this paper we prove the unique solution of the problem with a shift to a Lykov's type system of differential equations of first order. The proof is given for different values of the generalized operators of fractional integro-differentiation included in the boundary condition.Piecewise-polynomial signal segmentation using convex optimization.https://zbmath.org/1449.940412021-01-08T12:24:00+00:00"Rajmic, Pavel"https://zbmath.org/authors/?q=ai:rajmic.pavel"Novosadová, Michaela"https://zbmath.org/authors/?q=ai:novosadova.michaela"Daňková, Marie"https://zbmath.org/authors/?q=ai:dankova.marieSummary: A method is presented for segmenting one-dimensional signal whose independent segments are modeled as polynomials, and which is corrupted by additive noise. The method is based on sparse modeling, the main part is formulated as a convex optimization problem and is solved by a proximal splitting algorithm. We perform experiments on simulated and real data and show that the method is capable of reliably finding breakpoints in the signal, but requires careful tuning of the regularization parameters and internal parameters. Finally, potential extensions are discussed.On the nonlinear \(varPsi\)-Hilfer fractional differential equations.https://zbmath.org/1449.340232021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-d"Sousa, J. Vanterler da C."https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: We consider the nonlinear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve the nonlinear Cauchy problem for differential equations with \(\varPsi\)-Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and generalized (Kilbas-Saigo) Mittag-Leffler function respectively.Boundedness of self-map composition operators for two types of weights on the upper half-plane.https://zbmath.org/1449.470502021-01-08T12:24:00+00:00"Ardalani, Mohammad Ali"https://zbmath.org/authors/?q=ai:ardalani.mohammad-aliSummary: In this paper we find conditions for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for two kinds of weights which are of moderate growth.Approximately 2-local derivations on the semi-finite von Neumann algebras.https://zbmath.org/1449.470722021-01-08T12:24:00+00:00"Zhao, Xingpeng"https://zbmath.org/authors/?q=ai:zhao.xingpeng"Fang, Xiaochun"https://zbmath.org/authors/?q=ai:fang.xiaochun"Yang, Bing"https://zbmath.org/authors/?q=ai:yang.bingSummary: The definition of approximately 2-local derivation on von Neumann algebras is introduced based on the definitions of approximately local derivation and 2-local derivation. Approximately 2-local derivations on semi-finite von Neumann algebras are studied. Let \(M\) be a von Neumann algebra and \(\Delta: {M} \to {M}\) be an approximately 2-local derivation. It is easy to obtain that \(\Delta\) is homogeneous and \(\Delta\) satisfies \(\Delta ({x^2}) = \Delta (x)x + x\Delta (x)\) for any \(x \in {M}\). Besides, if \(M\) is a von Neumann algebra with a faithful normal semi-finite trace \(\tau\), then a sufficient condition for \(\Delta\) to be additive is given, that is, \(\Delta ({M}_\tau) \subseteq {M}_\tau\), where \(M_\tau = \{x \in M:\tau (|x|) < \infty\}\). In all, if \(\Delta\) is an approximately 2-local derivation on a semi-finite von Neumann algebra with a faithful normal semi-finite trace \(\tau\) and satisfies \(\Delta (M_\tau) \subseteq M_\tau\), where \(M_\tau = \{x \in M\}:\tau (|x|) < \infty\}\), by the conclusion that the Jordon derivation from a 2-torsion free semi-prime ring to itself is a derivation, it follows that \(\Delta\) is a derivation.The solution of the full matrix analogue of the generalized Abel equation with constant coefficients.https://zbmath.org/1449.470862021-01-08T12:24:00+00:00"Ismagilova, Rina Rinatovna"https://zbmath.org/authors/?q=ai:ismagilova.rina-rinatovnaSummary: The system of generalized integral Abel equations in the matrix form with constant coefficients on the segment is considered in terms of the integral Riemann-Liouville operators of matrix order. Its reduction to the system of singular integral equations was found. Solution of this system was found for the case of the commutative matrices of the simple structure in the explicit form.Existence and finite-time-stability of solutions for a class of fractional order fuzzy Cohen-Grossberg neural networks.https://zbmath.org/1449.342732021-01-08T12:24:00+00:00"Xiang, Hongjun"https://zbmath.org/authors/?q=ai:xiang.hongjun"Wang, Jinhua"https://zbmath.org/authors/?q=ai:wang.jinhua.1Summary: In this paper, a class of fractional-order fuzzy Cohen-Grossberg neural networks is discussed. By combining with the contraction mapping principle, the properties of fractional differential equations and inequality technique, the existence, uniqueness and finite-time-stability of the solutions for this model are studied. Additionally, an example is given to illustrate the main results.Mixed problems for degenerate abstract parabolic equations and applications.https://zbmath.org/1449.352562021-01-08T12:24:00+00:00"Shakhmurov, Veli. B."https://zbmath.org/authors/?q=ai:shakhmurov.veli-b"Sahmurova, Aida"https://zbmath.org/authors/?q=ai:sahmurova.aidaSummary: Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that these problems arise in fluid mechanics and environmental engineering.Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions.https://zbmath.org/1449.340842021-01-08T12:24:00+00:00"Ming, Zhongyang"https://zbmath.org/authors/?q=ai:ming.zhongyang"Zhang, Guowei"https://zbmath.org/authors/?q=ai:zhang.guowei|zhang.guowei.1"Li, Hongyu"https://zbmath.org/authors/?q=ai:li.hongyuSummary: In this paper, we investigate the derivative dependent second-order problem subject to Stieltjes integral boundary conditions
\[-u''(t)=f(t,u(t),u'(t)),\quad t\in[0,1],\]
\[au(0)-bu'(0)=\alpha[u],\, cu(1)+du'(1)=\beta[u],\]
where \(f\): \([0,1]\times \mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}^+\) is continuous, \(\alpha[u]\) and \(\beta[u]\) are linear functionals involving Stieltjes integrals. Some conditions on the nonlinearity \(f\) and the spectral radius of the linear operator are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index. The conditions allow that \(f(t,x_1,x_2)\) has superlinear or sublinear growth in \(x_1,x_2\). Two examples are provided to illustrate the theorems under multi-point and integral boundary conditions with sign-changing coefficients.Hardy type unique continuation properties for abstract Schrödinger equations and applications.https://zbmath.org/1449.353632021-01-08T12:24:00+00:00"Shakhmurov, Veli"https://zbmath.org/authors/?q=ai:shakhmurov.veli-bSummary: In this paper, Hardy's uncertainty principle and unique continuation properties of Schrödinger equations with operator potentials in Hilbert space-valued \(L^{2}\) classes are obtained. Since the Hilbert space \(H\) and linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrödinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.Linear operators and conjugations on a Banach space.https://zbmath.org/1449.470412021-01-08T12:24:00+00:00"Motoyoshi, Haruna"https://zbmath.org/authors/?q=ai:motoyoshi.harunaSummary: In this paper we study a conjugation on a Banach space \(\mathcal{X}\) and show properties of operators concerning conjugation \(C\) and show spectral properties of such operators. Next we show spectral properties of an \((m,C)\)-symmetry (isometry) operator \(T\) on a complex Banach space \(\mathcal{X}\). We prove that, for a \(C\)-doubly commuting pair \((T, S)\), if \(T\) is an \((m,C)\)-symmetry (isometry) and \(S\) is an \((n,C)\)-symmetry (isometry), then \(T + S\) and \(TS\) are \((m + n - 1,C)\)-symmetries (isometries).The relationship between similar invariant subspaces and invariant subspaces.https://zbmath.org/1449.470222021-01-08T12:24:00+00:00"Gu, Wen"https://zbmath.org/authors/?q=ai:gu.wen"Ni, Junna"https://zbmath.org/authors/?q=ai:ni.junnaSummary: The concept of ``similar invariant subspace'' is defined and the relationship between similar invariant subspace and invariant subspace under the conditions of reversible linear transformation and general linear transformation is discussed. Using the theory of vector space, it is proved that similar invariant subspace is equivalent to invariant subspace under the condition of reversible linear transformation. Furthermore, it is proved that for a linear transformation \(\sigma\) of vector space \(V\), if \(W\) is a similar invariant subspace, then \(W\) must be an invariant subspace.Duplex selections, equilibrium points, and viability tubes.https://zbmath.org/1449.340572021-01-08T12:24:00+00:00"Kánnai, Zoltán"https://zbmath.org/authors/?q=ai:kannai.zoltanSummary: Existence of viable trajectories to nonautonomous differential inclusions are proven for time-dependent viability tubes. In the convex case we prove a double-selection theorem and a new Scorza-Dragoni type lemma. Our result also provides a new and palpable proof for the equilibrium form of Kakutani's fixed point theorem.A class of compact integral-type operators on Bloch-Orlicz type spaces.https://zbmath.org/1449.470702021-01-08T12:24:00+00:00"He, Zhonghua"https://zbmath.org/authors/?q=ai:he.zhonghua"Jiang, Jiantao"https://zbmath.org/authors/?q=ai:jiang.jiantaoSummary: Combining the properties of the \(\mu\)-Bloch spaces and the function \({z^n}\), it is proved that if \(\varphi \in {\Delta_2}\) and \(\phi \in {\mathcal{B}^\varphi}\), then the integral type operator \({C_\phi}{I_g}\) is a compact operator on Bloch-Orlicz type spaces \({\mathcal{B}^\varphi}\) defined with Young functions. Furthermore, the necessary and sufficient conditions of the compactness of the operator \({C_\phi}{I_g}\) from Bloch spaces to \(\alpha\)-Bloch spaces is given in terms of the function \({z^n}\).Existence of positive solutions for a class of singular second-order ordinary differential equations with nonlinear boundary condition.https://zbmath.org/1449.340872021-01-08T12:24:00+00:00"Su, Xiaoxiao"https://zbmath.org/authors/?q=ai:su.xiaoxiaoSummary: In this paper, we study the existence of positive solutions of a class of singular second-order ordinary differential equations with nonlinear boundary conditions
\[\begin{cases}
u'' + \rho^2 u = \lambda g (t)f (u),\, t \in [0, 2\pi],\\
u (0) = h (u (2\pi))u (2\pi),\, u' (0) = u' (2\pi),
\end{cases}\]
where \(\rho \in (0, 1/4)\), \(\lambda > 0\) is a parameter, \(g:(0,2\pi] \to (0,\infty)\), and \(f: (0,\infty) \to \mathbb{R}\), \(h:[0, \infty) \to [1, \infty)\) are continuous functions, \(f\) may be singular at 0 and superlinear at \(\infty\). The proof of the main results is based on the Krasnoselskii's fixed point theorem.Existence and multiplicity of positive solutions for a class of second-order boundary value problems.https://zbmath.org/1449.340822021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantang"Jia, Kaijun"https://zbmath.org/authors/?q=ai:jia.kaijunSummary: In this paper, existence and multiplicity of positive solutions of the nonlinear second-order boundary value problems
\[\begin{cases}
(q (t)u' (t))' + f (u' (t)) = 0,\, t \in (0, 1), \\
q (0)u' (0) = 0, cu (1) + dq (1)u' (1) = 0
\end{cases}\]
are considered, where \(f: (-\infty, 0] \to [0, \infty)\), \(q:[0, 1] \to (0, \infty)\) are continuous functions, \(c > 0\), \(d \geq 0\) are constants. When the nonlinear term \(f\) satisfies superlinear growth condition or sublinear growth condition, we show that there exists at least one positive solution to the problem. When the nonlinear term \(f\) satisfies \(f_0:= \lim\limits_{s\to 0^-} \frac{f (s)}{s} = f_\infty:= \lim\limits_{s\to -\infty} \frac{f (s)}{s} = 0\) or \(f_0:= \lim\limits_{s \to 0^-} \frac{f (s)}{s} = f_\infty: = \lim\limits_{s \to -\infty} \frac{f (s)}{s} = \infty\), we show that there are at least two positive solutions to the problem. The proof is based on the fixed point theorem on cones.The existence of solutions for quasi-variational inequalities by using the fixed point index approach.https://zbmath.org/1449.471012021-01-08T12:24:00+00:00"Zhu, Yaping"https://zbmath.org/authors/?q=ai:zhu.yaping"Qu, Guorong"https://zbmath.org/authors/?q=ai:qu.guorong"Fan, Jianghua"https://zbmath.org/authors/?q=ai:fan.jianghuaSummary: A class of generalized projection operator is defined in this paper, and some properties of the generalized projection operator are obtained in reflexive, locally uniformly convex, smooth Banach spaces. The equivalence between the quasi-variational inequality problem and the fixed point problem is established. A concept of fixed point index of quasi-variational inequality is introduced and the fixed point index approach is applied to obtain the existence results for solutions of quasi-variational inequality problem under some conditions.Existence of positive solutions for a class of nonlinear fourth-order ordinary differential equations with boundary values.https://zbmath.org/1449.340952021-01-08T12:24:00+00:00"Zhang, Yali"https://zbmath.org/authors/?q=ai:zhang.yaliSummary: In this paper, we study the existence of positive solutions for a class of nonlinear fourth-order ordinary differential equations with boundary values:
\[\begin{cases}
u^{ (4)} (t) = \lambda f (t, u (t)),\, t \in (0, 1),\\
u (0) = u'' (0) = u''' (1) = 0,\\
u' (1) + C (u (1))u (1) = 0,
\end{cases}\]
where \(\lambda\) is a positive parameter, \(f:[0,1] \times \mathbb{R} \to [0,\infty)\) satisfies \({L^1}\)-Caratheodory conditions, \(C:[0, \infty) \to [0, \infty)\) is continuous. The proof of the main results is based on the fixed-point theorem of cone expansion-compression.New iteration process for approximating fixed points in Banach spaces.https://zbmath.org/1449.471232021-01-08T12:24:00+00:00"Bhutia, J. D."https://zbmath.org/authors/?q=ai:bhutia.jigmi-dorjee"Tiwary, K."https://zbmath.org/authors/?q=ai:tiwary.kalishankarSummary: The object of this paper is to present a new iteration process. We will show that our process is faster than the known recent iterative schemes. We discuss stability results of our iteration and prove some results in the context of uniformly convex Banach space for Suzuki generalized nonexpansive mappings. We also present a numerical example for proving the rate of convergence of our results. Our results improve many known results of the existing literature.Existence of mild solutions for a class of fractional semilinear integro-differential equation of mixed type.https://zbmath.org/1449.342772021-01-08T12:24:00+00:00"Zhu, Bo"https://zbmath.org/authors/?q=ai:zhu.bo"Han, Baoyan"https://zbmath.org/authors/?q=ai:han.baoyan"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishanSummary: In this paper, the authors studied the existence results of the mild solutions for a class of fractional semilinear integro-differential equation of mixed type by using the measure of noncompactness, \(k\)-set contraction and \(\beta\)-resolvent family. It is well known that the \(k\)-set contraction requires additional condition to ensure the contraction coefficient \(0 < k < 1\). We don't require additional condition to ensure the contraction coefficient \(0 < k < 1\). An example is introduced to illustrate the main results of this paper.The existence and stability of almost periodic solution for a class of neutral neural networks.https://zbmath.org/1449.342422021-01-08T12:24:00+00:00"Fang, Congna"https://zbmath.org/authors/?q=ai:fang.congna"Xie, Huiqin"https://zbmath.org/authors/?q=ai:xie.huiqinSummary: We study the almost periodic solution for a class of neutral neural networks with distributed time delays and time-varying delays. Using the fixed point theorem in Banach space, we obtain some new results on the existence and uniqueness and stability of the almost periodic solution. Finally, the validity of our results is illustrated by an example.Existence of positive periodic solutions of second-order differential equation with weak singularity.https://zbmath.org/1449.341212021-01-08T12:24:00+00:00"Miao, Liangying"https://zbmath.org/authors/?q=ai:miao.liangying"Liu, Xilan"https://zbmath.org/authors/?q=ai:liu.xilan"He, Zhiqian"https://zbmath.org/authors/?q=ai:he.zhiqianSummary: The existence of positive periodic solutions of the following second-order differential equation
\[u'' + a (t)u = f (t, u) + c (t)\]
is considered via Schauder's fixed point theorem, where \(a \in L^1(\mathbb{R}/T\mathbb{Z}; \mathbb{R}_+)\), \(c \in L^1 (\mathbb{R}/T\mathbb{Z}; \mathbb{R})\), \(f\) is a Carathéodory function. Our main results generalize some known results.Property \( (H)\) and perturbations.https://zbmath.org/1449.470112021-01-08T12:24:00+00:00"Chen, Lihong"https://zbmath.org/authors/?q=ai:chen.lihong"Su, Weigang"https://zbmath.org/authors/?q=ai:su.weigangSummary: This paper introduces two new spectral properties \( (H)\) and \( (gH)\), and investigates the two properties in connection with Weyl type theorems. Also, the preservations of the two properties are studied under commuting nilpotent, quasi-nilpotent, finite rank, or Riesz perturbation.A class of non-global derivable maps on prime rings.https://zbmath.org/1449.160842021-01-08T12:24:00+00:00"Kong, Liang"https://zbmath.org/authors/?q=ai:kong.liang"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(\mathscr{R}\) be a unital prime ring containing a nontrivial idempotent, \(\mathscr{Q} =\{T \in \mathscr{R}: {T^2} = 0\}\), and \(\mathscr{R} \to \mathscr{R}\) be a map (without additive assumption). Using the method of algebraic decomposition, we prove that if \(\delta (AB) = \delta (A)B + A\delta (B)\) for any \(A,B \in \mathscr{R}\) with \([A, B]B \in \mathscr{Q}\), then \(\delta\) is an additive derivation, where \([A, B] = AB - BA\) is the Lie product.Introducing \(p\)-eigenvectors; exact solutions for some simple matrices.https://zbmath.org/1449.150562021-01-08T12:24:00+00:00"Lócsi, Levente"https://zbmath.org/authors/?q=ai:locsi.leventeSummary: A common way to define a norm of a matrix is to take the supremum of the fraction of the vector norms of the matrix-vector product and the nonzero vector, with respect to a given vector norm, i.e. the least upper bound for the norm of the vectors of the transformed unit sphere. In this paper we examine the above mentioned fraction, defining induction curves and surfaces, we show that there exist some vectors, such that this fraction is independent of the applied \(p\)-norm (and are not eigenvectors). These are to be called \(p\)-eigenvectors. Exact solutions are constructed for some simple matrices. No previous work was found in this topic so far.On the invertibility of the sum of operators.https://zbmath.org/1449.470032021-01-08T12:24:00+00:00"Mortad, M. H."https://zbmath.org/authors/?q=ai:mortad.mohammed-hichemA few results about the invertibility of sums and linear combinations of operators on a Hilbert space are presented. The case of normal and self-adjoint operators is considered.
