Recent zbMATH articles in MSC 39https://zbmath.org/atom/cc/392022-11-17T18:59:28.764376ZWerkzeugTweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéryhttps://zbmath.org/1496.110972022-11-17T18:59:28.764376Z"Dougherty-Bliss, Robert"https://zbmath.org/authors/?q=ai:dougherty-bliss.robert"Koutschan, Christoph"https://zbmath.org/authors/?q=ai:koutschan.christoph"Zeilberger, Doron"https://zbmath.org/authors/?q=ai:zeilberger.doronThe authors present a general strategy of a computer assisted proof of irrationality of numbers in the form \[C=\int_0^1 K(x)\,\mathrm dx\] for some function~\(K\). For another function~\(S\) the authors introduce the sequence of numbers \[I(n)=\int_0^1 K(x)\bigl(x(1-x)S(x)\bigr)^n\,\mathrm dx\] (Similarly for multidimensional integrals.) In some cases it might happen that there are polynomials~\(p_i(n)\) such that the sequence~\(I(n)\) satisfies the recurrence relation \[\sum_{i=0}^L p_i(n)I(n+i)=0.\] It can then happen that \[I(n)=b_n C - a_n\] for some sequences of rational numbers~\(a_n\), \(b_n\) with a suitable asymptotic behaviour which then implies that the number~\(C\) is irrational.
In this paper, the authors generate examples of some numbers such that the above mentioned cases occur. For instance they prove the irrationality of \(\frac{\sqrt{\pi}\Gamma(7/3)}{\Gamma(-1/6)}\) or \(\frac{\sqrt{\pi}\Gamma(8/3)}{\Gamma(19/6)}\).
Reviewer: Jan Šustek (Ostrava)Positivity of integrals for higher order \(\nabla\)-convex and completely monotonic functionshttps://zbmath.org/1496.260112022-11-17T18:59:28.764376Z"Mehmood, Faraz"https://zbmath.org/authors/?q=ai:mehmood.faraz"Khan, Asif Raza"https://zbmath.org/authors/?q=ai:khan.asif-raza"Adnan, Muhammad"https://zbmath.org/authors/?q=ai:adnan.muhammad-abdullahSummary: We extend the definitions of \(\nabla\)-convex and completely monotonic functions for two variables. Some general identities of Popoviciu type integrals \(\int P(y)f(y) dy\) and \(\int \int P(y,z) f(y,z) dy dz\) are deduced. Using obtained identities, positivity of these expressions are characterized for higher order \(\nabla\)-convex and completely monotonic functions. Some applications in terms of generalized Cauchy means and exponential convexity are given.Constructing discrete harmonic functions in wedgeshttps://zbmath.org/1496.310042022-11-17T18:59:28.764376Z"Hoang, Viet Hung"https://zbmath.org/authors/?q=ai:hoang.viet-hung"Raschel, Kilian"https://zbmath.org/authors/?q=ai:raschel.kilian"Tarrago, Pierre"https://zbmath.org/authors/?q=ai:tarrago.pierreThis article proposes a systematic method for the construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, the authors prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.
Reviewer: Marius Ghergu (Dublin)Sums of two-parameter deformations of multiple polylogarithmshttps://zbmath.org/1496.330012022-11-17T18:59:28.764376Z"Kato, Masaki"https://zbmath.org/authors/?q=ai:kato.masakiSummary: We introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by \(\Phi_2(a;p,q)\), and study a \(q\)-difference equation satisfied by it. We show that this \(q\)-difference equation can be solved by expanding \(\Phi_2(a;p,q)\) into power series of the parameter \(p\) and then using the method of variation of constants. By letting \(p \rightarrow 0\) in the main theorem, we find that the generating function of sums of \(q\)-interpolated multiple zeta values can be written in terms of the \(q\)-hypergeometric function \(_3 \phi_2\), which is due to Li-Wakabayashi.Comment for ``Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel''https://zbmath.org/1496.340162022-11-17T18:59:28.764376Z"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyanSummary: In a published paper [\textit{A. Khan} et al., Chaos Solitons Fractals 127, 422--427 (2019; Zbl 1448.34046)], some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green's function of the fractional differential equation studied in this paper; The others are for the properties for Green's function, these miss prints affected the deriving of the main results in Khan's paper, some corrections for Khan's paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.The solution theory for the fractional hybrid \(q\)-difference equationshttps://zbmath.org/1496.340172022-11-17T18:59:28.764376Z"Ma, Kuikui"https://zbmath.org/authors/?q=ai:ma.kuikui"Gao, Lei"https://zbmath.org/authors/?q=ai:gao.lei(no abstract)Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearitieshttps://zbmath.org/1496.340292022-11-17T18:59:28.764376Z"Kuang, Juhong"https://zbmath.org/authors/?q=ai:kuang.juhong"Chen, Weiyi"https://zbmath.org/authors/?q=ai:chen.weiyi(no abstract)Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite latticeshttps://zbmath.org/1496.351252022-11-17T18:59:28.764376Z"Li, Congcong"https://zbmath.org/authors/?q=ai:li.congcong"Li, Chunqiu"https://zbmath.org/authors/?q=ai:li.chunqiu"Wang, Jintao"https://zbmath.org/authors/?q=ai:wang.jintaoSummary: In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-\(\mathcal{D}\) attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-\(\mathcal{D}\) attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.Failure of Fatou type theorems for solutions to PDE of \(p\)-Laplace type in domains with flat boundarieshttps://zbmath.org/1496.352142022-11-17T18:59:28.764376Z"Akman, Murat"https://zbmath.org/authors/?q=ai:akman.murat"Lewis, John"https://zbmath.org/authors/?q=ai:lewis.john-l"Vogel, Andrew"https://zbmath.org/authors/?q=ai:vogel.andrew-lIn [\textit{T. H. Wolff}, J. Anal. Math. 102, 371--394 (2007; Zbl 1213.35218)] (a posthumous publication of an unpublished result from 1984), highly oscillatory bounded solutions of \(div(\nabla u|\nabla u|^{p-2})=0\) in the semi-plane \(\mathbb R^2_+\) were constructed for \(p>2\) and Fatou's theorem was shown to fail for these solutions.
In this paper an analogue of the above result is obtained in domains of the form \(\mathbb R^n\setminus \Lambda_k\), where \(\Lambda_k\) is a \(k\)-dimensional subspace (\(1\leq k<n-1\)).
The same problem is also addressed for solutions to a more general class of PDE's modeled on the \(p\)-Laplacian, called \(\mathcal A\)-harmonic functions, but the obtained results are not as complete as in the \(p\)-harmonic case.
Reviewer: Eugenio Massa (São Carlos)Eigenfunctions of a discrete elliptic integrable particle model with hyperoctahedral symmetryhttps://zbmath.org/1496.370612022-11-17T18:59:28.764376Z"van Diejen, Jan Felipe"https://zbmath.org/authors/?q=ai:van-diejen.jan-felipe"Görbe, Tamás"https://zbmath.org/authors/?q=ai:gorbe.tamas-fThe main purpose of the authors is to carry out a finite-dimensional reduction of the eigenvalue problem for a second-order difference operator describing the quantum Hamiltonian of an elliptic Ruijsenaars type \(n\)-particle model on the circle with hyperoctahedral symmetry.
Reviewer: Mohammed El Aïdi (Bogotá)Ground states for infinite lattices with nearest neighbor interactionhttps://zbmath.org/1496.370782022-11-17T18:59:28.764376Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Hu, Die"https://zbmath.org/authors/?q=ai:hu.die"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan.1Summary: \textit{J. Sun} and \textit{S. Ma} [J. Differ. Equations 255, No. 8, 2534--2563 (2013; Zbl 1318.37026)]
proved the existence of a nonzero \(T\)-periodic solution for a class of one-dimensional lattice dynamical systems,
\[
\ddot{q_i}=\varPhi_{i-1}^\prime(q_{i-1}-q_i)- \varPhi_i^\prime(q_i-q_{i+1}),\quad i\in \mathbb{Z},
\] where \(q_i\) denotes the co-ordinate of the \(i\)th particle and \(\varPhi_i\) denotes the potential of the interaction between the \(i\)th and the \((i+1)\)th particle. We extend their results to the case of the least energy of nonzero \(T\)-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.Monotonic solutions of second order nonlinear difference equationshttps://zbmath.org/1496.390012022-11-17T18:59:28.764376Z"Wang, Lianwen"https://zbmath.org/authors/?q=ai:wang.lianwen"Willett, Anthony C."https://zbmath.org/authors/?q=ai:willett.anthony-cThe paper is focused on the second-order nonlinear difference equation
\[
\Delta(a_nf(\Delta x_n))=b_ng(x_{n+1}),\quad n\ge 1,
\]
where \(\Delta\) is the forward difference operator \(\Delta x_n=x_{n+1}-x_n\), \((a_n)_{n\ge 1}\) and \((b_n)_{n\ge 1}\) are positive real sequences, \(g:\mathbb{R}\to\mathbb{R}\) is an increasing continuous function with \(rg(r)>0\) for \(r\not=0\), \(f:\mathbb{R}\to\mathbb{R}\) is a strictly increasing continuous function with \(rf(r)>0\) for \(r\not=0\) satisfying some additional assumptions. The authors study the monotonicity and the classification of its solutions. Then they present necessary and sufficient conditions for the boundedness of all solutions. The existence of different types of monotonic solutions is finally investigated.