Reviewer: José Bonet (Valencia)Boundedness of the multilinear fractional integral operators on the Hardy spaces.https://zbmath.org/1449.420142021-01-08T12:24:00+00:00"Lin, Xiansheng"https://zbmath.org/authors/?q=ai:lin.xiansheng"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiechengSummary: In this paper we considered the boundedness of the multilinear fractional integral operators on a Hardy space. By virtue of the atomic decomposition of a Hardy space and the Hölder inequality, the boundedness of bilinear fractional integral operators and triple multilinear fractional integral operators on a Hardy space was obtained. The results extended some known conclusions.The discreteness of the spectrum of \(2N\)-th order one term vector differential operators.https://zbmath.org/1449.343092021-01-08T12:24:00+00:00"Qian, Zhixiang"https://zbmath.org/authors/?q=ai:qian.zhixiangSummary: The discreteness of the spectrum of vector differential operators generated by the \(2N\)-th order one term differential expression with matrix coefficients is considered and some sufficient conditions are obtained for ensuring the discreteness of the spectrum of these operators in the cases of self-adjoint and \(J\)-self-adjoint, respectively.On generalized harmonic vector variational inequalities using \(HC_*\)-condition.https://zbmath.org/1449.470992021-01-08T12:24:00+00:00"Mishra, S. N."https://zbmath.org/authors/?q=ai:mishra.suyash-narayan|mishra.swami-nath|mishra.satya-narayan"Das, P. K."https://zbmath.org/authors/?q=ai:das.pramod-kumar|das.pankaj-k|das.pankaj-kumar|das.prodip-kumar|das.pradip-kumar|das.pramode-k|das.ponkog-kumar|das.pranab-k-ii|das.prasanta-k|das.pratap-kumar|das.prakash-kumar"Mishra, S. K."https://zbmath.org/authors/?q=ai:mishra.sabin-kumar|mishra.sudib-kumar|mishra.saroj-kumar|mishra.surya-kant|mishra.shashi-kant|mishra.shailendra-kumar|mishra.sanjeev-kumar|mishra.satyendra-kumar|mishra.saroj-kanta|mishra.shambhu-kumar|mishra.surendra-kumarSummary: In this paper some results on \(HC_*\)-condition are established in the harmonic invex set and are used to establish the existence theorem of the solution of the generalized harmonic variational inequalities and its dual problem using generalized harmonically monotone property of the operator.Positive periodic solutions for second-order singular differential equations with damping terms.https://zbmath.org/1449.340722021-01-08T12:24:00+00:00"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: This paper studies the existence of positive periodic solutions of
\[u'' + p(t)u' + q (t)u = f (t,u) + c (t),\]
where \(p\), \(q\), \(c \in L^1 (\mathbb{R}/T\mathbb{Z};\mathbb{R})\), \(f\) is a Carathéodory function and is singular when \(u = 0\). By means of the fixed point theory, several existence theorems are established for the above equation, and some recent results in the literature are generalized and improved.Some results on second transpose of a dual valued derivation.https://zbmath.org/1449.460392021-01-08T12:24:00+00:00"Essmaili, Morteza"https://zbmath.org/authors/?q=ai:essmaili.mortezaSummary: Let \(A\) be a Banach algebra and \(X\) be an arbitrary Banach \(A\)-module. In this paper, we study the second transpose of derivations with value in dual Banach \(A\)-module \(X^*\). Indeed, for a continuous derivation \(D: A \longrightarrow X^*\) we obtain a necessary and sufficient condition such that the bounded linear map \(\Lambda \circ D^{\prime\prime} : A^{**} \longrightarrow X^{***}\) to be a derivation, where \(\Lambda\) is composition of restriction and canonical injection maps. This characterization generalizes some well known results in [\textit{M. Amini} et al., New York J. Math. 22, 265--275 (2016; Zbl 1354.46049)].A new Halpern-type algorithm for a generalized mixed equilibrium problem and a countable family of generalized nonexpansive-type maps.https://zbmath.org/1449.471052021-01-08T12:24:00+00:00"Chidume, C. E."https://zbmath.org/authors/?q=ai:chidume.charles-ejike"Nnnakwe, M. O."https://zbmath.org/authors/?q=ai:nnnakwe.m-oSummary: Let \(K\) be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space with dual space \(E^*\). In this paper, a new iterative algorithm of Halpern-type is constructed and used to approximate a common element of a generalized mixed equilibrium problem and a common fixed point for a countable family of ``generalized nonexpansive-type maps''. Application of our theorem, in the case of real Hilbert spaces, complements, extends and improves several important recent results. Finally, we give numerical experiments to illustrate the convergence of our sequence.Some convergence theorems for new iteration scheme in CAT(0) spaces.https://zbmath.org/1449.471172021-01-08T12:24:00+00:00"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In this paper, we construct an iteration scheme involving a hybrid pair of nonexpansive mappings. For this scheme, we prove some convergence theorems in CAT(0) spaces. In process, we remove a restricted condition (also called end-point condition) in previous existing results. Thus, we generalize and improve several relevant results cited in the literature.Existence of \(L_p(\varphi,\psi)\)-solutions of linear differential equations with generalized dichotomy in a Banach space.https://zbmath.org/1449.342072021-01-08T12:24:00+00:00"Kiskinov, H."https://zbmath.org/authors/?q=ai:kiskinov.hristo"Kostadinov, S."https://zbmath.org/authors/?q=ai:kostadinov.stepan-iSummary: A generalization of the well known dichotomies for a class of homogeneous linear differential equations in an arbitrary Banach space is used. By the help of them there are found sufficient conditions for the existence of \(L_p(\varphi,\psi)\)-solutions of the nonhomogeneous equation.Properties of inversion operator of the Abel matrix equation.https://zbmath.org/1449.470852021-01-08T12:24:00+00:00"Ismagilova, Rina Rinatovna"https://zbmath.org/authors/?q=ai:ismagilova.rina-rinatovnaSummary: Generalization of integral-differential Riemann-Liouville operator on the matrix order is reviewed and its properties are studied. Theorem of the composition of operators of the matrix of integration and differentiation can be proved. The necessary and sufficient conditions for the unique solvability of the matrix Abel equation in a special class of functions are obtained.Existence and uniqueness of positive solutions for a class of fractional impulsive differential equations with boundary value problems.https://zbmath.org/1449.340982021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"Gu, Chuanyun"https://zbmath.org/authors/?q=ai:gu.chuanyunSummary: By using the fixed point theorem for mixed monotone operator, a new criterion for the existence and uniqueness of positive solution of boundary value problems of a class of fractional impulsive differential equations
\[\begin{cases}
{}^CD_{0^+}^q u(t) = f(t, u(t), u(t)), t \in J' = J\backslash \{t_1,t_2,\cdots, t_m\}, J = [0,1],\\
\Delta u(t_k) = I_k(u(t_k), u(t_k)), \Delta u'(t_k) = J_k(u(t_k), u(t_k)), k = 1, 2, \cdots, m,\\
au(0) - bu(1) = 0,\, au'(0) - bu'(1) = 0
\end{cases}\]
is established, where \(1 < q < 2\), \({}^CD_{0^+}^q\) is the Caputo fractional derivative.Intertwining relation of integral-type operators on logarithmic Bloch-type spaces.https://zbmath.org/1449.470832021-01-08T12:24:00+00:00"Han, Xuehong"https://zbmath.org/authors/?q=ai:han.xuehong"Zeng, Honggang"https://zbmath.org/authors/?q=ai:zeng.honggangSummary: In this paper, we study the intertwining relation of a class of integral-type operators by composition operators on logarithmic Bloch-type space and give the equivalent conditions of \({C_\varphi}\) (compact) intertwining \({I_g}\) and \({I_h}\).Existence and uniqueness of solutions for Caputo-Hadamard type fractional differential equations.https://zbmath.org/1449.340322021-01-08T12:24:00+00:00"Shi, Linfei"https://zbmath.org/authors/?q=ai:shi.linfei"Li, Chengfu"https://zbmath.org/authors/?q=ai:li.chengfuSummary: In this paper, we study a class of Caputo-Hadamard fractional differential equations with boundary value problems. By using Banach fixed point theorem and the method of upper and lower solutions, the existence and uniqueness results of the solutions are obtained, which generalize some results about ordinary differential equations with boundary value problems. As an application, two examples are given to illustrate our main results.Approximating a common fixed point of a finite family of nonlinear mappings in modular function spaces.https://zbmath.org/1449.471282021-01-08T12:24:00+00:00"Wega, Getahun Bekele"https://zbmath.org/authors/?q=ai:wega.getahun-bekele"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: In this study, it is our purpose to investigate an algorithm for approximating a common fixed point of a finite family of \(\rho \)-quasi-nonexpansive mappings. In addition, we propose and analyze a scheme which estimates a common fixed point of a finite family of multivalued mappings in modular function spaces. As a consequence, we establish the \(\rho \)-convergence of the proposed algorithms under some mild conditions. In addition, some numerical examples which support our main results are presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.Dynamic total slip-rate-dependent frictional contact problem for a nonlinear viscoelastic material with long memory.https://zbmath.org/1449.741552021-01-08T12:24:00+00:00"Brahim, Nouiri"https://zbmath.org/authors/?q=ai:brahim.nouiri"Benyattou, Benabderrahmane"https://zbmath.org/authors/?q=ai:benyattou.benabderrahmaneSummary: In this paper, we consider a mathematical model which describes the dynamic frictional contact between a deformable body and rigid foundation. We assume that the behavior of the body is described by a nonlinear viscoelastic constitutive law with long memory. The friction condition is modeled with a simplified version of Coulomb's law in which the normal stress is prescribed and the coefficient of friction depends on the total slip-rate. We present the classical formulation of the problem, and derive a variational formulation which consists a second order evolutionary quasi-variational inequality for the displacement field. Then, we establish the existence and uniqueness result of weak solution. The proof is based on the Faedo-Galerkin method and Banach's fixed point theorem. Finally, we show a convergence result when the relaxation coefficients of long memory tend to zero.The structure of integral representations in topological vector spaces.https://zbmath.org/1449.460272021-01-08T12:24:00+00:00"Meziani, Lakhdar"https://zbmath.org/authors/?q=ai:meziani.lakhdarSummary: The subject of the present work deals with integral representations of bounded operators, acting on linear spaces of vector valued functions.
First we will consider operators on the space \(C_0(S,X)\) of all continuous functions \(f:S\to X\) vanishing at infinity, endowed with the uniform topology, \(S\) being a locally compact space and \(X\) a Banach space. We will give a complete characterization of operators \(T:C_0(S,X) \to X\) which enjoy an integral form with respect to a scalar measure \(\mu\) on \(S\).
Next we consider integral representations for operators on \(L_1\) type spaces, with values in a Banach space or a locally convex space. The main setting is the Bochner integration process with respect to finite abstract measure. The integral representations obtained may be considered as generalizations of the classical Riesz Theorem.Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms.https://zbmath.org/1449.471192021-01-08T12:24:00+00:00"Yao, Yonghong"https://zbmath.org/authors/?q=ai:yao.yonghong"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chih"Liou, Yeong-Cheng"https://zbmath.org/authors/?q=ai:liou.yeongcheng"Postolache, Mihai"https://zbmath.org/authors/?q=ai:postolache.mihaiSummary: The split common fixed points problem for demicontractive operators has been studied in Hilbert spaces. An iterative algorithm is considered and the weak convergence result is given under some mild assumptions.A bijection preserving \(*\)-quasi-product on \(B (H)\).https://zbmath.org/1449.470732021-01-08T12:24:00+00:00"Song, Xianhua"https://zbmath.org/authors/?q=ai:song.xianhuaSummary: Let \(B (H)\) be the algebra of all bounded linear operators on a complex Hilbert space \(H\) with dim \(H \ge 2\). For any \(A, B \in B (H)\), we define the quasi-product of \(A\) and \(B\) as \(A \circ B = A + B - AB\). It is proved that a bijection \(\phi\) on \(B (H)\) satisfies \(\phi ({A^*}\circ B) = \phi (A)^* \circ \phi (B)\) for all \(A, B\) in \(B (H)\) if and only if there is a unitary or an anti-unitary operator \(U\) on \(H\) such that \(\phi (A) = UA{U^*}\) for all \(A\) in \(B (H)\). When dim \(H \ge 3\) or there is a unitary operator \(U\) on \(H\) such that \(\phi (A) = U{A_\tau}{U^*}\) for all \(A\) in \(B (H)\) when dim \(H = 2\), where \(\tau\) is a ring automorphism on \(C\) and \({A_\tau} = \tau ({a_{ij}})\) for all \(A = ({a_{ij}})\) in \({M_2}\).\(k\)-quasi-homogeneous Toeplitz operators on pluriharmonic Bergman space of the unit ball.https://zbmath.org/1449.470592021-01-08T12:24:00+00:00"Dai, Xin"https://zbmath.org/authors/?q=ai:dai.xin"Dong, Xingtang"https://zbmath.org/authors/?q=ai:dong.xingtang"Zhang, Yingying"https://zbmath.org/authors/?q=ai:zhang.yingyingSummary: In this paper, we study some basic properties of the \(k\)-quasi-homogeneous Toeplitz operators on pluriharmonic Bergman space \(b_\alpha^2\) of the unit ball, and obtain two symmetric properties of the commutator and semi-commutator consisting of two such operators on \(b_\alpha^2\). Additionally, we obtain the necessary and sufficient conditions for the finite rank of commutator and semi-commutator of two monomial-type Toeplitz operators on \(b_\alpha^2\).Positive solutions for fractional boundary value problems of a class in ordered Banach spaces.https://zbmath.org/1449.340792021-01-08T12:24:00+00:00"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolongSummary: The existence of positive solutions for the boundary value problem of a class of fractional differential equations
\[{}^CD_{0^+}^\alpha u (t) = f (t,u (t)), 0 \leq t \leq 1,\, u (0) = u' (1) = u'' (0) = \theta\]
in an ordered Banach space \(E\) is discussed, where \(2<\alpha\leq 3\), and \({}^CD_{0^+}^\alpha\) is the Caputo fractional differentive, \(f:[0,1] \times P \to P\) is continuous, and \(P\) is the cone of positive elements in \(E\). An existence result of positive solutions is obtained by employing a new estimate of noncompactness measure and the fixed point index theory of condensing mapping.Product of Volterra type integral operator and composition operators between generalized Fock spaces.https://zbmath.org/1449.470672021-01-08T12:24:00+00:00"Luo, Xiaojuan"https://zbmath.org/authors/?q=ai:luo.xiaojuan"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jinSummary: In this paper, equivalent characterizations for the boundedness, compactness, and Schatten-\(p\) class properties of the product of a Volterra type integral operator and a composition operator between generalized Fock spaces \(F_\phi^p\) and generalized Fock spaces \(F_\phi^q\) are proposed in terms of certain Berezin integral transformations on the complex plane \(\textbf{C}\), where \(0 < p, q < \infty \). We also obtain some estimates on the essential norms of these operators.On a property of Toeplitz operators on Bergman space with a logarithmic weight.https://zbmath.org/1449.470622021-01-08T12:24:00+00:00"Sadraoui, Houcine"https://zbmath.org/authors/?q=ai:sadraoui.houcine"Halouani, Borhen"https://zbmath.org/authors/?q=ai:halouani.borhenSummary: An operator \(T\) on a Hilbert space is hyponormal if \(T^*T-TT^*\) is positive. In this work, we consider hyponormality of Toeplitz operators on the Bergman space with a logarithmic weight. Under a smoothness assumption, we give a necessary condition when the symbol is of the form \(f+\overline{g}\) with \(f,g\) analytic on the unit disk. We also find a sufficient condition when \(f\) is a monomial and \(g\) a polynomial.Periodic solutions and stability of nonlinear differential system with delays.https://zbmath.org/1449.342342021-01-08T12:24:00+00:00"Huang, Minghui"https://zbmath.org/authors/?q=ai:huang.minghui"Zhao, Guorui"https://zbmath.org/authors/?q=ai:zhao.guorui"Jin, Chuhua"https://zbmath.org/authors/?q=ai:jin.chuhuaSummary: By using Krasnoselskii's fixed point theorem, the existence of periodic solutions for nonlinear neutral differential system with delays are given. Some sufficient conditions for the uniqueness of periodic solutions and stability of zero solutions are obtained by the using contraction mapping principle. The conclusions generalize corresponding results in the literature.Second-order multiple-point boundary value problems with upper and lower solutions in the reverse order.https://zbmath.org/1449.340602021-01-08T12:24:00+00:00"Li, Haiyan"https://zbmath.org/authors/?q=ai:li.haiyan"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.2Summary: In this paper, we discuss second-order multiple-point boundary value problems with upper and lower solutions in the reverse order. We get the existence of the maximal solution and the minimal solution by introducing an increasing operator and giving a monotone iterative sequence. The uniqueness is proved by using the contraction mapping principle.A modified shrinking projection methods for numerical reckoning fixed points of \(G\)-nonexpansive mappings in Hilbert spaces with graphs.https://zbmath.org/1449.471242021-01-08T12:24:00+00:00"Hammad, A. A."https://zbmath.org/authors/?q=ai:hammad.a-a"Cholamjiak, W."https://zbmath.org/authors/?q=ai:cholamjiak.watcharaporn"Yambangwai, D."https://zbmath.org/authors/?q=ai:yambangwai.damrongsak"Dutta, H."https://zbmath.org/authors/?q=ai:dutta.hemen|dutta.himadri-sekhar|dutta.h-nr|dutta.harinarayan|dutta.haimontiSummary: In this paper, we introduce four new iterative schemes by modifying the shrinking projection method with Ishikawa iteration and \(S\)-iteration. Strong convergence theorems are given for obtaining a common fixed point of two \(G\)-nonexpansive mappings in a Hilbert space with a directed graph. We also give some numerical experiments for supporting our main theorems and compare the convergence rate between them.Coincidence point results in B-metric spaces via \(C_F\)-\(s\)-simulation function.https://zbmath.org/1449.540612021-01-08T12:24:00+00:00"Gupta, Anuradha"https://zbmath.org/authors/?q=ai:gupta.anuradha"Rohilla, Manu"https://zbmath.org/authors/?q=ai:rohilla.manuSummary: The notion of \(C_F\)-\(s\)-simulation function is introduced and the existence and uniqueness of coincidence point of two self mappings in the framework of b-metric spaces is investigated. An example with a corresponding numerical simulation is also provided to support the obtained result.Almost automorphic solutions for shunting inhibitory cellular neural networks with leakage delays on time scales.https://zbmath.org/1449.343182021-01-08T12:24:00+00:00"Dai, Lihua"https://zbmath.org/authors/?q=ai:dai.lihua"Hui, Yuanxian"https://zbmath.org/authors/?q=ai:hui.yuanxianSummary: Shunting inhibitory cellular neural networks with time-varying delays in the leakage term and continuously distributed delays on a time scale \(T\) are proposed. Based on the exponential dichotomy of linear dynamic equation on time scales, fixed point theorems on time scales, we obtain some new sufficient conditions for the existence and global exponential stability of almost automorphic solution for the class of neural networks. Moreover, we give convictive numerical examples to show the feasibility of our results. This paper studies several classes of functional differential equations, including the existence of solutions and the stability of this solution on time scales.A periodic solution of the coupled matrix Riccati differential equations.https://zbmath.org/1449.340652021-01-08T12:24:00+00:00"Goodarzi, Zahra"https://zbmath.org/authors/?q=ai:goodarzi.zahra"Razani, Abdolrahman"https://zbmath.org/authors/?q=ai:razani.abdolrahman"Mokhtarzadeh, M. R."https://zbmath.org/authors/?q=ai:mokhtarzadeh.mohammad-rezaSummary: Here, we generalize the martix Riccati differential equation to the coupled matrix Riccati differential equation. Using Schauder's fixed point theorem, the existence of at least one periodic solution of the coupled matrix Riccati equation with \(n\times n\) matrix coefficients is proved. Finally, two numerical examples are presented.Additivity of biderivable maps on generalized matrix algebras.https://zbmath.org/1449.160782021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: Let \(\mathcal{G}\) be a generalized matrix algebra, \(\varphi:\mathcal{G} \times \mathcal{G} \to \mathcal{G}\) be a mapping of \(\mathcal{G}\) (without assumption of additivity on each argument). If \(\varphi\) satisfies \(\varphi (XY, Z) = \varphi (X, Z)Y + X\varphi (Y, Z)\) and \(\varphi (X, YZ) = \varphi (X, Y)Z + Y\varphi (X, Z)\) for all \(X, Y, Z \in \mathcal{G}\), then \(\varphi\) is a biderivation.Recent development in fixed-point theory, optimization, and their applications.https://zbmath.org/1449.000042021-01-08T12:24:00+00:00"Li, Jinlu (ed.)"https://zbmath.org/authors/?q=ai:li.jinlu"Li, Chong (ed.)"https://zbmath.org/authors/?q=ai:li.chong.2"Wong, Ngai-Ching (ed.)"https://zbmath.org/authors/?q=ai:wong.ngai-ching"Yao, Jen-Chih (ed.)"https://zbmath.org/authors/?q=ai:yao.jen-chihFrom the text: This special volume was originally conceived to provide authors and readers a publication to present the most recent advances in the study of applications of various fixed point theorems.Non-classic 3D Goursat problem for one hyperbolic equation with discontinuous coefficients.https://zbmath.org/1449.354722021-01-08T12:24:00+00:00"Mamedov, Il'gar Gurdam"https://zbmath.org/authors/?q=ai:mamedov.ilgar-gurdamSummary: For a differential equation of hyperbolic type with discontinuous coefficients a 3D Goursat problem with nonclassical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions boundary condition is substantiated classical, in the case if the solution of the problem in the anisotropic S. L. Sobolev's space is found.On the triviality of domains of powers and adjoints of closed operators.https://zbmath.org/1449.470042021-01-08T12:24:00+00:00"Mortad, Mohammed Hichem"https://zbmath.org/authors/?q=ai:mortad.mohammed-hichemMotivated by results of \textit{K. Schmüdgen} [J. Oper. Theory 9, 53--75 (1983; Zbl 0507.47009)] about trivial domains of unbounded self-adjoint operators, the author presents several explicit (counter)examples concerning both the powers of an operator as well as the powers of their adjoints. Matrices of unbounded operators are utilized.