Reviewer: Rodica Luca (Iaşi)Global behavior of two third order rational difference equations with quadratic termshttps://zbmath.org/1496.390022022-11-17T18:59:28.764376Z"Abo-Zeid, R."https://zbmath.org/authors/?q=ai:abo-zeid.raafatSummary: In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations
\[ x_{n+1}= \frac{ax_nx_{n-1}}{bx_{n-1}+cx_{n-2}},\quad n=0,1,\ldots . \]
where \(a,b,c\) are positive real numbers and the initial conditions \(x_{-2},x_{-1},x_0\) are real numbers.Finite-time stability of ABC type fractional delay difference equationshttps://zbmath.org/1496.390032022-11-17T18:59:28.764376Z"Chen, Yuting"https://zbmath.org/authors/?q=ai:chen.yuting"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyan"Liu, Song"https://zbmath.org/authors/?q=ai:liu.songSummary: In this paper, finite-time stability of fractional delay difference equations with discrete Mittag-Leffler kernel are studied. Firstly, we establish a new generalized Gronwall inequality in sense of Atangana-Baleanu fractional difference sum operator. Then, based on this new generalized Gronwall inequality and the method of steps, finite-time stability criteria of fractional delay difference equations with discrete Mittag-Leffler kernel are induced respectively. Finally, examples are presented to illustrate the validity of main results.\(q\)-Hamiltonian systemshttps://zbmath.org/1496.390042022-11-17T18:59:28.764376Z"Paşaoğlu, Bilender"https://zbmath.org/authors/?q=ai:pasaoglu.bilender-p"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyinSummary: In this paper, we develop the basic theory of linear \(q\)-Hamiltonian systems. In this context, we establish an existence and uniqueness result. Regular spectral problems are studied. Later, we introduce the corresponding maximal and minimal operators for this system. Finally, we give a spectral resolution.New generalization involving convex functions via \(\hbar\)-discrete \(\mathcal{AB}\)-fractional sums and their applications in fractional difference equationshttps://zbmath.org/1496.390052022-11-17T18:59:28.764376Z"Rashid, Saima"https://zbmath.org/authors/?q=ai:rashid.saima"Khalid, Aasma"https://zbmath.org/authors/?q=ai:khalid.aasma"Karaca, Yeliz"https://zbmath.org/authors/?q=ai:karaca.yeliz"Hammouch, Zakia"https://zbmath.org/authors/?q=ai:hammouch.zakia"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yumingLink theorem and distributions of solutions to uncertain Liouville-Caputo difference equationshttps://zbmath.org/1496.390062022-11-17T18:59:28.764376Z"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Mohammed, Pshtiwan Othman"https://zbmath.org/authors/?q=ai:mohammed.pshtiwan-othman"Guirao, Juan L. G."https://zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Hamed, Y. S."https://zbmath.org/authors/?q=ai:hamed.yasser-sSummary: We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their \(\varrho\)-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its \(\varrho\)-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their \(\varrho\)-paths.On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stabilityhttps://zbmath.org/1496.390072022-11-17T18:59:28.764376Z"Nesterov, Pavel"https://zbmath.org/authors/?q=ai:nesterov.pavel-nikolaevichSummary: We construct the asymptotics for solutions of the higher-order scalar difference equation that is equivalent to the linear delay difference equation \(\Delta y(n) = -g(n)y(n-k)\). We assume that the coefficient of this equation oscillates at the certain level and the oscillation amplitude decreases as \(n\to\infty\). Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotic formulae. The obtained results are applied to the oscillation and stability problems for the solutions of the considered equation.Necessary and sufficient conditions for oscillation of nonlinear neutral difference systems of dim-2https://zbmath.org/1496.390082022-11-17T18:59:28.764376Z"Tripathy, Arun K."https://zbmath.org/authors/?q=ai:tripathy.arun-kumar"Das, Sunita"https://zbmath.org/authors/?q=ai:das.sunitaSummary: This work is concerned about the necessary and sufficient conditions for oscillation of solutions of 2-dimensional nonlinear neutral delay difference systems of the form:
\[
\varDelta \begin{bmatrix} x(n) + p(n)x(n-m) \\ y(n) + p(n)y(n-m) \end{bmatrix} = \begin{bmatrix} a(n) & b(n) \\ c(n) & d(n) \end{bmatrix} \begin{bmatrix} f(x(n - \alpha)) \\ g(y(n - \beta)) \end{bmatrix},
\]