Reviewer: José Bonet (Valencia)Existence of positive solutions for boundary value problems of third-order delay differential equations.https://zbmath.org/1449.342182021-01-08T12:24:00+00:00"Luo, Qiang"https://zbmath.org/authors/?q=ai:luo.qiang"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaoling"Yang, Zhonggui"https://zbmath.org/authors/?q=ai:yang.zhongguiSummary: By applying the fixed point theorem on the cone, this paper studies the existence of positive solutions for boundary value problems of third-order delay differential equation \[\begin{cases}
u''' (t) + \lambda a (t)f (t, u (t-\tau)) = 0, & t \in (0, 1),\, \tau > 0, \\
u (t) = 0, & -\tau \leq t \leq 0, \\
u (0) = u'' (0) = 0,\, u (1) = \alpha u (\eta),
\end{cases}\]
where \(\lambda\) is parameter, and \(0 < \eta < 1\), \(0 < \alpha < \frac{1}{\eta}\), \(f:[0,1] \times [0, \infty] \to [0, \infty)\) is continuous.Isometries on certain non-complete vector-valued function spaces.https://zbmath.org/1449.470712021-01-08T12:24:00+00:00"Mojahedi, Mojtaba"https://zbmath.org/authors/?q=ai:mojahedi.mojtaba"Sady, Fereshteh"https://zbmath.org/authors/?q=ai:sady.fereshtehFor compact Hausdorff spaces \(X,Y\) and Banach spaces \(E,F\), the authors consider dense linear subspaces \(A,B\) of \(C(X, E)\) and \(C(Y, F)\), respectively, equipped with norms \(\|\cdot\|_A = \max\left(\|\cdot\|_\infty, p(\cdot) \right)\), \(\|\cdot\|_B = \max\left(\|\cdot\|_\infty, q(\cdot) \right)\), where \(p\) and \(q\) are seminorms that vanish on constant functions. The paper is devoted to sufficient conditions on the spaces involved and on a bijective isometry \(T : A \to B\) ensure the existence of a representation of \(T\) of the form of weighted composition operator \[(Tf)(y) = V_y(f(\Phi(y)),\] where \(\Phi: Y \to X\) is a homeomorphism, and each \(V_y : E \to F\) is an isometry. The conditions on the spaces are of geometric nature and they are weaker than strict convexity. The results of the paper complement previously known resuls for isometries of Lipschitz spaces and of spaces of absolutely continuous vector-valued functions.
Reviewer: Vladimir Kadets (Kharkiv)A inertial hybrid proximal extragradient method for solving monotone inclusions.https://zbmath.org/1449.471092021-01-08T12:24:00+00:00"He, Mingming"https://zbmath.org/authors/?q=ai:he.mingming"Peng, Jianwen"https://zbmath.org/authors/?q=ai:peng.jianwenSummary: In this paper, we study a new inertial hybrid proximal extragradient method for solving monotone inclusion problems. By using the Opial theorem, we obtain the weak convergence and the non-asymptotic global convergence rate of the inertial hybrid proximal extragradient method. In the framework of inertial hybrid proximal extragradient method, we propose and analyze the convergence and non-asymptotic global convergence rate of an inertial Tseng's forward-backward method and an inertial inexact Spingarn's partial inverse method.Invertibility of functions of operators and existence of hyperinvariant subspaces.https://zbmath.org/1449.470212021-01-08T12:24:00+00:00"Gamal, Maria F."https://zbmath.org/authors/?q=ai:gamal.maria-fThis paper provides condition under which an absolutely continuous, polynomially bounded operator will have nontrivial hyperinvariant subspaces.
A bounded linear operator \(T\) on a complex, separable, infinite-dimensional Hilbert space \(\mathcal{H}\) is said to be \textit{polynomially bounded} if there is a constant \(M\) such that, for every polynomial \(p\), \[\|p(T)\|\leq M\,\sup\{|p(z)|:|z|\leq 1\}.\] For example, by von Neumann's inequality, every contraction is polynomially bounded, with \(M=1\). Every polynomially bounded operator may be decomposed as a direct sum of a singular part and an absolutely continuous part. If both parts are nontrivial, then the operator will have nontrivial hyperinvariant subspaces. If the singular part is trivial, then the operator is said to be \textit{absolutely continuous.} The paper under review seeks to find conditions under which an absolutely continuous, polynomially bounded operator has nontrivial hyperinvariant subspaces.
The principal result is given in Theorem 2.5. There, \(T\) is an absolutely continuous, polynomially bounded operator and \(\mu\) is a finite, positive, singular Borel measure on the unit circle \(\mathbb{T}\) such that \(\mu(\sigma(T)\cap\mathbb{T})>0\) and \(\mu\) has no point masses on \(\sigma(T)\cap\mathbb{T}\). Define the singular inner function \(\theta_{\mu}\) by \[ \theta_{\mu}(z)=\exp\, \int_{\mathbb{T}}{\frac{z+\zeta}{z-\zeta}\,d\mu(\zeta)}\text{ for }\ |z|<1.\] If the operator \(\theta_{\mu}(T)\) is invertible, then there are nontrivial hyperinvariant subspaces \(\mathcal{M}_1\) and \(\mathcal{M}_2\) for \(T\) such that \(\sigma(T|_{\mathcal{M}_1})\cap \sigma(T|_{\mathcal{M}_2})=\emptyset\). The proof relies heavily on earlier results, given here as a set of lemmas.
Theorems 2.6 and 2.7 provide refinements of Theorem 2.5 to the cases where the spectrum \(\sigma(T)\) is equal to the unit circle or equal to an arc of the unit circle, respectively.
The later sections of the paper discuss the invertibility of inner functions of quasianalytic operators and of weighted shift operators.
Reviewer: Timothy Feeman (Villanova)Spectra of composition operators on weighted Bergman spaces.https://zbmath.org/1449.470552021-01-08T12:24:00+00:00"Pons, Matthew A."https://zbmath.org/authors/?q=ai:pons.matthew-aSummary: We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of \textit{B. MacCluer} and \textit{K. Saxe} [Isr. J. Math. 128, 325--354 (2002; Zbl 1024.47009)].Relative position of three subspaces in a Hilbert space.https://zbmath.org/1449.460232021-01-08T12:24:00+00:00"Enomoto, Masatoshi"https://zbmath.org/authors/?q=ai:enomoto.masatoshi"Watatani, Yasuo"https://zbmath.org/authors/?q=ai:watatani.yasuoSummary: We study the relative position of three subspaces in an infinite dimensional Hilbert space. In the finite-dimensional case over an arbitrary field, \textit{S. Brenner} [J. Algebra 6, 100--114 (1967; Zbl 0229.16020)] described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space over the complex numbers.Toeplitz type operators on the derivative Hardy space \(S^2(\mathbb{D})\).https://zbmath.org/1449.470602021-01-08T12:24:00+00:00"Gupta, Anuradha"https://zbmath.org/authors/?q=ai:gupta.anuradha"Singh, Shivam Kumar"https://zbmath.org/authors/?q=ai:singh.shivam-kumarSummary: A Toeplitz type operator \(T_{\phi}\) with co-analytic symbol \(\phi\) which can be seen as the adjoint of the multiplication operator on \(S^2(\mathbb{D})\) is introduced and studied on the derivative Hardy space \(S^2(\mathbb{D})\). The characterizations for the operator \(T_{\phi}\) to be normal, self-adjoint and isometric on \(S^2(\mathbb{D})\) have been obtained. In addition, it has been shown that the operator \(T_{\overline{z}^k}\) for a fixed non-negative integer \(k\) is a Fredholm operator and its point spectrum is the closed unit disk.Existence of multiple positive solutions for a class of fractional differential equations with integral boundary value conditions.https://zbmath.org/1449.340862021-01-08T12:24:00+00:00"Sun, Rui"https://zbmath.org/authors/?q=ai:sun.rui"Zhou, Wenxue"https://zbmath.org/authors/?q=ai:zhou.wenxueSummary: In this paper, using the fixed point theorem of Guo-Krasnosellskii's cone expansion and cone compression, the existence of multiple positive solutions of fractional differential equations with integral boundary value conditions is studied in case of uniform fractional derivative.Characterization of self-adjoint domains for regular even order \(C\)-symmetric differential operators.https://zbmath.org/1449.470822021-01-08T12:24:00+00:00"Sun, Jiong"https://zbmath.org/authors/?q=ai:sun.jiong"Bao, Qinglan"https://zbmath.org/authors/?q=ai:bao.qinglan"Hao, Xiaoling"https://zbmath.org/authors/?q=ai:hao.xiaoling"Zettl, Anton"https://zbmath.org/authors/?q=ai:zettl.antonSummary: Let \(C\) be a skew-diagonal constant matrix satisfying \(C^{-1}=-C=C^{\ast}\). We characterize the self-adjoint domains for regular even order \(C\)-symmetric differential operators with two-point boundary conditions. The previously known characterizations are a special case of this one.Existence and uniqueness of solutions for boundary value problems of fractional Langevin equation.https://zbmath.org/1449.340342021-01-08T12:24:00+00:00"Wang, Wenqian"https://zbmath.org/authors/?q=ai:wang.wenqian"Sun, Rui"https://zbmath.org/authors/?q=ai:sun.rui"Chai, Jianhong"https://zbmath.org/authors/?q=ai:chai.jianhong"Zhou, Yuqun"https://zbmath.org/authors/?q=ai:zhou.yuqunSummary: In this paper, the existence and uniqueness of solutions for boundary value problems of fractional Langevin equations are studied by using Leray-Schauder fixed point theorem. Then, the existence theorem of solutions is obtained.Solvability of infinite system of nonlinear singular integral equations in the \(C(I \times I, c)\) space and modified semi-analytic method to find a closed-form of solution.https://zbmath.org/1449.450072021-01-08T12:24:00+00:00"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.rezaSummary: In this article, we discuss about solvability of infinite systems of singular integral equations with two variables in the Banach sequence space \(C(I \times I, c)\) by applying measure of noncompactness and Meir-Keeler condensing operators. By presenting an example, we have illustrated our results. For validity of the results we introduce a modified semi-analytic method in the case of two variables to make an iteration algorithm to find a closed-form of solution for the above problem. The numerical results show that the produced sequence for approximating the solution of example is in the \(c\) space with a high accuracy.On the generalization of nonlinear fractional differential equations with non-separated boundary conditions.https://zbmath.org/1449.340372021-01-08T12:24:00+00:00"Xing, Yanyuan"https://zbmath.org/authors/?q=ai:xing.yanyuan"Xiao, Huafeng"https://zbmath.org/authors/?q=ai:xiao.huafengSummary: In this paper, the existence and uniqueness results of nonlinear fractional differential equations with non-separated boundary conditions are investigated. The Banach's fixed point theorem and Leray-Schauder degree theory are applied to establish the results. Some examples are given to illustrate the main result. Relevant results are generalized and improved.Derived cones to reachable sets of a second-order evolution inclusion.https://zbmath.org/1449.930032021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: We consider a class of second-order evolution inclusions and we prove that the reachable set of a certain second-order variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. This result allows to obtain a sufficient condition for local controllability along a reference trajectory.Commutators close to the identity.https://zbmath.org/1449.470372021-01-08T12:24:00+00:00"Tao, Terence"https://zbmath.org/authors/?q=ai:tao.terence-cSummary: Let \(D, X\in B(H)\) be bounded operators on an infinite dimensional Hilbert space \(H\). If the commutator \([D,X]=DX-XD\) lies within \(\epsilon\) in operator norm of the identity operator \(1_{B(H)}\), then it was observed by \textit{S. Popa} [Oper. Theory Adv. Appl. 6, 195--207 (1982; Zbl 0529.46043)] that one has the lower bound \(\Vert D\Vert\Vert X\Vert\geq\frac{1}{2}\log\frac{1}{\epsilon}\) on the product of the operator norms of \(D,X\). This is a quantitative version of the Wintner-Wielandt theorem that \(1_{B(H)}\) cannot be expressed as the commutator of bounded operators. On the other hand, it follows from the work of \textit{A. Brown} and \textit{C. Pearcy} [Ann. Math. (2) 82, 112--127 (1965; Zbl 0131.12302)] that one can construct examples in which \(\Vert D\Vert\Vert X\Vert=O(\epsilon^{-2})\). In this paper, the author improves the Brown-Pearcy construction to obtain examples od \(D\), \(X\) with \(\Vert[D,X]-1_{B(H)}\Vert\leq\epsilon\) and \(\Vert D\Vert \Vert X\Vert=O(\log^5\frac{1}{\epsilon})\).Hankel operators on generalized Fock spaces.https://zbmath.org/1449.470652021-01-08T12:24:00+00:00"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jin"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjunSummary: We characterize boundedness and compactness of Hankel operators on a very general class of weighted Fock spaces over \({\mathbb{C}^n}\) in terms of a certain notion of bounded and vanishing mean oscillation. The analogous description holds for the commutators \([{M_f}, P]\) where \({{M_f}}\) denotes the multiplication operator with symbol \(f\) and \(P\) is the Toeplitz projection. We also give geometric descriptions for the spaces BMO and VMO which are defined in terms of the Berezin transform.Alternative iterative technique.https://zbmath.org/1449.340522021-01-08T12:24:00+00:00"Avery, Richard"https://zbmath.org/authors/?q=ai:avery.richard-i"Anderson, Douglas"https://zbmath.org/authors/?q=ai:anderson.douglas-robert"Henderson, Johnny"https://zbmath.org/authors/?q=ai:henderson.johnnySummary: The standard methods of applying iterative techniques do not apply when the nonlinear term is neither monotonic (corresponding to an increasing or decreasing operator) nor Lipschitz (corresponding to a condensing operator). However, by applying the Layered Compression-Expansion Theorem in conjunction with an alternative inversion technique, we show how one can apply monotonicity techniques to a right focal boundary value problem.An integral-type operators from weighted Bergman space to Zygmund type spaces on the unit ball.https://zbmath.org/1449.320042021-01-08T12:24:00+00:00"Zhao, Yanhui"https://zbmath.org/authors/?q=ai:zhao.yanhui"Liao, Chunyan"https://zbmath.org/authors/?q=ai:liao.chunyan"Deng, Chunhong"https://zbmath.org/authors/?q=ai:deng.chunhongSummary: Some questions of integral-type operator were studied on Zygmund type spaces on the unit ball. By the methods of functional analysis and several complex variables, the necessary and sufficient conditions are given for integral-type operators to be bounded and compact on Zygmund type spaces on the unit ball. At the same time, the corresponding conclusions are obtained on the disk \(D\) and \(\varphi (z) = z\), respectively.Qualitative properties of solutions for mixed type functional-differential equations with maxima.https://zbmath.org/1449.342162021-01-08T12:24:00+00:00"Otrocol, Diana"https://zbmath.org/authors/?q=ai:otrocol.dianaSummary: In this paper, we study some properties of the solutions of a second order system of functional-differential equations with maxima, of mixed type, with ``boundary'' conditions. We use the weakly Picard operator technique.On existence and asymptotic behavior of the time-dependent solution of the \(\mathrm{M}/\mathrm{G}/1\) queueing model with optional deterministic server vacations.https://zbmath.org/1449.601302021-01-08T12:24:00+00:00"Kasim, Ehmet"https://zbmath.org/authors/?q=ai:kasim.ehmet"Gupur, Geni"https://zbmath.org/authors/?q=ai:gupur.geniSummary: In this paper, we consider the \(\mathrm{M}/\mathrm{G}/1\) queueing model with optional deterministic server vacations. Firstly, we convert the system into an abstract Cauchy problem, then we prove well-posedenss of the system by using the operator semigroup methods. Next, we investigate asymptotic behavior of its time-dependent solution by studying spectral properties of the corresponding operator. Therefore, we conclude that the time-dependent solution of the model strongly converges to its steady-state solution.Existence and stability results for nonlocal initial value problems for differential equations with Hilfer fractional derivative.https://zbmath.org/1449.340122021-01-08T12:24:00+00:00"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Bouriah, Soufyane"https://zbmath.org/authors/?q=ai:bouriah.soufyane"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of nonlocal initial value problems for differential equations with Hilfer's fractional derivative. The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results.Boundedness of the fractional integral operator with rough kernel and its commutator in vanishing generalized variable exponent Morrey spaces on unbounded sets.https://zbmath.org/1449.420182021-01-08T12:24:00+00:00"Mo, Huixia"https://zbmath.org/authors/?q=ai:mo.huixia"Wang, Xiaojuan"https://zbmath.org/authors/?q=ai:wang.xiaojuan"Han, Zhe"https://zbmath.org/authors/?q=ai:han.zhe|han.zhe.1Summary: In this paper, we study the boundedness of fractional integral operators and their commutators in vanishing generalized Morrey spaces with variable exponent on unbounded sets. Using the properties of variable exponent functions and the pointwise estimates of operators \(T_{\Omega, \alpha}\) and their commutators \([b, T_{\Omega, \alpha}]\) in Lebesgue spaces with variable exponent, we obtain the boundedness of fractional integral operators \(T_{\Omega, \alpha}\) and their commutators \([b, T_{\Omega, \alpha}]\) in vanishing generalized Morrey spaces with variable exponents on unbounded sets, which extend the previous results.Existence of entropy solutions to a doubly nonlinear integro-differential equation.https://zbmath.org/1449.450212021-01-08T12:24:00+00:00"Scholtes, Martin"https://zbmath.org/authors/?q=ai:scholtes.martin"Wittbold, Petra"https://zbmath.org/authors/?q=ai:wittbold.petraThe authors consider a class of doubly nonlinear problems with memory. They consider kernels of the type \(k(t)=t^{-\alpha}/\Gamma(1-\alpha)\). Doing so, the time-derivatives side becomes the fractional derivative of order \(\alpha\in(0,1)\) in the sense of Riemann-Liouville. The uniqueness of entropy solutions has been shown in a previous work. In this paper, the authors prove the existence of entropy solutions for general \(L^1\)-data and Dirichlet boundary conditions. The main idea of the existence proof is a modification of the regularization method by \textit{R. Landes} [J. Reine Angew. Math. 393, 21--38 (1989; Zbl 0664.35027)].
Reviewer: Vincenzo Vespri (Firenze)The mixed Lie triple \(\xi\)-derivation on prime \(*\)-algebras.https://zbmath.org/1449.170322021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Yang, Zhujun"https://zbmath.org/authors/?q=ai:yang.zhujun"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: The aim of this paper is to characterize the nonlinear mixed Lie triple \(\xi \)-derivation \( (\xi \ne 1)\) of a prime \(*\)-algebra. By using Peirce decomposition and the main proposition of mixed Lie triple \(\xi\)-derivation, it is proved that the nonlinear mixed Lie triple \(\xi\)-derivation \( (\xi \ne 1)\) of a prime \(*\)-algebra with unit and non-trivial projection is an additive \(*\)-derivation and linear about \(\xi\).Essential norm estimates for little Hankel operators with anti holomorphic symbols on weighted Bergman spaces of the unit ball.https://zbmath.org/1449.470632021-01-08T12:24:00+00:00"Tanaka, K."https://zbmath.org/authors/?q=ai:tanaka.keigo|tanaka.kohei|tanaka.kazuyuki|tanaka.keisuke|tanaka.kazuhito|tanaka.kanji|tanaka.ken-ichi|tanaka.kazunori|tanaka.katsunori|tanaka.koichiro|tanaka.kazunaga|tanaka.katsuhiko|tanaka.kaori|tanaka.katsuaki|tanaka.kazuyo|tanaka.kazuhiko|tanaka.kokoro|tanaka.katsuto|tanaka.keiji|tanaka.kenichiro|tanaka.kazuaki|tanaka.koumei|tanaka.keiichi|tanaka.kazuhide|tanaka.katsuhiro|tanaka.kiyoki|tanaka.kiyoshi|tanaka.kengo|tanaka.katsuki|tanaka.katsuya|tanaka.kotaro|tanaka.koichi|tanaka.kikuaki|tanaka.koji|tanaka.kazuo|tanaka.kakuji|tanaka.kenji|tanaka.kentaro|tanaka.k.4|tanaka.kanya|tanaka.katsuyuki|tanaka.kimiyuki|tanaka.kensuke|tanaka.katsumi|tanaka.k.3|tanaka.kazuhiro|tanaka.kiyoaki"Yamaji, S."https://zbmath.org/authors/?q=ai:yamaji.satoshiSummary: We give estimates for the essential norm of a little Hankel operator with anti holomorphic symbol on weighted Bergman spaces of the unit ball in terms of the Bloch semi-norm of its symbol function.Multiple positive solutions to singular fractional differential system with Riemann-Stieltjes integral boundary condition.https://zbmath.org/1449.340932021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Li, Yaohong"https://zbmath.org/authors/?q=ai:li.yaohongSummary: In this paper, we study a class of singular fractional differential systems with Riemann-Stieltjes integral boundary condition by constructing a new cone and using the Leggett-Williams fixed point theorem. The existence of multiple positive solutions is obtained. An example is presented to illustrate our main results.Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints.https://zbmath.org/1449.490202021-01-08T12:24:00+00:00"Mahmudov, Elimhan N."https://zbmath.org/authors/?q=ai:mahmudov.elimhan-nSummary: The present paper studies a new class of problems of optimal control theory with differential inclusions described by fourth order Sturm-Liouville type differential operators (SLDOs). Then, there arises a rather complicated problem with simultaneous determination of the SLDOs with variable coefficients and a Mayer functional depending of high order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and ``transversality'' conditions are derived. Formulation of the transversality conditions at the endpoints \(t = 0\) and \(t = 1\) of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with fourth order SLDO is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem, for which the Weierstrass-Pontryagin maximum principle is obtained.Variation inequality for heat semigroup related to Schrödinger operator on the weighted Morrey spaces.https://zbmath.org/1449.471002021-01-08T12:24:00+00:00"Yu, Jinxia"https://zbmath.org/authors/?q=ai:yu.jinxia"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.5|zhang.jing.9|zhang.jing.11|zhang.jing.7|zhang.jing.10|zhang.jing.12|zhang.jing.6|zhang.jing.8|zhang.jing.1|zhang.jing.3|zhang.jing.2Summary: This paper is devoted to investigate variation inequality of heat semigroup related to Schrödinger operator on the weighted Morrey spaces. By estimating the kernel function and using weights and weight's properties, it is proved that the variation operator is bounded on the weighted Morrey spaces.Different types of solutions for nonlinear fractional integral boundary value problems with two parameters.https://zbmath.org/1449.340352021-01-08T12:24:00+00:00"Wang, Wenxia"https://zbmath.org/authors/?q=ai:wang.wenxia"Mi, Fang"https://zbmath.org/authors/?q=ai:mi.fangSummary: This paper is concerned with the existence of different types of solutions for a class of nonlinear fractional differential equations with two parameters under integral boundary conditions. By using a fixed point theorem and analytic technique, we divide the range of these parameters for the existence of positive solutions, negative solutions and sign-changing solutions for the boundary value problem and obtain some new results.Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives.https://zbmath.org/1449.340202021-01-08T12:24:00+00:00"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waell"Harikrishnan, Sugumaran"https://zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Kanagarajan, Kuppusamy"https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamySummary: In this note, we discuss the existence, uniqueness and stability results for a general class of Langevin equations. We suggest the generalization via the Hilfer-Katugampola fractional derivative. We introduce some conditions for existence and uniqueness of solutions. We utilize the concept of fixed point theorems (Krasnoselskii fixed point theorem (KFPT), Banach contraction principle (BCP)). Moreover, we illustrate definitions of the Ulam type stability. These definitions generalize the fractional Ulam stability.Boundary triples for integral systems on the half-line.https://zbmath.org/1449.450172021-01-08T12:24:00+00:00"Strelnikov, D."https://zbmath.org/authors/?q=ai:strelnikov.dmytro|strelnikov.d-iThe author studies the following integro-differential system of Sturm-Liouville type on the half-line \([0,\infty)\): \[ J\vec{f}(x)-J\vec{a}=\int\limits_0^x\begin{pmatrix} \lambda dW-dQ & 0\\ 0 & dP\end{pmatrix} \vec{f}(t),\quad J=\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}. \] Here \(P,Q,W\) are real functions of locally bounded variation on \([0,\infty )\), and \(W\) is non-decreasing. The author defines minimal and maximal linear relations associated with the system in the limit point and limit circle cases, and finds boundary triplets and corresponding Weyl functions. The case of a compact interval was studied by the author earlier [\textit{D. Strelnikov}, J. Math. Sci., New York 231, No. 1, 83--100 (2018; Zbl 1401.45002)].
Reviewer: Anatoly N. Kochubei (Kyïv)Property \( (\omega_1)\) and the single-valued extension property.https://zbmath.org/1449.470162021-01-08T12:24:00+00:00"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.lei"Huang, Xiaojing"https://zbmath.org/authors/?q=ai:huang.xiaojing"Guo, Qi"https://zbmath.org/authors/?q=ai:guo.qiSummary: A bounded linear operator \(T\) satisfies property \( (\omega_1)\), if the complement in the approximate point spectrum \({\sigma_a} (T)\) of the upper semi-Weyl spectrum \({\sigma_{ea}} (T)\) is contained in the set of all isolated points of the spectrum \(\sigma (T)\) which are finite eigenvalues. In this paper, by means of the new spectrum defined in view of the single-valued extension property, the sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space satisfying the property \( (\omega_1)\) are established. As an application, the property \( (\omega_1)\) for hypercyclic (or supercyclic) operators are characterized.Sharp inequalities for the numerical radii of block operator matrices.https://zbmath.org/1449.470182021-01-08T12:24:00+00:00"Ghaderi Aghideh, M."https://zbmath.org/authors/?q=ai:ghaderi-aghideh.masoomeh"Moslehian, M. S."https://zbmath.org/authors/?q=ai:moslehian.mohammad-sal"Rooin, J."https://zbmath.org/authors/?q=ai:rooin.jamalLet \(A,B,C,D\) be bounded linear Hilbert space operators. The authors obtain sharp upper bounds for the numerical radii \(w(T)\) and \(w(S)\) of the block operator matrices \(T = \left(\begin{smallmatrix} A & O \\ O & D \end{smallmatrix}\right)\) and \(S = \left(\begin{smallmatrix} 0 & B \\ C & 0 \end{smallmatrix}\right)\). For example, it is shown that, if \(f(t),g(t)\) are continuous non-negative functions on \([0,\infty)\) with \(f(t)g(t)=t\), then, for any non-negative non-decreasing convex function \(h(t)\) on \([0,\infty)\), the following inequality holds:
\[h(w(T))\le \frac{1}{2} \max \left(||h(f^{2} (|A|))+h(g^{2} (|A|))||,\, ||h(f^{2} (|A|))+h(g^{2} (|A|))||\right).\]
A similar inequality is proved for the numerical radius \(w(S)\). The paper also contains other variations of these inequalities.
Reviewer: Khristo N. Boyadzhiev (Ada)Duals of Hardy amalgam spaces and norm inequalities.https://zbmath.org/1449.420382021-01-08T12:24:00+00:00"Ablé, Z. V. P."https://zbmath.org/authors/?q=ai:able.z-v-p"Feuto, J."https://zbmath.org/authors/?q=ai:feuto.justinThere are many generalizations of the classical Hardy spaces by taking the norm of the maximal function in certain spaces rather than in the Lebesgue ones. The author's choice for replacing is the Wiener amalgam spaces. In the paper under review, they first study characterizations of such spaces, including the atomic ones. Then they characterize the dual spaces of the generalized Hardy spaces defined in the above way. Finally, they prove that in these generalized Hardy spaces some classical singular operators, such as Calderón-Zygmund, convolution and Riesz potential operators, are bounded.
Reviewer: Elijah Liflyand (Ramat-Gan)\((\alpha ,\beta )\)-\(A\)-normal operators in semi-Hilbertian spaces.https://zbmath.org/1449.470422021-01-08T12:24:00+00:00"Benali, Abelkader"https://zbmath.org/authors/?q=ai:benali.abelkader"Ould Ahmed Mahmoud, Sid Ahmed"https://zbmath.org/authors/?q=ai:sid-ahmed.ould-ahmed-mahmoudSummary: Let \({\mathcal{H}}\) be a Hilbert space and let \(A\) be a positive bounded operator on \({\mathcal{H}}\). The semi-inner product \(\langle u\,|\,v \rangle _A:=\langle Au\,|\,v\rangle\), \(u,v \in{\mathcal{H}}\), induces a semi-norm \(\left\| .\right\| _A\) on \({\mathcal{H}}.\) This makes \({\mathcal{H}}\) into a semi-Hilbertian space. In this paper, we introduce a new class of operators called \((\alpha ,\beta )\)-\(A\)-normal operators in semi-Hilbertian spaces. Some structural properties of this class of operators are established.Extension of factorization theorems of Maurey to \(s\)-positively homogeneous operators.https://zbmath.org/1449.470392021-01-08T12:24:00+00:00"Tiaiba, Abdelmoumen"https://zbmath.org/authors/?q=ai:tiaiba.abdelmoumenSummary: In the present work, we prove that the class of \(s\)-positively homogeneous operators is a Banach space. As application, we give the generalization of some Maurey factorization theorems to \(T\) which is a \(s\)-positively homogeneous operator from \(X\) a Banach space into \(L_p\). We establish necessary and sufficient conditions to prove that \(T\) factors through \(L_q\). After this we extend the dual factorization theorem to the same class of operators above.The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative.https://zbmath.org/1449.340402021-01-08T12:24:00+00:00"Zhang, Shuqin"https://zbmath.org/authors/?q=ai:zhang.shuqin"Hu, Lei"https://zbmath.org/authors/?q=ai:hu.lei"Sun, Sujing"https://zbmath.org/authors/?q=ai:sun.sujingSummary: In this paper, we study some results about the expression of solutions to some linear differential equations for the Caputo-Fabrizio fractional derivative. Furthermore, by the Banach contraction principle, the unique existence of the solution to an initial value problem for nonlinear differential equation involving the Caputo-Fabrizio fractional derivative is obtained.Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms.https://zbmath.org/1449.150672021-01-08T12:24:00+00:00"Li, Chi-Kwong"https://zbmath.org/authors/?q=ai:li.chi-kwong"Tsai, Mimg-Cheng"https://zbmath.org/authors/?q=ai:tsai.mimg-cheng"Wang, Ya-Shu"https://zbmath.org/authors/?q=ai:wang.ya-shu"Wong, Ngai-Ching"https://zbmath.org/authors/?q=ai:wong.ngai-ching|wong.ngaichingThis paper offers solutions to some new linear preserver problems (for some historical background, see [the first author and \textit{N.-K. Tsing}, Linear Algebra Appl. 162--164, 217--235 (1992; Zbl 0762.15016); the first author and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)]). Let \(M_{m,n}\) be the \(\mathbb{F}\)-space of all \(m\times n\) matrices over \(\mathbb{F}\) where \(\mathbb{F=R}\) or \(\mathbb{C}\). We say that \(A,B\in M_{m,n}\) are disjoint (written \(A\bot B\)) if \(AB^{\ast}=0\) and \(BA^{\ast}=0\). An \(\mathbb{F}\)-linear mapping \(\varphi : M_{m,n}\rightarrow M_{r,s}\) is said to preserve disjointness if \(A\bot B\) \(\implies\varphi(A)\bot\varphi(B)\) for all \(A,B\in M_{m,n}\). The main theorem of this paper is that \(\varphi\) preserves disjointness if and only if there exist diagonal matrices \(Q_{1},Q_{2}\) with positive diagonal entries such that for some unitary matrices \(U\) and \(V\) of suitable sizes we have \[ \varphi(A)=U\left[ \begin{matrix} A\otimes Q_{1} & 0 & 0\\ 0 & A\otimes Q_{2} & 0\\ 0 & 0 & 0 \end{matrix} \right] V\text{ for all }A\in M_{m,n} \] (\(Q_{1}\) or \(Q_{2}\) may be vacuous). A linear mapping \(\varphi:M_{m,n} \rightarrow M_{r,s}\) is called a \(JB^{\ast}\)-triple homomorphism if \(\varphi(AB^{\ast}C+CB^{\ast}A)=\varphi(A)\varphi(B)^{\ast}\varphi (C)+\varphi(C)\varphi(B)^{\ast}\varphi(A)\) for all \(A,B,C\in M_{m,n}\). Using the main theorem, the authors show that there is a similar description for \(\varphi\) when \(\varphi\) is a \(JB^{\ast}\)-triple homomorphism (an extra hypothesis on the size of the zero block in the bottom right-hand corner of the displayed matrix is required). Further results are obtained for linear preservers \(\varphi\) for the Schatten \(p\)-norm and the Ky-Fan \(k\)-norm. However in the latter case, the result only holds for the case \(\mathbb{F} =\mathbb{C}\) since real isometries for Ky-Fan \(k\)-norms do not preserve disjointness.