where \(m > 0, \alpha \geq 0, \beta \geq 0\) are integers, \(a(n), b(n), c(n), d(n), p(n)\) are real sequences and \(f, g \in \mathcal{C}(\mathbb{R}, \mathbb{R})\).Bounded solutions of difference equations in a Banach space with input data from subspaceshttps://zbmath.org/1496.390092022-11-17T18:59:28.764376Z"Chaikovs'kyi, A. V."https://zbmath.org/authors/?q=ai:chajkovskyj.andrij-v"Lagoda, O. A."https://zbmath.org/authors/?q=ai:lagoda.o-aSummary: We study the problem of existence and uniqueness of a bounded solution to a difference equation of the first order with constant operator coefficient in a Banach space. We establish necessary and sufficient conditions for the case where the initial condition and the input sequence belong to certain subspaces. These results are applied to the case of difference equations with a jump of the operator coefficient and difference equations of higher orders.A discrete time delayed neural network with potential for associative memory revisitedhttps://zbmath.org/1496.390102022-11-17T18:59:28.764376Z"Kosovalić, N."https://zbmath.org/authors/?q=ai:kosovalic.nemanja"Wu, J."https://zbmath.org/authors/?q=ai:wu.jianhongSummary: We consider a nonlinear difference equation with a time delay which represents the simplest possible discretely updated neural network. We show that provided that the integral time delay is large enough, this self feedback system exhibits a huge number of asymptotically stable periodic orbits. Therefore, such a network has potential for associative memory and pattern recognition.On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm-Liouville equationshttps://zbmath.org/1496.390112022-11-17T18:59:28.764376Z"Aygar Küçükevcilioğlu, Yelda"https://zbmath.org/authors/?q=ai:aygar-kucukevcilioglu.yelda"Bayram, Elgiz"https://zbmath.org/authors/?q=ai:bayram.elgiz"Özbey, Güher Gülçehre"https://zbmath.org/authors/?q=ai:ozbey.guher-gulcehreSummary: This work develops scattering and spectral analysis of a discrete impulsive Sturm-Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit.Existence of nonnegative solutions for discrete Robin boundary value problems with sign-changing weighthttps://zbmath.org/1496.390122022-11-17T18:59:28.764376Z"Zhu, Yan"https://zbmath.org/authors/?q=ai:zhu.yanSummary: In this paper,~we are concerned with the following discrete problem first
\[
\begin{cases}
-\Delta^2u(t-1) = \lambda p(t)f(u(t)), \quad t\in[1, N-1]_{\mathbb{Z}},\\
\Delta u(0) = u(N) = 0,
\end{cases}
\]
where \(N>2\) is an integer, \(\lambda>0\) is a parameter, \(p: [1,N-1]_{\mathbb{Z}}\rightarrow\mathbb{R}\) is a sign-changing function, \(f: [0,+\infty)\rightarrow[0, +\infty)\) is a continuous and nondecreasing function. \(\Delta u(t) = u(t+1)-u(t)\), \(\Delta^2u(t) = \Delta(\Delta u(t))\). By using the iterative method and Schauder's fixed point theorem, we will show the existence of nonnegative solutions to the above problem. Furthermore, we obtain the existence of nonnegative solutions for discrete Robin systems with indefinite weights.Finite time stability of fractional delay difference systems: a discrete delayed Mittag-Leffler matrix function approachhttps://zbmath.org/1496.390132022-11-17T18:59:28.764376Z"Du, Feifei"https://zbmath.org/authors/?q=ai:du.feifei"Jia, Baoguo"https://zbmath.org/authors/?q=ai:jia.baoguoSummary: A discrete delayed Mittag-Leffler matrix function is developed in this paper. Based on this function, an explicit formula of the solution of fractional delay difference system (FDDS) is derived. Furthermore, a criterion on finite time stability (FTS) of FDDS with constant coefficients is obtained by use of this formula. However, it can't be directly used to investigate the FTS of FDDS with variable coefficients. To overcome this difficulty, a comparison theorem of FDDS is established to obtain a criterion of the FTS of FDDS with variable coefficients. Finally, a numerical example is given to show the effectiveness of the proposed results.Ternary biderivations and ternary bihomorphisms in \(C^\ast\)-ternary algebrashttps://zbmath.org/1496.390142022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In [Rocky Mt. J. Math. 49, No. 2, 593--607 (2019; Zbl 1417.39078)], the second author et al. introduced the following bi-additive \(s\)-functional inequality
\[
\begin{aligned}
\| f(x&+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \\
\quad \le \, &\bigg\|s \left(2f \left(\frac{x+y}{2}, z-w \right) + 2f \left(\frac{x-y}{2}, z+w \right) - 2f(x,z)+ 2 f(y, w) \right) \bigg\|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\). Using the fixed point method, we prove the Hyers-Ulam stability of ternary biderivations and ternary bihomomorphism in \(C^\ast\)-ternary algebras, associated with the bi-additive \(s\)-functional inequality (1).
For the entire collection see [Zbl 1485.65002].Functional inequalities for multi-additive-quadratic-cubic mappingshttps://zbmath.org/1496.390152022-11-17T18:59:28.764376Z"Bodaghi, Abasalt"https://zbmath.org/authors/?q=ai:bodaghi.abasalt"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In this chapter, a new version of multi-quadratic mappings are characterized. By this characterization, every multi-additive-quadratic-cubic mapping which is defined as system of functional equations can be unified as a single equation. In addition, by applying two fixed point theorems, the generalized Hyers-Ulam stability of multi-additive-quadratic-cubic mappings in normed and non-Archimedean normed spaces are studied. A few corollaries corresponding to some known stability and hyperstability outcomes for multi-additive, multi-quadratic, multi-cubic, and multi-additive-quadratic-cubic mappings (functional equations) are presented.
For the entire collection see [Zbl 1485.65002].Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equationhttps://zbmath.org/1496.390162022-11-17T18:59:28.764376Z"Thuc, Nguyen Dat"https://zbmath.org/authors/?q=ai:thuc.nguyen-dat"Ngoc, Le Thi Phuong"https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Long, Nguyen Thanh"https://zbmath.org/authors/?q=ai:nguyen-thanh-long.Summary: This paper is devoted to the study of the following nonlinear functional integral equation
\[
f(x)=\sum\limits_{i=1}^q\alpha_i(x)f(\tau_i(x)) + \int_0^{\sigma_1(x)}\Psi\left(x, t, f(\sigma_2(t)), \int_0^{\sigma_3(t)}f(s)ds\right) dt + g(x),\;\forall x\in [0,1], \tag{E}
\]
where \(\tau_i, \sigma_1, \sigma_2, \sigma_3 :[0,1]\rightarrow [0,1]\); \(\alpha_i, g: [0,1]\rightarrow \mathbb{R}\); \(\Psi: [0,1]\times [0,1]\times\mathbb{R}^2\rightarrow \mathbb{R}\) are the given continuous functions and \(f:[0,1]\,\rightarrow\mathbb{R}\) is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of \(\Psi\in C^2([0, 1]\times [0,1]\times \mathbb{R}^2; \mathbb{R})\), we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.Approximate generalized Jensen mappings in 2-Banach spaceshttps://zbmath.org/1496.390172022-11-17T18:59:28.764376Z"Almahalebi, Muaadh"https://zbmath.org/authors/?q=ai:almahalebi.muaadh"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-m"Al-Ali, Sadeq"https://zbmath.org/authors/?q=ai:al-ali.sadeq-a-a"Hryrou, Mustapha E."https://zbmath.org/authors/?q=ai:hryrou.mustapha-esseghyrSummary: Our aim is to investigate the generalized Hyers-Ulam-Rassias stability for the following general Jensen functional equation:
\[
\sum_{k=0}^{n-1} f(x+ b_ky)=nf(x),
\]
where \(n \in \mathbb{N}_2\), \(b_k=\exp (\frac{2i\pi k}{n})\) for \(0 \leq\) k \(\leq\) n \(- 1\), in 2-Banach spaces by using a new version of Brzdȩk's fixed point theorem. In addition, we prove some hyperstability results for the considered equation and the general inhomogeneous Jensen equation
\[
\sum_{k=0}^{n-1} f(x+ b_ky)=nf(x)+G(x,y).
\]
For the entire collection see [Zbl 1485.65002].Some hyperstability results in non-Archimedean 2-Banach space for a \(\sigma\)-Jensen functional equationhttps://zbmath.org/1496.390182022-11-17T18:59:28.764376Z"El Ghali, Rachid"https://zbmath.org/authors/?q=ai:el-ghali.rachid"Kabbaj, Samir"https://zbmath.org/authors/?q=ai:kabbaj.samirSummary: By combining the two versions of Brzdȩk's fixed point theorem in non-Archimedean Banach spaces [\textit{J. Brzdȩk} and \textit{K. Ciepliński}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 6861--6867 (2011; Zbl 1237.39022)] and that in 2-Banach spaces [\textit{J. Brzdȩk} and \textit{K. Ciepliński}, Acta Math. Sci., Ser. B, Engl. Ed. 38, No. 2, 377--390 (2018; Zbl 1399.39063)], we will investigate the hyperstability of the following \(\sigma\)-Jensen functional equation:
\[
f(x+y)+f(x+\sigma (y))=2f(x),
\]
where \(f : X \to Y\) such that \(X\) is a normed space, \(Y\) is a non-Archimedean 2-Banach space, and \(\sigma\) is a homomorphism of \(X\). In addition, we prove some interesting corollaries corresponding to some inhomogeneous outcomes and particular cases of our main results in \(C^\ast\)-algebras.