Reviewer: John D. Dixon (Ottawa)Spectra of weighted composition operators induced by automorphisms.https://zbmath.org/1449.470482021-01-08T12:24:00+00:00"Gao, Yong-Xin"https://zbmath.org/authors/?q=ai:gao.yongxin"Zhou, Ze-Hua"https://zbmath.org/authors/?q=ai:zhou.zehuaLet \(\varphi\) be an automorphism (that is, a bijective holomorphic selfmap) of the unit disk \(\mathbb{D}\) and \(\psi\) just holomorphic on \(\mathbb{D}\). The authors determine the spectra for non-invertible weighted composition operators \(C_{\psi,\varphi}: f\mapsto (f\circ\varphi) \psi\) on the usual Hardy space \(H^2(\mathbb{D})\) and discuss the situation for the weighted Bergman spaces \(A^2_\alpha(\mathbb{D})\). As usual, the result splits into the three cases of hyperbolic, elliptic and parabolic automorphisms. This work complements that by \textit{O. Hyvärinen} et al. [J. Funct. Anal. 265, No. 8, 1749--1777 (2013; Zbl 1325.47054)]. Birkhoff's ergodic theorem plays an important role in the proofs.
Reviewer: Raymond Mortini (Metz)Proof of the \( (\omega)\) property.https://zbmath.org/1449.470142021-01-08T12:24:00+00:00"Yan, Huihuang"https://zbmath.org/authors/?q=ai:yan.huihuang"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: Using the new spectrum we define in this paper, we characterize the necessary and sufficient conditions for the bounded linear operators on Hilbert spaces satisfying \( (\omega_1)\) property and \( (\omega)\) property. Moreover, using this spectrum, we prove the \( (\omega)\) property of the functional calculus for operators.A note on the range of the product of two operators.https://zbmath.org/1449.470052021-01-08T12:24:00+00:00"Qin, Mengjie"https://zbmath.org/authors/?q=ai:qin.mengjie"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangSummary: Let \(H\) and \(K\) be two Hilbert spaces, and let \(A\in {\mathrm{B}} (H)\), \(B\in {\mathrm{B}} (K,H)\) be two bounded linear operators such that \({\mathrm{ind}} (A) \le 1\), \({\mathrm{R}} (AB)\subseteq {\mathrm{R}} (B)\) and \({\mathrm{R}} (B)\) is closed in \(H\). A sufficient condition is given under which \({\mathrm{R}} (AB) = {\mathrm{R}} (A) \cap {\mathrm{R}} (B)\). Furthermore, a counterexample is constructed such that \({\mathrm{R}} (AB) \ne {\mathrm{R}} (A) \cap {\mathrm{R}} (B)\).Inverse spectral problem for a pair of self-adjoint Hankel operators.https://zbmath.org/1449.350322021-01-08T12:24:00+00:00"Gérard, Patrick"https://zbmath.org/authors/?q=ai:gerard.patrick"Grellier, Sandrine"https://zbmath.org/authors/?q=ai:grellier.sandrineSummary: We give a precise inverse spectral result for compact self-adjoint Hankel operators. From Megretskii-Peller-Treil, a necessary and sufficient condition on a sequence of non zero real numbers, finite or infinite but tending to zero, to be a sequence of eigenvalues of some self-adjoint and compact Hankel operator is that the multiplicity of an eigenvalue \(\lambda\) should differ from the multiplicity of \(-\lambda\) at most by one. Under this condition, we describe precisely the set of symbols for which the Hankel operator has a given sequence of eigenvalues. This theorem is a consequence of a general inverse spectral result that we proved for non-necessarily self-adjoint Hankel operators. As a by-product, we show how we recover the Megretskii-Peller-Treil condition.
For the entire collection see [Zbl 1404.42002].Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems.https://zbmath.org/1449.340812021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantangSummary: We consider the existence of positive solutions for the periodic boundary value problems of nonlinear second-order systems \[\begin{cases}u'' + A (t)u = \Lambda G (t)F (u),\, 0 < t < 1, \\ u (0) = u (1), u' (0) = u' (1),\end{cases}\] where \(u = ({u_1}, \cdots, {u_n})^{\mathrm{T}}\), \(A (t) = {\mathrm{diag}}[{a_1} (t),\cdots,{a_n} (t)]\), \({a_i} (t)\) can change the sign in \([0,1] (i = 1,\cdots, n)\), \(G (t) = {\mathrm{diag}}[{g_1} (t),\cdots, {g_n} (t)]\), \(F (u) = ({f_1} (u),\cdots,{f_n} (u))^{\mathrm{T}}\), \(\Lambda = {\mathrm{diag}} ({\lambda_1}, \cdots, {\lambda_n})\), \({\lambda_i}\) is a positive parameter \( (i = 1, \cdots, n)\). Under the assumption that the nonlinear term \(F\) satisfies superlinear, sublinear and asymptotic growth condition, the existence of positive solutions of the problem is obtained by using the fixed-point theorem of cone expansion-compression. The conclusions in this paper generalize and improve related results.Nonlocal integral boundary value problem of Bagley-Torvik type fractional differential equations and inclusions.https://zbmath.org/1449.340142021-01-08T12:24:00+00:00"Chen, Lizhen"https://zbmath.org/authors/?q=ai:chen.lizhen"Ibrahim, Badawi Hamza Eibadawi"https://zbmath.org/authors/?q=ai:ibrahim.badawi-hamza-eibadawi"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8Summary: In this article, we consider the Bagley-Torvik type fractional differential equation \[{}^cD^{v_1}l (t)-a{}^cD^{v_2}l (t) = g (t,l (t))\]
and the differential inclusion
\[{}^cD^{v_1}l (t)-a{}^cD^{v_2}l (t) \in G (t,l (t)), t \in (0,1)\]
subject to
\[l (0) = {l_0}, \text{ and }l (1) = \lambda'\int_0^\omega \frac{(\omega-s)^{\chi-1}l (s)}{\Gamma (\chi)}ds,\]
where \(1 < v_1 \leq 2\), \(1 \leq v_2 < v_1\), \(0 < \omega \leq 1\), \(\chi = v_1 - v_2 > 0\), \(a\), \(\lambda'\) are given constants. By using the Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend existence theorems for the classical Bagley-Torvik equation and some related models.Characterizations of incoherent operators and strongly incoherent operators.https://zbmath.org/1449.470382021-01-08T12:24:00+00:00"Tian, Rui"https://zbmath.org/authors/?q=ai:tian.rui"Chen, Zhengli"https://zbmath.org/authors/?q=ai:chen.zhengli"Li, Rui"https://zbmath.org/authors/?q=ai:li.rui.1|li.rui.4|li.rui|li.rui.3|li.rui.2Summary: We study the Kraus operator decomposition corresponding to incoherent quantum operations and strictly incoherent quantum operations by means of operator algebra and matrix theory. Firstly, we give the equivalence conditions of bounded linear operators as incoherent operators and the concrete forms of incoherent operators, and give the method of characterizing incoherent quantum operations with incoherent operators. Secondly, we give the concrete forms of strongly incoherent operators and the method of characterizing strictly incoherent quantum operations with strongly incoherent operators, and give the relationship between strictly incoherent quantum operations and incoherent quantum operations.Boundedness of a class of singular integral operators and commutators on Herz-Morrey-Hardy spaces with variable exponent.https://zbmath.org/1449.420272021-01-08T12:24:00+00:00"Zhao, Huan"https://zbmath.org/authors/?q=ai:zhao.huan"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangSummary: Let \(\Omega \in {L^s} (S^{n - 1})\) for \({s \ge 1}\) be a homogeneous function of degree zero and \(b\) be BMO functions. Using the atomic decomposition theorem, we obtain the boundedness of the Calderón-Zygmund singular integral operator \({T_\Omega}\) and its commutator \([b, {T_\Omega}]\) on Herz-Morrey-Hardy spaces with variable exponent.Topological algebras of locally solid vector subspaces of order bounded operators.https://zbmath.org/1449.460422021-01-08T12:24:00+00:00"Zabeti, Omid"https://zbmath.org/authors/?q=ai:zabeti.omidSummary: Let \(E\) be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on \(E\). With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras.Positive periodic solutions of general nonlinear third-order ordinary differential equations.https://zbmath.org/1449.341192021-01-08T12:24:00+00:00"Deng, Zhengping"https://zbmath.org/authors/?q=ai:deng.zhengping"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: Using the fixed point index theory of cones, we consider the existence of positive \(2\pi\)-periodic solutions for general third-order ordinary differential equation
\[Lu (t) = f (t, u (t), u' (t), u'' (t)) (t\in\mathbb{R}),\]
where
\[Lu (t) = u''' (t) + a_2 u'' (t) + {a_1}u' (t) + {a_0}u (t)\]
is a third-order ordinary differential operator, \(f:\mathbb{R} \times [0,\infty) \times {\mathbb{R}^2} \to [0,\infty)\) is a continuous function and \(f (t, x, y, z)\) is \(2\pi\)-periodic with respect to \(t\). Under the conditions that the nonlinear term \(f\) satisfies some easily verifiable inequalities, some existence results for positive \(2\pi\)-periodic solutions of the equation are obtained that allow \(f (t, x, y, z)\) satisfies superlinear or sublinear growth with respect to \(x\), \(y\), \(z\).Anti-periodic solutions for time scale dynamic equations with exponential dichotomy.https://zbmath.org/1449.343222021-01-08T12:24:00+00:00"Meng, Xin"https://zbmath.org/authors/?q=ai:meng.xin"Lv, Xin"https://zbmath.org/authors/?q=ai:lv.xinSummary: We consider anti-periodic solutions for a class of time scale dynamic equations with exponential dichotomy. By applying the Banach fixed point theorem, we give sufficient conditions for the existence of anti-periodic solutions for nonhomogeneous linear time scale dynamic equations and semi-linear time scale dynamic equations, and give some examples to illustrate the applicability of the main results in practical problems.Consistent invertibility and the proof of Weyl's theorem.https://zbmath.org/1449.470012021-01-08T12:24:00+00:00"Liu, Ying"https://zbmath.org/authors/?q=ai:liu.ying.2|liu.ying.5|liu.ying.3|liu.ying.1|liu.ying.4|liu.ying.6|liu.ying"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: \(H\) is an infinite dimensional separable complex Hilbert space and \(B (H)\) is the algebra of all bounded linear operators on \(H\). An operator \(T \in B (H)\) is said to be ``consistent in invertibility'' provided that for each \(S \in B (H)\), \(TS\) and \(ST\) are both or neither invertible. Based on the property of consistency in invertibility, we give the necessary and sufficient conditions for \(T\) and its functional calculus satisfying the Weyl's theorem.Nonlinear maps preserving mixed Lie triple \(\xi \)-product on factor von Neumann algebras.https://zbmath.org/1449.460532021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: In this paper, we prove that every bijective map preserving mixed Lie triple \(\xi\)-products with \(\xi \ne 1\) from a factor von Neumann algebra \(\mathcal{M}\) with dim \(\mathcal{M} > 1\) into another factor von Neumann algebra \(\mathcal{N}\) with dim \(\mathcal{N} > 1\) is of the form \(A \to \varepsilon \Psi (A)\), where \(\varepsilon \in \{1, -1\}\) and \(\Psi:\mathcal{M} \to \mathcal{N}\) is a linear or conjugate linear \(*\)-isomorphism when \(\xi \in \text bf{R}\) and \(\Psi\) is a linear \(*\)-isomorphism when \(\xi \in \text bf{C}\backslash \text bf{R}\).Multiple positive solutions for a class of integral boundary value problem.https://zbmath.org/1449.340922021-01-08T12:24:00+00:00"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.1|yang.yang.5|yang.yang.4|yang.yang.3|yang.yang|yang.yang.2"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Liu, Kepan"https://zbmath.org/authors/?q=ai:liu.kepanSummary: In this paper, the existence and multiplicity of positive solutions for a class of non-resonant fourth-order integral boundary value problems
\[\begin{cases}
u^{ (4)} (t)+\beta u'' (t)-\alpha u (t) = f (t,u (t),u'' (t)), \, t \in (0,1),\\
u'' (0) = u'' (1) = 0,\\
u (0) = 0,\, u (1) = \left (\frac{1}{\lambda_2}-\frac{1}{\lambda_1}\right)\int_0^1 q (s)f (s,u (s),u'' (s))\mathrm{d}s,
\end{cases}\]
with two parameters are established by using Guo-Krasnoselskii's fixed point theorem, where \(f\in C ([0,1]\times [0,+\infty)\times [-\infty,0), [0,+\infty))\), \(q (t)\in {L^1}[0,1]\) is nonnegative, \(\alpha\), \(\beta \in \mathbb{R}\) and satisfy \(\beta< 2\pi^2\), \(\alpha > 0\), \(\alpha/\pi^4 +\beta/\pi^2 < 1\), \(\lambda_{1,2} = (-\beta\mp \sqrt{\beta^2+4\alpha})/2\). Corresponding examples illustrate the results we obtained.Coincidence point theorem and common fixed point theorem for nonself single-valued almost contractions.https://zbmath.org/1449.470962021-01-08T12:24:00+00:00"Berinde, Vasile"https://zbmath.org/authors/?q=ai:berinde.vasile"Sridarat, Phikul"https://zbmath.org/authors/?q=ai:sridarat.phikul"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthepUsing an idea given by the first author and \textit{M. Pacurar} [Fixed Point Theory 14, No. 2, 301--312 (2013; Zbl 1292.47036)], the authors prove the existence of coincidence points and common fixed points of nonself almost contractions in a nonempty closed subset of a Banach space. An example supports the theorems showing the superior generality with respect to other previous results.