For the entire collection see [Zbl 1485.65002].Hyers-Ulam stability of an additive-quadratic functional equationhttps://zbmath.org/1496.390192022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of Lie biderivations and Lie bihomomorphisms in Lie Banach algebras, associated with the bi-additive functional inequality
\[
\begin{aligned}
\| f(x+&y, z+w) + f(x+y, z-w) + f(x-y, z+w)\\
+ &f(x-y, z-w) -4f(x,z)\|\\
\le \| &s (2f(x+ y, z-w) + 2f(x-y, z+ w) - 4f(x,z)+ 4 f(y, w)) \|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\).
For the entire collection see [Zbl 1485.65002].Hyers-Ulam stability of symmetric biderivations on Banach algebrashttps://zbmath.org/1496.390202022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In [Indian J. Pure Appl. Math. 50, No. 2, 413--426 (2019; Zbl 1428.39031)], the second author introduced the following bi-additive \(s\)-functional inequality:
\[
\begin{aligned}
\| f(x&+y, z-w) + f(x-y, z+w) -2f(x, z)+2 f(y, w)\|\\
\le &\bigg\|s \left(2f \left(\frac{x+y}{2}, z-w\right) + 2f \left(\frac{x-y}{2}, z+w\right) - 2f(x, z)+ 2 f(y, w)\right) \bigg\|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\). Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of symmetric biderivations and a skew-symmetric biderivation on Banach algebras and unital \(C^\ast\)-algebras, associated with the bi-additive \(s\)-functional inequality (1).
For the entire collection see [Zbl 1483.00042].On Hyers-Ulam-Rassias stability of a Volterra-Hammerstein functional integral equationhttps://zbmath.org/1496.450042022-11-17T18:59:28.764376Z"Ciplea, Sorina Anamaria"https://zbmath.org/authors/?q=ai:ciplea.sorina-anamaria"Lungu, Nicolaie"https://zbmath.org/authors/?q=ai:lungu.nicolaie"Marian, Daniela"https://zbmath.org/authors/?q=ai:marian.daniela"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: The aim of this paper is to study the Hyers-Ulam-Rassias stability for a Volterra-Hammerstein functional integral equation in three variables via Picard operators.
For the entire collection see [Zbl 1485.65002].A new aspect of generalized integral operator and an estimation in a generalized function theoryhttps://zbmath.org/1496.450132022-11-17T18:59:28.764376Z"Al-Omari, Shrideh"https://zbmath.org/authors/?q=ai:al-omari.shrideh-khalaf-qasem"Almusawa, Hassan"https://zbmath.org/authors/?q=ai:almusawa.hassan"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this paper we investigate certain integral operator involving Jacobi-Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi-Dunkl integral operator [\textit{N. B. Salem} and \textit{A. O. A. Salem}, Ramanujan J. 12, No. 3, 359--378 (2006; Zbl 1122.44002)] is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi-Dunkl operator are discussed. An inversion formula and certain convergence with respect to \(\delta\) and \(\Delta\) convergences are also introduced.Certain properties of Jordan homomorphisms, \(n\)-Jordan homomorphisms and \(n\)-homomorphisms on rings and Banach algebrashttps://zbmath.org/1496.460462022-11-17T18:59:28.764376Z"Honary, Taher Ghasemi"https://zbmath.org/authors/?q=ai:ghasemi-honary.taherSummary: We investigate under what conditions \(n\)-Jordan homomorphisms between rings are \(n\)-homomorphism, or homomorphism; and under what conditions, \(n\)-Jordan homomorphisms are continuous.
One of the main goals in this work is to show that every \(n\)-Jordan homomorphism \(f : A \rightarrow B\), from a unital ring \(A\) into a ring \(B\) with characteristic greater than \(n\), is a multiple of a Jordan homomorphism and hence, it is an \(n\)-homomorphism if every Jordan homomorphism from \(A\) into \(B\) is a homomorphism. In particular, if \(B\) is an integral domain whose characteristic is greater than \(n\), then every \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is an \(n\)-homomorphism.
Along with some other results, we show that if \(A\) and \(B\) are unital rings such that the characteristic of \(B\) is greater than \(n\), then every unital \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is a Jordan homomorphism and hence, it is an \(m\)-Jordan homomorphism for any positive integer \(m \geq 2\).
We also investigate the automatic continuity of \(n\)-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the \(n\)-Jordan homomorphism has dense range.Hyperstability of orthogonally 3-Lie homomorphism: an orthogonally fixed point approachhttps://zbmath.org/1496.460782022-11-17T18:59:28.764376Z"Keshavarz, Vahid"https://zbmath.org/authors/?q=ai:keshavarz.vahid"Jahedi, Sedigheh"https://zbmath.org/authors/?q=ai:jahedi.sedighehSummary: In this chapter, by using the orthogonally fixed point method, we prove the Hyers-Ulam stability and the hyperstability of orthogonally 3-Lie homomorphisms for additive \(\rho\)-functional equation in 3-Lie algebras. Indeed, we investigate the stability and the hyperstability of the system of functional equations
\[
\begin{cases}
f(x+y)-f(x)-f(y)= \rho (2f(\frac{x+y}{2})+ f(x)+ f(y)),\\
f([[u,v],w])=[[f(u),f(v)],f(w)]
\end{cases}
\]
in 3-Lie algebras where \(\rho \neq 1\) is a fixed real number.
For the entire collection see [Zbl 1485.65002].Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacianshttps://zbmath.org/1496.470112022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arneSummary: We consider the discrete Laplacian on \(\mathbb{Z}^d\), and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if \(d\) is odd, and a logarithm branching if \(d\) is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensionshttps://zbmath.org/1496.470122022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.2|ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arne-m|jensen.arne-skov|jensen.arneThe authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\dots-2\cos(\theta_d)-z}d\theta.\]
Such investigation was mainly restricted to dimensions \(d=1,2\). In the Introduction, they start to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \( \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To get rid of the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G(\frac{-2d\epsilon}{1-\epsilon},n)\), for values of \(\epsilon\in (0,1]\).
In this way, they succeed in representing \(G(z,n)\) as a convergent series (see, e.g., Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past.
As a whole, this paper is complementary to the thors' previous paper [J. Funct. Anal. 277, No. 4, 965--993 (2019; Zbl 1496.47011)] in which the authors have shown that \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\dots,n\), the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) -- the singular part of \(G(z,n)\) -- was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\), \(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix~A.
Reviewer: Nelson Faustino (Alfeizerão)Essential spectrum of a weighted geometric realizationhttps://zbmath.org/1496.470422022-11-17T18:59:28.764376Z"Hatim, Khalid"https://zbmath.org/authors/?q=ai:hatim.khalid"Baalal, Azeddine"https://zbmath.org/authors/?q=ai:baalal.azeddineSummary: In this present article, we construct a new framework that's we call the weighted geometric realization of 2 and 3-simplexes. On this new weighted framework, we construct a nonself-adjoint 2-simplex Laplacian \(L\) and a self-adjoint 2-simplex Laplacian \(N\). We propose general conditions to ensure sectoriality for our new nonself-adjoint 2-simplex Laplacian \(L\). We show the relation between the essential spectra of \(L\) and \(N\). Finally, we prove the absence of the essential spectrum for our 2-simplex Laplacians \(L\) and \(N\).Multiplicative operator functions and abstract Cauchy problemshttps://zbmath.org/1496.470692022-11-17T18:59:28.764376Z"Früchtl, Felix"https://zbmath.org/authors/?q=ai:fruchtl.felixSummary: We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including \(C_{0}\)-groups and cosine operator functions, and more generally, Sturm-Liouville operator functions.Decay of harmonic functions for discrete time Feynman-Kac operators with confining potentialshttps://zbmath.org/1496.600872022-11-17T18:59:28.764376Z"Cygan, Wojciech"https://zbmath.org/authors/?q=ai:cygan.wojciech"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Śliwiński, Mateusz"https://zbmath.org/authors/?q=ai:sliwinski.mateuszSummary: We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in a countably infinite space. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.The discrete analogue of the differential operator \( \frac{\mathrm{d}^{2m}}{\mathrm{d}\,x^{2m}}+2\omega^2\frac{\mathrm{d}^{2m-2}}{\mathrm{d}\,x^{2m-2}}+\omega^4\frac{\mathrm{d}^{2m-4}}{\mathrm{d}\,x^{2m-4}} \)https://zbmath.org/1496.650152022-11-17T18:59:28.764376Z"Hayotov, A. R."https://zbmath.org/authors/?q=ai:hayotov.abdullo-rakhmonovich(no abstract)The numerical solution of semidiscrete linear evolution problems on the finite interval using the unified transform methodhttps://zbmath.org/1496.651452022-11-17T18:59:28.764376Z"Cisneros, Jorge"https://zbmath.org/authors/?q=ai:cisneros.jorge"Deconinck, Bernard"https://zbmath.org/authors/?q=ai:deconinck.bernardSummary: We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval \(x\in (0,L)\). The semidiscrete method is applied to various spatial discretizations of several first and second-order linear equations, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. From these solutions, we derive alternative series representations that are better suited for numerical computations. In addition, we show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and we propose the notion of ``natural'' discretizations. We consider the continuum limit of the semidiscrete solutions and compare with standard finite-difference schemes.Differential recurrences for the distribution of the trace of the \(\beta\)-Jacobi ensemblehttps://zbmath.org/1496.810912022-11-17T18:59:28.764376Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.2|kumar.santosh.1|kumar.santosh.4|kumar.santosh|kumar.santosh.3Summary: Examples of the \(\beta\)-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scattering problems. For this application, the trace is of relevance as determining the conductance. Earlier, in the case \(\beta = 1\), the trace statistic was isolated in studies of covariance matrices in multivariate statistics. There, Davis showed that for \(\beta = 1\) the trace statistic could be characterised by \((N + 1) \times (N + 1)\) matrix differential equations, now understood for general \(\beta > 0\) as part of the theory of Selberg correlation integrals. However the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameters \(b\) and Dyson index \(\beta\) non-negative integers. The distribution then has the functional form of a series of piecewise power functions times a polynomial, and our characterisation gives a recurrence for the computation of the polynomials. For all \(\beta > 0\) we express the Fourier-Laplace transform of the trace statistic in terms of a generalised hypergeometric function based on Jack polynomials.The asymptotic approach to the continuum of lattice QCD spectral observableshttps://zbmath.org/1496.810952022-11-17T18:59:28.764376Z"Husung, Nikolai"https://zbmath.org/authors/?q=ai:husung.nikolai"Marquard, Peter"https://zbmath.org/authors/?q=ai:marquard.peter"Sommer, Rainer"https://zbmath.org/authors/?q=ai:sommer.rainerSummary: We consider spectral quantities in lattice QCD and determine the asymptotic behaviour of their discretization errors. Wilson fermion with \(\mathrm{O}(a)\)-improvement, (Möbius) Domain wall fermion (DWF), and overlap Dirac operators are considered in combination with the commonly used gauge actions. Wilson fermions and DWF with domain wall height \(M_5 = 1 + \mathrm{O}(g_0^2)\) have the same, approximate, form of the asymptotic cutoff effects: \(Ka^2 [\bar{g}^2(a^{-1})]^{0.760}\). A domain wall height \(M_5 = 1.8\), as often used, introduces large mass-dependent \(K^\prime(m) a^2 [\bar{g}^2(a^{-1})]^{0.518}\) effects. Massless twisted mass fermions have the same form as Wilson fermions when the Sheikholeslami-Wohlert term [\textit{B. Sheikholeslami} and \textit{R. Wohlert}, ``Improved continuum limit lattice action for QCD with Wilson fermions'', Nucl. Phys., B 259, No. 4, 572--596 (1985; \url{doi:10.1016/0550-3213(85)90002-1})] is included. For their mass-dependent cutoff effects we have information on the exponents \(\hat{\Gamma}_i\) of \(\bar{g}^2(a^{-1})\) but not for the pre-factors. For staggered fermions there is only partial information on the exponents.
We propose that tree-level \(\mathrm{O}(a^2)\) improvement, which is easy to do [\textit{M. Alford}, \textit{T. R. Klassen} and \textit{G. P. Lepage}, ``Improving lattice quark actions'', Nucl. Phys., B 496, No. 1--2, 377--407 (1997; \url{doi:10.1016/S0550-3213(97)00249-6})], should be used in the future -- both for the fermion and the gauge action. It improves the asymptotic behaviour in all cases.Permanence for nonautonomous scalar delay difference population modelshttps://zbmath.org/1496.920962022-11-17T18:59:28.764376Z"Saito, Yasuhisa"https://zbmath.org/authors/?q=ai:saito.yasuhisaSummary: A general class of nonautonomous single-species delay difference population models is considered. Such models are shown to be permanent not only under some biologically realistic assumptions but also without assuming continuity on the right-hand side of their models. It is also shown that any type of delay is harmless for the permanence.