Reviewer: Salvatore Sessa (Napoli)Multiplicity of positive solutions to a class of multi-point boundary value problem.https://zbmath.org/1449.340962021-01-08T12:24:00+00:00"Zhao, Bao"https://zbmath.org/authors/?q=ai:zhao.bao"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Zhou, Yonghui"https://zbmath.org/authors/?q=ai:zhou.yonghuiSummary: In this paper, the existence of multiple positive solutions to a class of nonlinear multi-point boundary value problems is established by using Guo-Krasnoselskii's fixed-point theorem. At the same time, an example is given to illustrate our conclusion.Existence of periodic solution for a kind of \( (m,n)\)-order generalized neutral differential equation.https://zbmath.org/1449.342362021-01-08T12:24:00+00:00"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, we consider the following higher-order \(p\)-Laplacian generalized neutral differential equation with variable parameter \[ (\varphi_p(x (t)-c (t)x (t-\sigma))^{(n)})^{(m)}+g (t,x (t),x (t-\tau (t)), x' (t),\cdots, x^{(m)} (t)) = e (t).\] By the coincidence degree theory, sufficient conditions for the existence of periodic solutions are established.Hypercyclic multiplication composition operators on weighted Banach space.https://zbmath.org/1449.470562021-01-08T12:24:00+00:00"Wang, Cui"https://zbmath.org/authors/?q=ai:wang.cui"Lu, Huiqiang"https://zbmath.org/authors/?q=ai:lu.huiqiangSummary: This paper characterizes some sufficient and necessary conditions for the hypercyclicity of multiple composition operators on \(H_{\log, 0}^\infty\).Global existence of mild solutions for the elastic system with structural damping.https://zbmath.org/1449.351872021-01-08T12:24:00+00:00"Shi, Wei"https://zbmath.org/authors/?q=ai:shi.weiSummary: In this paper, we study the global existence of mild solutions for the semilinear initial-value problems of second order evolution equations, which can model an elastic system with structural damping. This discussion is based on the operator semigroups theory and the Leray-Schauder fixed point theorem. In addition, an example is presented to illustrate our theoretical result.The existence of extremal solutions for fractional \(p\)-Laplacian problems with the right-handed Riemann-Liouville fractional derivative.https://zbmath.org/1449.340382021-01-08T12:24:00+00:00"Xue, Tingting"https://zbmath.org/authors/?q=ai:xue.tingting"Fan, Xiaolin"https://zbmath.org/authors/?q=ai:fan.xiaolin"Xu, Jiabo"https://zbmath.org/authors/?q=ai:xu.jiaboSummary: In this paper, we study the solvability of fractional \(p\)-Laplacian problems involving the right-hand Riemann-Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, we obtain sufficient conditions for the existence and uniqueness of extremal solutions, and extend the existing results. Finally, we provide an examples to illustrate the results.Existence of positive solutions for a class of second order periodic boundary value problems with one parameter.https://zbmath.org/1449.340802021-01-08T12:24:00+00:00"Li, Zhaoqian"https://zbmath.org/authors/?q=ai:li.zhaoqianSummary: In this paper, we consider the existence of positive solutions for the following nonlinear second-order ordinary differential equation with periodic boundary values:
\[\begin{cases}
u'' + a (t,u)u = \lambda g (t)f (u),\; t \in [0, T],\\
u (0) = u (T), u' (0) = u' (T),
\end{cases}\]
where \(\lambda\) is an positive parameter, \(a:[0, T] \times [0, \infty) \to \mathbb{R}^+\) is a \(L^p\)-Caratheodory function, \(g:[0, T] \to [0, \infty)\), \(f:[0, \infty) \to [0, \infty)\) are continuous functions. The proof of the main results is based on the fixed point index theory on cones.On the theorem for a generalized concave operator in differential equations involving a fractional order and impulsive boundary conditions.https://zbmath.org/1449.341032021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"Xiao, Weizhong"https://zbmath.org/authors/?q=ai:xiao.weizhong"Xie, Maosen"https://zbmath.org/authors/?q=ai:xie.maosenSummary: By using a fixed point theorem for a generalized concave operator, a new criterion for the existence and uniqueness of solutions involving a fractional order and impulsive boundary conditions is established. Finally, an example is given to illustrate the main results.On the essential spectrum of a model operator associated with the system of three particles on a lattice.https://zbmath.org/1449.810212021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovichSummary: A model operator \(H\) associated with the system of three-identical particles on a lattice \(\mathbb{Z}^3\) is considered. The location of the essential spectrum of \(H\) is described by the spectrum of the corresponding Friedrichs model, that is, the two-particle and three-particle branches of the essential spectrum of \(H\) are singled out. It is proved that the essential spectrum of \(H\) consists of no more than three bounded closed intervals. An appearance of two-particle branches on the both sides of the three-particle branch is shown. Moreover, we obtain an analogue of the Faddeev equation and its symmetric version, for the eigenfunctions of \(H\).Solvability for fractional \(p\)-Laplacian differential equation with integral boundary conditions at resonance on infinite interval.https://zbmath.org/1449.340262021-01-08T12:24:00+00:00"Liu, Zongbao"https://zbmath.org/authors/?q=ai:liu.zongbao"Liu, Wenbin"https://zbmath.org/authors/?q=ai:liu.wenbin.1"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.12|zhang.wei.9|zhang.wei.7|zhang.wei.3|zhang.wei.6|zhang.wei.15|zhang.wei.2|zhang.wei.18|zhang.wei.4|zhang.wei.5|zhang.wei.17|zhang.wei.11|zhang.wei.1|zhang.wei.13|zhang.wei.10|zhang.wei.14|zhang.wei.16Summary: In this paper, we investigate the existence of solutions for a class of fractional integral boundary value problems with \(p\)-Laplacian operator at resonance on infinite interval by using Mawhin's continuation theorem. An example is given to show the applicability of our main result.Explicit solution of Cauchy problem for the linearized system of phase field equations.https://zbmath.org/1449.351752021-01-08T12:24:00+00:00"Umarov, Khasan Galsanovich"https://zbmath.org/authors/?q=ai:umarov.khasan-galsanovichSummary: The explicit solution of Cauchy problem for the linearized system of phase field equations is received by reduction it to the abstract Cauchy problem in Banach space.Existence and stability of periodic solution for a Lasota-Wazewska model with discontinuous harvesting.https://zbmath.org/1449.342352021-01-08T12:24:00+00:00"Yang, Chao"https://zbmath.org/authors/?q=ai:yang.chao.3|yang.chao.2|yang.chao.1"Li, Runjie"https://zbmath.org/authors/?q=ai:li.runjieSummary: In this paper, we study a class of mixed time-varying delayed Lasota-Wazewska model with discontinuous harvesting, which is described by a periodic nonsmooth dynamical system. Based on nonsmooth analysis, Kakutani's fixed point method and the generalized Lyapunov method, easily verifiable delay-independent criteria are established to ensure the existence and exponential stability of positive periodic solutions. Finally, we give an example to further illustrate the effectiveness of our main results.Algebraic properties of Toeplitz operators on cutoff harmonic Bergman space.https://zbmath.org/1449.470662021-01-08T12:24:00+00:00"Yang, Jingyu"https://zbmath.org/authors/?q=ai:yang.jingyu"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huoSummary: In this paper, we first investigate the finite-rank product problems of several Toeplitz operators with quasihomogeneous symbols on the cutoff harmonic Bergman space \(b_n^2\). Next, we characterize the finite rank commutators and semi-commutators of two Toeplitz operators with quasihomogeneous symbols on \(b_n^2\).Global exponential periodicity of complex-valued neural networks with discontinuous activation functions.https://zbmath.org/1449.342402021-01-08T12:24:00+00:00"Zou, Yao"https://zbmath.org/authors/?q=ai:zou.yao"Zeng, Chunna"https://zbmath.org/authors/?q=ai:zeng.chunna"Hu, Jin"https://zbmath.org/authors/?q=ai:hu.jinSummary: In this paper, we investigate a type of complex-valued neural networks with discontinuous activation functions. By using Filippov differential inclusion theory, Leray-Schauder alternative theorem and Lyapunov function, we obtain the sufficient conditions for the global exponential periodicity of the neural network. The simulation shows the effectiveness of the results.A general viscosity approximation method for the implicit rule of nonexpansive mappings in Hilbert spaces.https://zbmath.org/1449.471142021-01-08T12:24:00+00:00"Tang, Tianguo"https://zbmath.org/authors/?q=ai:tang.tianguoSummary: The purpose of this paper is to introduce and study the general viscosity approximation methods for the implicit rule of nonexpansive mappings in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, a strong convergence theorem to a fixed point of the nonexpansive mapping is proved. Also, it is shown that the limit solves an additional variational inequality. The results presented in this paper are new and extend and improve some recent results.Strong \(k\)-commutativity preserving maps on prime rings with characteristic 2.https://zbmath.org/1449.160702021-01-08T12:24:00+00:00"Jia, Juan"https://zbmath.org/authors/?q=ai:jia.juan"Qi, Xiaofei"https://zbmath.org/authors/?q=ai:qi.xiaofeiSummary: Let \(\mathcal{R}\) be a unital prime ring of characteristic 2 containing a nontrivial idempotent. Assume that \(f:\mathcal{R} \to \mathcal{R}\) is a surjective map and \(k = 2,3\). Then \(f\) satisfies \([f (x), f (y)]_k = [x,y]_k = [[x,y]_{k-1}, y]\) for all \(x,y \in \mathcal{R}\) if and only if there exist a map \(\mu:\mathcal{R} \to \mathcal{C}\) and an element \(\lambda \in \mathcal{C}\) with \(\lambda^{k+1} = 1\) such that \(f (x) = \lambda x + \mu (x)\) for all \(x \in \mathcal{R}\), where \(\mathcal{C}\) is the extended centroid of \(\mathcal{R}\).Orthotropic strip with central semi-infinite crack under arbitrary loads applied far apart from the crack tip.https://zbmath.org/1449.741782021-01-08T12:24:00+00:00"Ustinov, Konstantin Borisovich"https://zbmath.org/authors/?q=ai:ustinov.konstantin-borisovich"Lisovenko, Dmitriĭ Sergeevich"https://zbmath.org/authors/?q=ai:lisovenko.dmitrii-sergeevich"Chentsov, Aleksandr Viktorovich"https://zbmath.org/authors/?q=ai:chentsov.aleksandr-viktorovichSummary: The exact analytical solution has been obtained for a problem of orthotropic strip with central semi-infinite crack loaded normally with self-balanced system of forces applied far enough from the crack tip to be considered as applied at infinity. The general solution is expressed as a superposition of solutions for two modes of loading: (i) symmetrically applied moments; (ii) symmetrically applied transverse forces with compensating moments. The exact expressions for stress intensity factor (SIF) have been obtained. Due to symmetry only the opening mode of SIF is present for each case of loading. For both cases of loading the stress states are determined by two dimensionless parameters composed by four elastic constants. Expression for SIF for the case of loading with symmetrically applied moments is obtained in terms of elementary functions and coincides with the elementary solution due to beam theory. Expression for SIF for the case of loading with symmetrically applied transverse forces with compensating moments has been obtained in terms of one function of one of the parameters expressed as a single integral, multiplied by a power function of the second parameter. The solution for this case demonstrated good agreement with the existing numerical solution for the range of parameters, for which the latter had been obtained. The obtained solution covers all possible range of parameters.New perturbation bounds for factor R of the SR factorization.https://zbmath.org/1449.150362021-01-08T12:24:00+00:00"Li, Ping"https://zbmath.org/authors/?q=ai:li.ping.1|li.ping.2|li.ping|li.ping.3|li.ping.5|li.ping.4"Yu, Xiaofei"https://zbmath.org/authors/?q=ai:yu.xiaofeiSummary: The SR factorization is a useful tool in the computation of some optimal control problems, such as algebraic Riccati equation. In this paper, combining the block matrix-vector equation approach with the technique of Lyapunov control function and the Banach fixed point principle, we obtain some new rigorous perturbation bounds and first-order perturbation bound for the factor R of the SR factorization under normwise perturbations, which improved the existing results.On \(n\)-hyponormal of weighted shifts operators.https://zbmath.org/1449.470432021-01-08T12:24:00+00:00"Dong, Yanwu"https://zbmath.org/authors/?q=ai:dong.yanwu"Zheng, Guijun"https://zbmath.org/authors/?q=ai:zheng.guijun"Li, Xiaopei"https://zbmath.org/authors/?q=ai:li.xiaopeiSummary: The \(n\)-hyponormal operators for more general sequences than Bergman weighted shifts are obtained by using the positivity of infinite dimension matrix, which extend some known results.Non-additive Lie centralizer of infinite strictly upper triangular matrices.https://zbmath.org/1449.160542021-01-08T12:24:00+00:00"Aiat Hadj, D. A."https://zbmath.org/authors/?q=ai:aiat-hadj.driss-ahmedSummary: Let \(\mathcal{F}\) be an field of zero characteristic and \(N_\infty(\mathcal{F})\) be the algebra of infinite strictly upper triangular matrices with entries in \(\mathcal{F}\), and \(f:N_\infty(\mathcal{F})\rightarrow N_\infty(\mathcal{F})\) be a non-additive Lie centralizer of \(N_\infty(\mathcal{F})\), that is, a map satisfying that \(f([X,Y])=[f(X),Y]\) for all \(X,Y\in N_\infty(\mathcal{F})\). We prove that \(f(X)=\lambda X\), where \(\lambda \in \mathcal{F}\).Path connected components in the spaces of nonzero weighted composition operators with the strong operator topology. I.https://zbmath.org/1449.470532021-01-08T12:24:00+00:00"Izuchi, Kei Ji"https://zbmath.org/authors/?q=ai:izuchi.keiji"Izuchi, Yuko"https://zbmath.org/authors/?q=ai:izuchi.yukoSummary: It is determined path connected components in the space of weighted composition operators on \(H^{\infty}\) and the disk algebra with the strong operator topology.Existence of periodic solutions of the first-Order non-Autonomous systems at resonance.https://zbmath.org/1449.341172021-01-08T12:24:00+00:00"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: This paper studies the existence of periodic solutions of the first-order non-autonomous systems at resonance, where nonlinear terms are periodic continuous functions. Several new existence results are established by means of Miranda's theorem and Schauder's fixed point theorem. Our results enrich and complement those available in the literature.Completions of quantum group algebras in certain norms and operators which commute with module actions.https://zbmath.org/1449.460602021-01-08T12:24:00+00:00"Nemati, Mehdi"https://zbmath.org/authors/?q=ai:nemati.mehdiSummary: Let \(L^1_{\text{cb}}(\mathbb{G})\) (respectively \(L^1_{\text{M}}(\mathbb{G})\)) denote the closure of the quantum group algebra \(L^1(\mathbb{G})\) of a locally compact quantum group \(\mathbb{G}\), in the space of completely bounded (respectively bounded) double centralizers of \(L^1(\mathbb{G}\)). In this paper, we study quantum group generalizations of various results from Fourier algebras of locally compact groups. In particular, left invariant means on \(L^1_{\text{cb}}(\mathbb{G})^*\) and on \(L^1_{\text{M}}(\mathbb{G})^*\) are defined and studied. We prove that the set of left invariant means on \(L^\infty(\mathbb{G})\) and on \(L^1_{\text{cb}}(\mathbb{G})^*(L^1_{\text{M}}(\mathbb{G})^*\)) have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. Finally, for a Banach \(\mathcal{A}\)-bimodule \(\mathcal{X}\) of a Banach algebra \(\mathcal{A}\) we establish a contractive and injective representation from the dual of a left introverted subspace of \(\mathcal{A}^*\) into \(B_\mathcal{A}(\mathcal{X}^*)\). As an application of this result we show that if the induced representation \(\varPhi:L\mathcal{U}C_{\text{cb}}(\mathbb{G})^*\to B_{L^1_{\text{cb}}(\mathbb{G})}(L^\infty(\mathbb{G}))\) is surjective, then \(L^1_{\text{cb}}(\mathbb{G})\) has a bounded approximate identity. We also obtain a characterization of co-amenable quantum groups in terms of representations of quantum measure algebras \(M(\mathbb{G})\).Automatic continuity of almost conjugate Jordan homomorphism on Fréchet \(Q\)-algebras.https://zbmath.org/1449.460432021-01-08T12:24:00+00:00"Omidi, Mohammad Reza"https://zbmath.org/authors/?q=ai:omidi.mohammad-rezaSummary: In this paper, the notation of almost conjugate Jordan homomorphism between Fréchet algebras is introduced. It is proven that, under special hypotheses, every almost conjugate Jordan homomorphism on Fréchet algebras is an (weakly) almost homomorphism. Also, the automatic continuity of them is generalized.Taylor spectra and common invariant subspaces through the Duggal and generalized Aluthge transforms for commuting \(n\)-tuples of operators.https://zbmath.org/1449.470192021-01-08T12:24:00+00:00"Kim, Jaewoong"https://zbmath.org/authors/?q=ai:kim.jaewoong"Yoon, Jasang"https://zbmath.org/authors/?q=ai:yoon.jasangThe authors introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties, including hyponormality and norm-continuity. Futher, they study how the Taylor spectrum and Taylor essential spectrum of 2-variable weighted shifts behave under the Duggal transforms, including generalized Aluthge transforms. Also, they investigate nontrivial common invariant subspaces between the Duggal transform and the original \(n\)-tuple of bounded operators with dense ranges.
Reviewer: Sergei S. Platonov (Petrozavodsk)Essential spectrum and Fredholm properties for operators on locally compact groups.https://zbmath.org/1449.460582021-01-08T12:24:00+00:00"Măntoiu, Marius Laurenţiu"https://zbmath.org/authors/?q=ai:mantoiu.mariusSummary: We study the essential spectrum and Fredholm properties of certain integral and pseudo-differential operators associated to non-commutative locally compact groups~$G$. The techniques involve crossed product \(C^*\)-algebras. We extend previous results on the structure of the essential spectrum to self-adjoint operators belonging (or affiliated) to the Schrödinger representation of certain crossed products. When the group $G$ is unimodular and type~I, we cover a new class of global pseudo-differential differential operators with operator-valued symbols involving the unitary dual of~$G$. We use recent results of Nistor, Prudhon and Roch on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.Hvala's theorem for generalized left \(\sigma \)-derivation on \({C^*}\)-algebras.https://zbmath.org/1449.460592021-01-08T12:24:00+00:00"Jayalakshmi, K."https://zbmath.org/authors/?q=ai:jayalakshmi.k"Bharathi, M. V. L."https://zbmath.org/authors/?q=ai:bharathi.m-v-lSummary: Generalized left \(\sigma \)-derivations in noncommutative prime \({C^*}\)-algebras are classified. If \({f_1}\), \({f_2}\) are two generalized left \(\sigma \)-derivations of a 2-torsion free prime \({C^*}\)-algebra \(A\), then the product \({f_1}{f_2}\) is again generalized left \(\sigma \)-derivation if and only if one of the following possibilities holds: there exists \(\tau \in C\) such that either \({f_1} (x) = \sigma (x)\tau \) or \({f_2} (x) = \sigma (x)\tau \), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) such that \({f_1} (x) = m\sigma (x)\) and \({f_2} (x) = n\sigma (x)\), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) such that \({f_1} (x) = \sigma (x)m\) and \({f_2} (x) = \sigma (x)n\), there exists \(m\), \(n\) in \({Q_l} ({R_c})\) and \(\lambda, \mu \in C\) such that \({f_1} (x) = \sigma (x)m + n\sigma (x)\) and \({f_2} (x) = \sigma (x)\lambda + (\sigma (x)m - n\sigma (x))\mu\). Further if \(A\) is a noncommutative 2-torsion free \({C^*}\)-algebra and \({f_1}{f_2}:A \to \mathcal{M}\) is nonzero generalized left \(\sigma \)-derivations satisfying \([{f_1} (x), {f_2} (x)] = 0\) for all \(x \in A\), then there exists \(\lambda \in C\) such that \({f_1} (x) = \lambda{f_2} (x)$, $x \in A\). Also \(f = 0\) where \(f:A \to \mathcal{M}\) provided that \(f (x)^n = 0\) for \(n > 1\) and \(A\) is \(2,\dots, n-1\)-torsion free using GPI ring theory.Three-dimensional integro-multipoint boundary value problem for loaded Volterra-hyperbolic integro-differential equations of Bianchi type.https://zbmath.org/1449.352972021-01-08T12:24:00+00:00"Mamedov, Il'gar Gurbam"https://zbmath.org/authors/?q=ai:mamedov.ilgar-gurbamSummary: In this paper the combined three-dimensional non-local boundary value problem with integro-multipoint conditions for loaded volterra-hyperbolic integro-differential equation of Bianchi type is explored. The matter of principle is the fact, that the considered equation has discontinuous coefficients which satisfy only some conditions of \(P\)-integrability type and boundedness, i.e. the considered hyperbolic differential operator has no traditional conjugate operator. In particular, for example, Riemann function under Goursat conditions for such equation cannot be constructed by classical method of characteristics.On the existence of positive solutions to a type of Riemann-Liouville fractional differential equations.https://zbmath.org/1449.340392021-01-08T12:24:00+00:00"Xue, Yimin"https://zbmath.org/authors/?q=ai:xue.yimin"Dai, Zhenxiang"https://zbmath.org/authors/?q=ai:dai.zhenxiang"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.7|liu.jie.2|liu.jie.3|liu.jie.1|liu.jie.5|liu.jie.4Summary: Using the properties of Green's function and Guo-Krasnosel'skii's fixed point theorem, the boundary value problem for the existence of positive solutions to a type of Riemann-Liouville fractional differential equations is studied: \[\begin{cases}{D^\alpha}u (t) + f (t,u (t)) = 0,\; (0 < t < 1), \\ u (0) = {D^\beta}u (0) = {D^\beta}u (1) = 0, \end{cases}\] where \(2 <\alpha \le 3\), \(1 < \beta \le 2\), \(1 + \beta \le \alpha\), \(f\in C ([0,1] \times [0,\infty), [0,\infty))\), \({D^\alpha}\) and \({D^\beta}\) are the standard Riemann-Liouville fractional derivative of order \(\alpha\) and \(\beta\), respectively. Two sufficient conditions for the existence of positive solutions are obtained, and one example is given to illustrate the applicability of the main result.Eigenvalues and virtual levels of a family of \(2\times 2\) operator matrices.https://zbmath.org/1449.810202021-01-08T12:24:00+00:00"Rasulov, Tulkin H."https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovich"Dilmurodov, Elyor B."https://zbmath.org/authors/?q=ai:dilmurodov.elyor-bThe authors consider a family of \(2\times 2\) operator matrices \(A_\mu (k)\), \(k\in \mathbb{T}^3=(-\pi ,\pi]^3\), \(\mu >0\), associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice \(\mathbb{Z}^3\), interacting via creation and annihilation operators. Spectral properties of the operators \(A_\mu (k)\) including the location of its virtual levels (resonances) are studied.
Reviewer: Anatoly N. Kochubei (Kyïv)Positive solutions for the boundary value problem of Laplacian-like equation with parameter and delay.https://zbmath.org/1449.342192021-01-08T12:24:00+00:00"Yan, Shulin"https://zbmath.org/authors/?q=ai:yan.shulin"Li, Zhiyan"https://zbmath.org/authors/?q=ai:li.zhiyanSummary: Using the fixed point theorem on cone, we study a class of Laplacian-like equation with a parameter and a delay, and obtain the existence of positive solutions for the boundary value problem.Unbounded translation invariant operators on commutative hypergroups.https://zbmath.org/1449.430062021-01-08T12:24:00+00:00"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishvesh"Kumar, N. Shiravan"https://zbmath.org/authors/?q=ai:kumar.n-shiravan"Sarma, Ritumoni"https://zbmath.org/authors/?q=ai:sarma.ritumoniLet \(K\) be a commutative hypergroup. The authors characterize translation invariant operators on \(L^1(K)\) and \(L^2(K)\) in terms of the Fourier transform. For these two cases, the space of all closed translation invariant operators forms a commutative algebra over \(\mathbb{C}\). An interpolation theorem for translation invariant operators on \(L^p(K)\), \(1\le p\le 2\), is proved.
Reviewer: Anatoly N. Kochubei (Kyïv)Some results on almost Banach-Saks operators.https://zbmath.org/1449.470762021-01-08T12:24:00+00:00"Hafidi, M."https://zbmath.org/authors/?q=ai:hafidi.m"H'Michane, J."https://zbmath.org/authors/?q=ai:hmichane.jawad"Sarih, M."https://zbmath.org/authors/?q=ai:sarih.maati|sarih.mustaphaAn operator \(T:\ E\to X\) from a Banach lattice \(E\) to a Banach space \(X\) is said to be almost Banach-Saks if, for each bounded disjoint sequence \((x_n)\) in \(E\), \((Tx_n)\) has a subsequence whose Cesàro means are norm convergent in \(X\). The authors characterize those Banach lattices for which each operator is almost Banach-Saks. The relationship with other classes of operators is studied.
Reviewer: Anatoly N. Kochubei (Kyïv)Scattering problem for Dirac system with nonlocal potentials.https://zbmath.org/1449.470252021-01-08T12:24:00+00:00"Cojuhari, P. A."https://zbmath.org/authors/?q=ai:cojuhari.petru-a"Nizhnik, L. P."https://zbmath.org/authors/?q=ai:nizhnik.leonid-pavlovichThe authors consider the problem \[ i\frac{d\psi_1 (x)}{dx}+v_1(x)\psi_+=\lambda \psi_1(x), \] \[ -i\frac{d\psi_2 (x)}{dx}+v_2(x)\psi_+=\lambda \psi_2(x), \] where \(v_1,v_2\in L_1(0,\infty)\cap L_2(0,\infty)\), \(\psi_+=\frac12 [\psi_1(0)+\psi_2(0)]\), with the boundary condition \[ \psi_1(0)-\psi_2(0)-i\int\limits_0^\infty [\psi_1(x)\overline{v_1(x)}+\psi_2(x)\overline{v_2(x)}]\,dx=0. \] An explicit expression for the scattering operator is described.
Reviewer: Anatoly N. Kochubei (Kyïv)Limited and Dunford-Pettis operators on Banach lattices.https://zbmath.org/1449.470752021-01-08T12:24:00+00:00"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"El Aloui, Abdennabi"https://zbmath.org/authors/?q=ai:el-aloui.abdennabi"Elbour, Aziz"https://zbmath.org/authors/?q=ai:elbour.azizAn operator \(T:\ X\to Y\) (\(X,Y\) are Banach spaces) is called a Dunford-Pettis operator if \(T\) carries weakly convergent sequences to norm convergent ones; \(T\) is called limited if \(T'\) carries weakly* convergent sequences in \(Y'\) to norm convergent sequences in \(X'\). The authors find conditions on a pair \(E,F\) of Banach lattices under which any positive Dunford-Pettis operator \(T:\ E\to F\) is limited. In particular, in this case the norm on \(E'\) is order continuous or \(\dim F<\infty\). Some sufficient conditions are also found.
Reviewer: Anatoly N. Kochubei (Kyïv)Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces.https://zbmath.org/1449.340172021-01-08T12:24:00+00:00"Graef, John R."https://zbmath.org/authors/?q=ai:graef.john-r"Guerraiche, Nassim"https://zbmath.org/authors/?q=ai:guerraiche.nassim"Hamani, Samira"https://zbmath.org/authors/?q=ai:hamani.samiraSummary: The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order \(\alpha\in(1, 2]\) in Banach spaces. Their approach uses Mönch's fixed point theorem and the Kuratowski measure of noncompacteness.Convergence of inexact orbits of monotone nonexpansive mappings.https://zbmath.org/1449.470902021-01-08T12:24:00+00:00"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeon"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study monotone nonexpansive self-mappings of a closed and convex cone in an ordered Banach space with particular emphasis on the asymptotic behavior of their inexact iterates.Subgradient algorithm for split hierarchical optimization problems.https://zbmath.org/1449.471112021-01-08T12:24:00+00:00"Nimana, Nimit"https://zbmath.org/authors/?q=ai:nimana.nimit"Petrot, Narin"https://zbmath.org/authors/?q=ai:petrot.narinSummary: In this paper we emphasize a split type problem of some integrating ideas of the split feasibility problem and the hierarchical optimization problem. Working on real Hilbert spaces, we propose a subgradient algorithm for approximating a solution of the introduced problem. We discuss its convergence results and present a numerical example.Fractional evolution equations with nonlocal conditions in partially ordered Banach space.https://zbmath.org/1449.351742021-01-08T12:24:00+00:00"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form \[\begin{cases} ^CD^\sigma_tx(t)+Ax(t)=f(t,x(t)),\quad t\in J:= [0,b],\\ x(0)=x_0\in X,\end{cases}\tag{A}\] where \({}^CD^\sigma_t\) denotes the Caputo fractional derivative of order \(\sigma\in(0,1),-A:D(A)\subset X\to X\) generates a positive \(C_0\)-semigroup \(T(t)(t\ge 0)\) of uniformly bounded linear operator in \(X,b >0\) is a constant, \(f\) is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone \(K\) is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.On existence of solution of a class of quadratic-integral equations using contraction defined by simulation functions and measure of noncompactness.https://zbmath.org/1449.470912021-01-08T12:24:00+00:00"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.rezaSummary: In this paper we have introduced a new type of contraction condition using a class of simulation functions, in the sequel using the new contraction definition, involving measure of noncompactness; we establish few results on existence of fixed points of continuous functions defined on a subset of Banach space. This result also generalizes other related results obtained by \textit{R. Arab} [Miskolc Math. Notes 18, No. 2, 595--610 (2017; Zbl 1399.54082)] and by \textit{J. Banaś} and \textit{K. Goebel} [Measures of noncompactness in Banach spaces. New York, Basel: Marcel Dekker, Inc. (1980; Zbl 0441.47056)]. The obtained results are used in establishing existence theorems for a class of nonlinear quadratic equations (which generalizes several types of fractional-quadratic integral equations such as Abel's integral equation) defined on a closed and bounded subset of \(\mathbb{R}\). The existence of solutions is established with the aid of a measure of noncompactness defined on the function space \(C(I)\) introduced by \textit{J. Banaś} and \textit{L. Olszowy} [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 41, 13--23 (2001; Zbl 0999.47041)].Higher order collocation methods for nonlocal problems and their asymptotic compatibility.https://zbmath.org/1449.653552021-01-08T12:24:00+00:00"Aksoylu, Burak"https://zbmath.org/authors/?q=ai:aksoylu.burak"Celiker, Fatih"https://zbmath.org/authors/?q=ai:celiker.fatih"Gazonas, George A."https://zbmath.org/authors/?q=ai:gazonas.george-aSummary: We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal diffusion are asymptotically compatible. We verify these findings through extensive numerical experiments.Commuting Toeplitz operators and Toeplitz operators with unbounded symbols on generalized Segal-Bargmann space.https://zbmath.org/1449.470642021-01-08T12:24:00+00:00"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.1"Xia, Jin"https://zbmath.org/authors/?q=ai:xia.jin"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjunSummary: We consider two Toeplitz operators \({T_u}\) and \({T_v}\) on the generalized Fock space over the complex plane \(\mathbb{C}\). Let's assume that \(u\) is a radial function and the two operators commute. Under a certain growth condition at infinity of \(u\) and \(v\), we prove that \(v\) must be a radial function as well. Finally, we also construct a \({S_p}\) class of Toeplitz operators on the generalized Fock space with symbols which are essentially unbounded on any point of the complex plane \(\mathbb{C}\).Commutative zero point \(\xi \)-Lie higher derivable maps on triangular algebras.https://zbmath.org/1449.160812021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Zhang, Haifang"https://zbmath.org/authors/?q=ai:zhang.haifang"Lu, Cuixian"https://zbmath.org/authors/?q=ai:lu.cuixianSummary: Let \(\mathcal{U}\) be a triangular algebra over a number field \(\mathcal{F}\). If \(D = \{d_k\}_{k \in \mathbb{N}}\) is a commutative zero point \(\xi\)-Lie \( (\xi \ne 1)\) higher derivable mapping from \(\mathcal{U}\) into itself with \({d_k} (1) = 0\) \((\forall k \in \mathbb{N}^+)\), then \(D\) is a higher derivation.Solving split equality common fixed point problem for infinite families of demicontractive mappings.https://zbmath.org/1449.471252021-01-08T12:24:00+00:00"Hanjing, Adisak"https://zbmath.org/authors/?q=ai:hanjing.adisak"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthepSummary: In this paper, we consider the split equality common fixed point problem of infinite families of demicontractive mappings in Hilbert spaces. We introduce a simultaneous iterative algorithm for solving the split equality common fixed point problem of infinite families of demicontractive mappings and prove strong convergence of the proposed algorithm under some control conditions.Structural and spectral properties of \(k\)-quasi class \(Q\) operators.https://zbmath.org/1449.470452021-01-08T12:24:00+00:00"Hamiti, Valdete Rexhëbeqaj"https://zbmath.org/authors/?q=ai:hamiti.valdete-rexhebeqaj"Lohaj, Shqipe"https://zbmath.org/authors/?q=ai:lohaj.shqipeSummary: An operator \(T\in\mathcal{L}(\mathcal{H})\) is said to be \(k\)-quasi class \(Q\) if \(\|T^{k+1}x\|^2\leq\frac{1}{2}(\|T^{k+2}x\|^2+\|T^kx\|^2)\), for all \(x\in\mathcal{H}\), where \(k\) is a natural number. In this paper, first we prove some results for the matrix representation of \(k\)-quasi class \(Q\) operators. Then, we give the inclusion of approximate point spectrum of \(k\)-quasi class \(Q\) operators. Also, we give the equivalence between Aluthge transformation and \(\ast\)-Aluthge transformation of \(k\)-quasi class \(Q\) operators.Weyl's theorem for class \(Q\) and \(k\)-quasi class \(Q\) operators.https://zbmath.org/1449.470462021-01-08T12:24:00+00:00"Parvatham, S."https://zbmath.org/authors/?q=ai:parvatham.s"Senthilkumar, D."https://zbmath.org/authors/?q=ai:senthilkumar.dharmapuri-vijayanSummary: In this paper, we give some properties of class \(Q\) operators. It is proved that every class \(Q\) operator satisfies Weyl's theorem under the condition that \(T^2\) is an isometry. Also we prove that every \(k\) quasi class \(Q\) operators is polaroid and the spectral mapping theorem holds for this class of operators. It will be proved that the single valued extension property, Weyl and generalized Weyl's theorem hold for every \(k\) quasi class \(Q\) operator.Wiener-Hopf operators admit triangular factorization.https://zbmath.org/1449.470582021-01-08T12:24:00+00:00"Bessonov, R. V."https://zbmath.org/authors/?q=ai:bessonov.roman-vIt is proved that a bounded positive Wiener-Hopf operator \(W_\psi\) with kernel \(\psi\) on \(L^2(\mathbb{R}_+)\) admits a factorization \(W_\psi=A^*A\) where \(A\) is a bounded invertible operator on \(L^2(\mathbb{R}_+)\) and \(AL^2[0,r]=L^2[0,r]\). The latter means that \(A\) is upper triangular with respect to the continuous chain \(L^2[0,r]\), \(r>0\). This is answering a question posed by \textit{L. A. Sakhnovich} [Ukr. Math. J. 46, No. 3, 304--317 (1994; Zbl 0848.47016); translation from Ukr. Mat. Zh. 46, No. 3, 293--304 (1994)]. Since one can consider \(\psi\) as the Fourier transform of a bounded and invertible weight \(w\) with \(0<c_1\le w(x)\le c_2\) a.e.\ on \(\mathbb{R}\), a variant of the previous result states that there is an operator \(\mathcal{F}_w:L^2(\mathbb{R}_+)\to L^2(w)\) that maps \(L^2[0,r]\) to the Paley-Wiener space \(PW_{[0,r]}\). It is yet another equivalent formulation that is the central approach of the paper. It states that under the previous conditions there exists a unique Hamiltonian \(\mathcal{H}\) on \(\mathbb{R}_+\) with spectral measure \(w\,\text{d} x\). The two previous results are easy consequences. The Hamiltonian approach is related to previous papers by the author, e.g., [Int. Math. Res. Not. 2018, No. 12, 3744--3768 (2018; Zbl 1422.42022)], where also a correction to the Sakhnovich paper was given.
Reviewer: Adhemar Bultheel (Leuven)Fixed point problems concerning contractive type operators on KST-Spaces.https://zbmath.org/1449.470942021-01-08T12:24:00+00:00"Ansari, Arslan H."https://zbmath.org/authors/?q=ai:ansari.arslan-hojat"Guran, Liliana"https://zbmath.org/authors/?q=ai:guran.liliana"Latif, Abdul"https://zbmath.org/authors/?q=ai:latif.abdulSummary: Using the concept of \(w\)-distance, we prove some results on the existence of fixed points for contractive type operators, namely, \((\alpha,\mu)\)-\(\psi\)-contractive operators. Applications are also presented. Our results improve and generalize a number of known results of fixed point theory, including the recent results of \textit{L. Guran} and \textit{M.-F. Bota} [``Ulam-Hyers stability problems and fixed point theorems concerning \(\alpha\)-\(\psi\)-type contractive operators on KST-spaces'' (submitted), see: Linear Nonlinear Anal. 5, No. 3, 379--390 (2019), \url{http://yokohamapublishers.jp/online2/oplna/vol5/p379.html}] and \textit{A. H. Ansari} and \textit{S. Shukla} [J. Adv. Math. Stud. 9, No. 1, 37--53 (2016; Zbl 1353.54030)].The iterates of positive linear operators with the set of constant functions as the fixed point set.https://zbmath.org/1449.470772021-01-08T12:24:00+00:00"Cătinaş, Teodora"https://zbmath.org/authors/?q=ai:catinas.teodora"Otrocol, Diana"https://zbmath.org/authors/?q=ai:otrocol.diana"Rus, Ioan A."https://zbmath.org/authors/?q=ai:rus.ioan-aSummary: Let \(\Omega \subset \mathbb{R}^r\), \(p \in \mathbb{N}^*\), be a nonempty subset and \(B(\Omega)\) be the Banach lattice of all bounded real functions on \(\Omega\), equipped with ``sup norm''. Let \(X \subset B(\Omega)\) be a linear sublattice of \(B(\Omega)\) and \(A:X \in X\) be a positive linear operator with the constant functions as the fixed point set. In this paper, using the weakly Picard operators technique, we study the iterates of the operator \(A\). Some relevant examples are also given.Existence of positive solutions of a class of first-order singular differential equations with nonlinear boundary conditions.https://zbmath.org/1449.340692021-01-08T12:24:00+00:00"Zhu, Yan"https://zbmath.org/authors/?q=ai:zhu.yanSummary: By using the Krasnoselskii fixed point theorem, the author proves the existence of positive solutions of the first-order differential equation with nonlinear boundary conditions: \[\begin{cases}u' (t) + a (t)u (t) = \lambda h (t)f (u (t)),\; t \in (0, 1), \\ u (0) = c (u (1))u (1),\end{cases}\] where \(\lambda\) is a positive parameter. \(a \in C ([0, 1], [0, \infty))\) and \(\int_0^1 a (t){\mathrm{d}}t > 0, h \in C ([0, 1], (0, \infty)), c \in C ([0, \infty), [1, \infty))\) and \(c < \exp \left\{\int_0^1 a (\theta){\mathrm{d}}\theta\right\}\), \(f: (0, \infty) \to \mathbb{R}\) is continuous, superlinear at \(\infty\) and is allowed to be singular at 0.The defect spectrum of bounded upper triangular operator matrices.https://zbmath.org/1449.470102021-01-08T12:24:00+00:00"Liu, Aichun"https://zbmath.org/authors/?q=ai:liu.aichunSummary: Perturbation properties of bounded upper triangular operator matrices on Hilbert space are investigated in this paper. A sufficient and necessary condition of the equality between the defect spectrum of a bounded upper triangular operator matrix and the union of defect spectra of the diagonal operators of the bounded upper triangular operator matrix is presented under the condition that the diagonal operator is given. The condition for the case of bounded upper triangular Hamilton-type operator is given as an application, and an example is given to verify the validity.Existence of solutions of implicit integral equations via \(Z\)-contraction.https://zbmath.org/1449.540892021-01-08T12:24:00+00:00"Patle, Pradip R."https://zbmath.org/authors/?q=ai:patle.pradip-ramesh"Patel, Deepesh Kumar"https://zbmath.org/authors/?q=ai:patel.deepesh-kumarSummary: The main focus of this work is to assure that the sum of a compact operator with a \(Z\)-contraction admits a fixed point. The concept of condensing mapping (in the sense of Hausdorff non-compactness measure) is used to establish the concerned result which generalizes some of the existing state-of-art in the literature. Presented result is used to verify the actuality of solutions of implicit integral equations.A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces.https://zbmath.org/1449.470202021-01-08T12:24:00+00:00"Chen, Cui"https://zbmath.org/authors/?q=ai:chen.cui"Zhou, Ze-Hua"https://zbmath.org/authors/?q=ai:zhou.zehua"Wang, Ya"https://zbmath.org/authors/?q=ai:wang.yaLet \(H\) be a separable infinite-dimensional Hilbert space and \(B(H)\) be the set of all bounded linear operators on. In the paper under review, the authors show that, if \(M\) is an almost-invariant subspace for \(T \in B(H)\), then every maximal almost-invariant subspace of \(M\) is of codimension \(1\) in \(M\). They describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most \(1\) on \(H\). They show that, for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension \(1\) in it.
Reviewer: Ömer Gök (Istanbul)On the Cauchy dual of closed range operators.https://zbmath.org/1449.470442021-01-08T12:24:00+00:00"Ezzahraoui, H."https://zbmath.org/authors/?q=ai:ezzahraoui.hamid"Mbekhta, M."https://zbmath.org/authors/?q=ai:mbekhta.mostafa"Zerouali, E. H."https://zbmath.org/authors/?q=ai:zerouali.el-hassanSummary: We extend in this paper the notion of Cauchy dual to operators with closed range. We then give several useful properties of Cauchy duals extending the case of left-invertible operators. As a consequence, we show that a weak concavity concept of an operator induces a corresponding weak hyponormality of its Cauchy dual.Some coincidence point theorems in ordered metric spaces via \(w\)-distances.https://zbmath.org/1449.540812021-01-08T12:24:00+00:00"Mongkolkeha, Chirasak"https://zbmath.org/authors/?q=ai:mongkolkeha.chirasak"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: The purpose of this paper is to prove some existence theorems of coincidence points for generalized weak contractions in the setting of partially ordered sets with a metric via \(w\)-distances and give some example to illustrate our main results.Characterizations of inner spaces under strongly \(E\)-convex set-valued mappings.https://zbmath.org/1449.470872021-01-08T12:24:00+00:00"Li, Ru"https://zbmath.org/authors/?q=ai:li.ru"Yu, Guolin"https://zbmath.org/authors/?q=ai:yu.guolin"Kong, Xiangyu"https://zbmath.org/authors/?q=ai:kong.xiangyuSummary: In this note, a kind of generalized strongly convex set-valued mappings, termed strongly \(E\)-convex set-valued mappings, is introduced in real normed spaces. Then, by employing Rådström cancellation law, some basic properties of strongly \(E\)-convex set-valued mappings are proposed. Finally, a characterization of inner product spaces involving the strongly \(E\)-convex set-valued mapping is presented.Application of DJ method to Itô stochastic differential equations.https://zbmath.org/1449.600992021-01-08T12:24:00+00:00"Deilami Azodi, H."https://zbmath.org/authors/?q=ai:deilami-azodi.hSummary: This paper develops iterative method described by \textit{V. Daftardar-Gejji} and \textit{H. Jafari} [J. Math. Anal. Appl. 316, No. 2, 753--763 (2006; Zbl 1087.65055)] to solve Itô stochastic differential equations. The convergence of the method for Itô stochastic differential equations is assessed. To verify efficiency of method, some examples are expressed.Multiple positive solutions to a \((2m)\)th-order boundary value problem.https://zbmath.org/1449.340702021-01-08T12:24:00+00:00"Boulaiki, Habiba"https://zbmath.org/authors/?q=ai:boulaiki.habiba"Moussaoui, Toufik"https://zbmath.org/authors/?q=ai:moussaoui.toufik"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.raduSummary: The aim of the present paper is to study the existence, localization and multiplicity of positive solutions for a \((2m)\)th-order boundary value problem subject to the Dirichlet conditions. Our approach is based on critical point theory in conical shells and Harnack type inequalities.Pseudo almost automorphic solutions of hematopoiesis model with mixed delays.https://zbmath.org/1449.342842021-01-08T12:24:00+00:00"Aouiti, Chaouki"https://zbmath.org/authors/?q=ai:aouiti.chaouki"Dridi, Farah"https://zbmath.org/authors/?q=ai:dridi.farah"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchaoSummary: This paper is concerned with a hematopoiesis model with mixed delays. Under new conditions, we study the existence, uniqueness and global exponential stability of pseudo almost automorphic solutions for the suggested model. Our approach is mainly based on the exponential dichotomy of linear differential equation, Banach's fixed-point principle and suitable Lyapunov functional. At the end, some numerical examples are presented to demonstrate the effectiveness of our findings.Triple positive solutions for a third-order three-point boundary value problem.https://zbmath.org/1449.340912021-01-08T12:24:00+00:00"Wu, Hongping"https://zbmath.org/authors/?q=ai:wu.hongpingSummary: In this paper, we study the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem \[\begin{cases}u''' (t) = -h (t)f (t,u (t),u' (t),u'' (t)),\; 0< t < 1, \\ u (0) = u' (1) = u'' (\eta) = 0,\end{cases}\] where \(\eta\in [0,\frac{1}{2})\) is a constant. By using a fixed-point theorem, we obtain the triple positive solutions to the boundary value problem, and an example is given to illustrate the importance of the result we obtained.Mixed-type reverse order laws associated to \(\{1, 3, 4\}\)-inverse.https://zbmath.org/1449.470072021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Deng, Chunyuan"https://zbmath.org/authors/?q=ai:deng.chunyuanSummary: In this paper, we study the mixed-type reverse order laws to \(\{1, 3, 4\}\)-inverses for closed range operators \(A, B\) and \(AB\). It is shown that \(B\{1, 3, 4\}A\{1, 3, 4\} \subseteq (AB)\{1, 3\}\) if and only if \(R({A^*}AB) \subseteq R (B)\). For every \(A^{(134)} \in A (1, 3, 4)\), we have \( (A^{(134)}AB)\{1, 3, 4\}A\{1, 3, 4\} = (AB)\{1, 3, 4\}\) if and only if \(R (A{A^*}AB) \subseteq R (AB)\). As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the \(\{1, 3, 4\}\)-inverse are established.Essential and Weyl spectra of \(2 \times 2\) bounded block operator matrices.https://zbmath.org/1449.470092021-01-08T12:24:00+00:00"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.1|li.lin.2|li.lin"Alatancang"https://zbmath.org/authors/?q=ai:alatancang.chen|chen.alatancangSummary: This paper is concerned with the necessary and sufficient conditions under which a class of bounded \(2 \times 2\) block operator matrices are Fredholm operators or Weyl operators. Some necessary and sufficient conditions are given under which the essential spectrum and the Weyl spectrum of the block operator matrix coincide with the essential spectrum and the Weyl spectrum of its entries.Suzuki \(\phi F\)-contractions and some fixed point results.https://zbmath.org/1449.470932021-01-08T12:24:00+00:00"Secelean, Nicolae-Adrian"https://zbmath.org/authors/?q=ai:secelean.nicolae-adrianSummary: The purpose of this paper is to combine and extend some recent fixed point results of \textit{T. Suzuki} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, 5313--5317 (2009; Zbl 1179.54071)] and \textit{N.-A. Secelean} and \textit{D. Wardowski} [Result. Math. 70, No. 3--4, 415--431 (2016; Zbl 1442.54048)]. The continuity and the completeness conditions are replaced by orbital continuity and orbital completeness, respectively. It is given an illustrative example of a Picard operator on a noncomplete metric space which is neither nonexpansive nor expansive and has a unique continuity point.On convergence of the iterative process for the third order pseudo-parabolic equation with nonlocal boundary value conditions in a multidimensional domain.https://zbmath.org/1449.354712021-01-08T12:24:00+00:00"Beshtokov, Murat Khamidbievich"https://zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievichSummary: In this paper the nonlocal boundary value problem for the pseudo-parabolic equation of the third-order in a multidimensional domain is considered. Using an iterative method, the solving process of the nonlocal boundary value problem is reduced to solving the series of some local problems. An a priori estimate for the convergence of the iterative method in the norm \(W^1_2(G)\) is obtained.Existence of a unique positive solution for a singular fractional boundary value problem.https://zbmath.org/1449.340762021-01-08T12:24:00+00:00"Karimov, E. T."https://zbmath.org/authors/?q=ai:karimov.erkinjon-tulkinovich"Sadarangani, K."https://zbmath.org/authors/?q=ai:sadarangani.kishin-bSummary: In the present work, we discuss the existence of a unique positive solution of a boundary value problem for a nonlinear fractional order equation with singularity. Precisely, order of equation
\[D^\alpha_0 u+u(t)= f(t,u(t))\]
belongs to \((3,4]\) and \(f\) has a singularity at \(t= 0\) and as boundary conditions we use \(u(0) =u(1) =u'(0) =u'(1) = 0\). Using a fixed point theorem, we prove the existence of unique positive solution of the considered problem.Some observations on generalized non-expansive mappings with an application.https://zbmath.org/1449.471222021-01-08T12:24:00+00:00"Ali, Faeem"https://zbmath.org/authors/?q=ai:ali.faeem"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: This study aimed at showing that the classes of generalized non-expansive mappings due to Hardy and Rogers and the mappings satisfying Suzuki's condition \((C)\) are independent and study some basic properties of generalized non-expansive mappings. Also, we introduce a new iterative scheme, called JF iterative scheme, and prove convergence results for generalized non-expansive mappings due to Hardy and Rogers in uniformly convex Banach spaces. Moreover, we show numerically that JF iterative scheme converges to a fixed point of generalized non-expansive mappings faster than some known and leading iterative schemes. As an application, we utilize newly defined iterative scheme to approximate the solution of a delay differential equation. Also, we present some nontrivial illustrative numerical examples to support main results. Our results are new and extend several relevant results in the existing literature.An iterative process for a hybrid pair of generalized \(I\)-asymptotically nonexpansive single-valued mappings and generalized nonexpansive multi-valued mappings in Banach spaces.https://zbmath.org/1449.471072021-01-08T12:24:00+00:00"Farajzadeh, Ali"https://zbmath.org/authors/?q=ai:farajzadeh.ali-p"Chuasuk, Preeyanuch"https://zbmath.org/authors/?q=ai:chuasuk.preeyanuch"Kaewcharoen, Anchalee"https://zbmath.org/authors/?q=ai:kaewcharoen.anchalee"Mursaleen, Mohammad"https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-aymanSummary: In this paper, an iterative process for a hybrid pair of a finite family of generalized \(I\)-asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multivalued mappings is established. Moreover, weak convergence theorems and strong convergence theorems of the proposed iterative process in Banach spaces are proven. The examples are established for supporting our main results. The obtained results can be viewed as an improvement and extension of several results in the literature.On some parameters in the space of regulated functions and their applications.https://zbmath.org/1449.260032021-01-08T12:24:00+00:00"Cichoń, Kinga"https://zbmath.org/authors/?q=ai:cichon.kinga"Cichoń, Mieczysław"https://zbmath.org/authors/?q=ai:cichon.mieczyslaw"Metwali, Mohamed M. A."https://zbmath.org/authors/?q=ai:metwali.mohamed-m-aSummary: In this paper, we study a class of discontinuous functions being a space of solutions for some differential and integral equations. We investigate functions having finite one-sided limits, i.e. regulated functions. In the space of such functions, we introduce some new concepts like a modulus of equi-regularity or a measure of noncompactness, allowing us to unify the proofs for the results about existence for both continuous and discontinuous solutions. An example of applications for quadratic integral equations, essentially improving earlier ones, completes the paper.On generalized Lie derivations.https://zbmath.org/1449.160772021-01-08T12:24:00+00:00"Bennis, Driss"https://zbmath.org/authors/?q=ai:bennis.driss"Vishki, Hamid Reza Ebrahimi"https://zbmath.org/authors/?q=ai:vishki.hamid-reza-ebrahimi"Fahid, Brahim"https://zbmath.org/authors/?q=ai:fahid.brahim"Bahmani, Mohammad Ali"https://zbmath.org/authors/?q=ai:bahmani.mohammad-aliSummary: In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by \textit{D. Benkovič} [Linear Algebra Appl. 434, No. 6, 1532--1544 (2011; Zbl 1216.16032)]. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras.On power finite rank operators.https://zbmath.org/1449.470062021-01-08T12:24:00+00:00"Zeng, Qingping"https://zbmath.org/authors/?q=ai:zeng.qingping"Wu, Zhenying"https://zbmath.org/authors/?q=ai:wu.zhenyingSummary: An operator \(F \in \mathcal{B} (X)\) is called power finite rank if \({F^n}\) is of finite rank for some \(n \in \mathbb{N}\). In this note, we provide several interesting characterizations of power finite rank operators. In particular, we show that the class of power finite rank operators is the intersection of the class of Riesz operators and the class of operators with eventual topological uniform descent.Asymptotic stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion.https://zbmath.org/1449.342792021-01-08T12:24:00+00:00"Cui, Jing"https://zbmath.org/authors/?q=ai:cui.jing"Liang, Qiuju"https://zbmath.org/authors/?q=ai:liang.qiuju"Bi, Nana"https://zbmath.org/authors/?q=ai:bi.nanaSummary: In this paper, we consider the asymptotic stability in the \(p\)-th moment of mild solutions of impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in a real separable Hilbert space. A fixed point approach is used to achieve the required result. A practical example is provided to illustrate the viability of the abstract result of this work.Angular derivatives and compactness of composition operators on Hardy spaces.https://zbmath.org/1449.470512021-01-08T12:24:00+00:00"Betsakos, Dimitrios"https://zbmath.org/authors/?q=ai:betsakos.dimitriosLet \(D_0\) be a simply connected domain included in the unit disc \(\mathbb{D}\) of the complex plane. Let \(D \subset D_0\) be a domain such that \(D_0 \setminus D\) is a compact subset of \(D_0\). Let \(\phi\) be a universal covering map of \(\mathbb{D}\) onto \(D\) and let \(\psi\) be a Riemann map of \(\mathbb{D}\) onto \(D_0\). The author proves that the following statements are equivalent: (1) the composition operator \(C_{\phi}\) is compact on the Hardy space \(H^p\), \(0<p \leq \infty\), (2) the composition operator \(C_{\psi}\) is compact on the Hardy space \(H^p\), \(0<p \leq \infty\), (3) \(\phi\) does not have an angular derivative at any point of the unit circle, (4) \(\psi\) does not have an angular derivative at any point of the unit circle. This result improves recent work by \textit{M. M. Jones} [J. Funct. Anal. 268, No. 4, 887--901 (2015; Zbl 1308.47030); Ill. J. Math. 59, No. 3, 707--715 (2015; Zbl 1353.47052)]. Different tools, such as Green functions, subordination, and prime ends, are used to prove the stronger result.
Reviewer: José Bonet (Valencia)Generalized spectra of convolution operators.https://zbmath.org/1449.460412021-01-08T12:24:00+00:00"Kumar, G. Krishna"https://zbmath.org/authors/?q=ai:kumar.g-krishna"Kulkarni, S. H."https://zbmath.org/authors/?q=ai:kulkarni.s-hSummary: The article introduces an algebra of integral operators namely convolution operators. The spectra, pseudospectra and condition spectra of convolution operators are described using Banach algebra techniques and the results developed are illustrated with examples and figures.Composition operator on the weighted analytic Lipschitz space in \(\mathbb{C}^n\).https://zbmath.org/1449.470572021-01-08T12:24:00+00:00"Yuan, Qianqian"https://zbmath.org/authors/?q=ai:yuan.qianqian"Lu, Zhenguo"https://zbmath.org/authors/?q=ai:lu.zhenguo"Du, Jinji"https://zbmath.org/authors/?q=ai:du.jinji"Qin, Chuangliang"https://zbmath.org/authors/?q=ai:qin.chuangliangSummary: The characterization of the weighted analytic Lipschitz space in \(\mathbb{C}^n\) is given, and the boundedness and compactness of the composition operator \({C_\varphi}\) in this space are obtained. Lastly, the sufficient and necessary conditions about them are proved.Standard versus strict bounded real lemma with infinite-dimensional state space. I: The state-space-similarity approach.https://zbmath.org/1449.470342021-01-08T12:24:00+00:00"Ball, Joseph A."https://zbmath.org/authors/?q=ai:ball.joseph-a"Groenewald, Gilbert J."https://zbmath.org/authors/?q=ai:groenewald.gilbert-j"ter Horst, Sanne"https://zbmath.org/authors/?q=ai:ter-horst.sanneSummary: The bounded real lemma, i.e., the state-space linear matrix inequality characterization (referred to as Kalman-Yakubovich-Popov or KYP-inequality) of when an input/state/output linear system satisfies a dissipation inequality, has recently been studied for infinite-dimensional discrete-time systems in a number of different settings: with or without stability assumptions, with or without controllability/observability assumptions, with or without strict inequalities. In these various settings, sometimes unbounded solutions of the KYP-inequality are required while in other instances bounded solutions suffice. In a series of reports, we show how these diverse results can be reconciled and unified. This first instalment focusses on the state-space-similarity approach to the bounded real lemma. We shall show how these results can be seen as corollaries of a new state-space-similarity theorem for infinite-dimensional linear systems.Random semilinear system of differential equations with state-dependent delay.https://zbmath.org/1449.342782021-01-08T12:24:00+00:00"Blouhi, Tayeb"https://zbmath.org/authors/?q=ai:blouhi.tayeb"Ferhat, Mohamed"https://zbmath.org/authors/?q=ai:ferhat.mohamedSummary: In this paper, we prove the existence of mild solutions for a first-order semilinear differential equation with state-dependent delay. The existence results are established by means of a new version of Perov's fixed point principles.Generalized Kato decomposition and Weyl type theorems.https://zbmath.org/1449.470262021-01-08T12:24:00+00:00"Chen, Lihong"https://zbmath.org/authors/?q=ai:chen.lihong"Su, Weigang"https://zbmath.org/authors/?q=ai:su.weigangSummary: Using the character of generalized Kato decomposition, this paper discusses the sufficient and necessary conditions under which Browder's theorem and Weyl's theorem hold from the view of generalized Kato spectrum for a bounded linear operator.Controllability for impulsive fractional evolution inclusions with state-dependent delay.https://zbmath.org/1449.342692021-01-08T12:24:00+00:00"Aissani, Khalida"https://zbmath.org/authors/?q=ai:aissani.khalida"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, sufficient conditions are provided for the controllability of impulsive fractional evolution inclusions with state-dependent delay in Banach spaces. We used a fixed-point theorem for condensing maps due to Bohnenblust-Karlin and the theory of semigroup for the achievement of the results. An illustrative example is presented.Taylor asymptotics of spectral action functionals.https://zbmath.org/1449.470322021-01-08T12:24:00+00:00"Skripka, Anna"https://zbmath.org/authors/?q=ai:skripka.annaSummary: We establish a Taylor asymptotic expansion of the spectral action functional on self-adjoint operators \(V \mapsto \tau(f(H+V))\) with remainder \(\mathcal{O}(\Vert f^{(n)}\Vert_\infty\Vert V\Vert^n)\) and derive an explicit representation for the remainder in terms of spectral shift functions. For this expansion, we assume only that \(H\) has \(\tau\)-compact resolvent and \(V\) is a bounded perturbation; in particular, neither summability of \(V\) nor of the resolvent of \(H\) is required.Existence of solutions for a new class of fuzzy differential inclusions with resolvent operators in Banach spaces.https://zbmath.org/1449.340072021-01-08T12:24:00+00:00"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Vo Minh Tam"https://zbmath.org/authors/?q=ai:vo-minh-tam."O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, a new class of fuzzy differential inclusions with resolvent operators in Banach spaces using \((H(\cdot ,\cdot ),\eta)\)-monotone operators is introduced and studied. A continuous selection theorem and fixed point theory are used to establish the existence of solutions. Finally, as applications, we consider special cases of fuzzy differential inclusions with general \(A\)-monotone operators. Some examples are given to illustrate our results.