Recent zbMATH articles in MSC 34Khttps://zbmath.org/atom/cc/34K2022-11-17T18:59:28.764376ZWerkzeugStability, boundedness and controllability of solutions of measure functional differential equationshttps://zbmath.org/1496.340042022-11-17T18:59:28.764376Z"Andrade da Silva, F."https://zbmath.org/authors/?q=ai:andrade-da-silva.f"Federson, M."https://zbmath.org/authors/?q=ai:federson.marcia"Toon, E."https://zbmath.org/authors/?q=ai:toon.eduardIn the present paper, converse Lyapunov results on uniform boundedness for the very general class of generalized differential equations are established. Relations between stability and boundedness of solutions are also obtained. Using Lyapunov techniques, asymptotic controllability is characterized as well. As the theory of measure functional differential equations is a particular case of this wide setting, corresponding theorems are derived for measure functional differential equations.
Reviewer: Bianca-Renata Satco (Suceava)Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditionshttps://zbmath.org/1496.340112022-11-17T18:59:28.764376Z"Bourguiba, Rim"https://zbmath.org/authors/?q=ai:bourguiba.rim"Cabada, Alberto"https://zbmath.org/authors/?q=ai:cabada.alberto"Kalthoum, Wanassi Om"https://zbmath.org/authors/?q=ai:kalthoum.wanassi-omSummary: In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green's function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions.Sufficient conditions for the existence of oscillatory solutions to nonlinear second order differential equationshttps://zbmath.org/1496.340222022-11-17T18:59:28.764376Z"Sethi, Abhay Kumar"https://zbmath.org/authors/?q=ai:sethi.abhay-kumar"Ghaderi, Mehran"https://zbmath.org/authors/?q=ai:ghaderi.mehran"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahram"Kaabar, Mohammed K. A."https://zbmath.org/authors/?q=ai:kaabar.mohammed"Inc, Mostafa"https://zbmath.org/authors/?q=ai:inc.mostafa"Masiha, Hashem Parvaneh"https://zbmath.org/authors/?q=ai:masiha.hashem-parvanehSummary: Some electrical events consist of mathematical models and due to the essence of electric current, their related differential equations have naturally an interesting property where all solutions are oscillatory. Many researchers have studied the necessary conditions for oscillatory solutions in the literature. But in this work, some sufficient conditions are investigated using Riccati transformation for nonlinear second order differential equations in which all solutions are oscillatory. Two illustrative examples are provided to validate our theoretical results. In addition, simulation results and numerical experiments are conducted to validate our obtained results.On Landesman-Lazer conditions and the fundamental theorem of algebrahttps://zbmath.org/1496.340752022-11-17T18:59:28.764376Z"Amster, Pablo"https://zbmath.org/authors/?q=ai:amster.pabloIn this paper, the author deals with the differential system
\[
u'(t)+g(u(t))=p(t), \tag{1}
\]
where \(g:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) is bounded and \(p\) is continuous and \(T\)-periodic. Two results for the existence of at least one \(T\)-periodic solution for system (1) are obtained when \(g\) satisfies Landesman-Lazer type conditions. The connection of the second result with the fundamental theorem of algebra is stated.
Furthermore, the author treats the following delay systems
\[
u'(t)=g(u(t))+p(t,u(t),u(t-\tau)), \tag{2}
\]
where \(\tau>0\) and \(p\) is bounded, continuous and \(T\)-periodic in the first coordinate. Under similar conditions, two theorems for the existence of at least one \(T\)-periodic solution for system (2) are proved.
Reviewer: Chun Li (Chongqing)Individual nonuniform dichotomy and admissibility for linear skew-products semiflows over a semiflowhttps://zbmath.org/1496.340912022-11-17T18:59:28.764376Z"Onofrei, Oana Romina"https://zbmath.org/authors/?q=ai:onofrei.oana-romina"Preda, Petre"https://zbmath.org/authors/?q=ai:preda.petreSummary: The aim of this paper is to give a new characterization of the admissibility of the pair \((L^1; L^\infty)\) to the case of linear skew-product semiflows over semiflows, which satisfy the following conditions: the cocycle \( \pi = (\Phi \sigma)\) has no exponential growth and \(K\) the constant from the ``boundedness'' theorem it depends on \(\theta \in \Theta\), by using the ``input-output'' technique.Quasi-synchronization of heterogenous fractional-order dynamical networks with time-varying delay via distributed impulsive controlhttps://zbmath.org/1496.340992022-11-17T18:59:28.764376Z"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.1"Zheng, Zhaowen"https://zbmath.org/authors/?q=ai:zheng.zhaowen"Yang, Yongqing"https://zbmath.org/authors/?q=ai:yang.yongqingSummary: This paper investigates the quasi-synchronization problem of a heterogeneous dynamical network. All nodes have fractional order dynamical behavior with time-varying delay. The distributed impulsive control strategy is applied to drive all the nodes to approximately synchronize with the target orbit within a nonzero error bound. A new comparison principle of impulsive fractional order functional differential equation has been built at first. Then, based on the Lyapunov stability theory, some basic theories of fractional order functional differential equation, and the definition of an average impulsive interval, some quasi-synchronization criteria are derived with explicit expressions of the error bound. Both synchronizing impulses and desynchronizing impulses are discussed in this paper. Finally, two numerical examples are presented to illustrate the validity of the theoretical analysis.Qualitative and quantitative analysis of functional-differential equations of Goodwin typehttps://zbmath.org/1496.341002022-11-17T18:59:28.764376Z"Khidirov, B. N."https://zbmath.org/authors/?q=ai:khidirov.b-n"Khidirova, M. B."https://zbmath.org/authors/?q=ai:khidirova.m-b"Shakarov, A. R."https://zbmath.org/authors/?q=ai:shakarov.a-r(no abstract)Topological structure of the solution sets for a nonlinear delay evolutionhttps://zbmath.org/1496.341012022-11-17T18:59:28.764376Z"Wang, Rong-Nian"https://zbmath.org/authors/?q=ai:wang.rongnian"Ma, Zhong-Xin"https://zbmath.org/authors/?q=ai:ma.zhong-xin"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-mIn this paper, the authors study the following nonlinear delay evolution equation with multivalued perturbation on a noncompact interval \[\begin{cases} u'(t)\in -A(t)u(t)+f(t), ~~~ t\in \mathbb{R}^+,\\
f(t)\in F(t,u_t), ~~~ t\in \mathbb{R}^+,\\
u(t)=\phi(t), ~~~ t\in [-r,0], \end{cases} \] where \(A(t), t\in \mathbb{R}^+\) is a family of possibly unbounded operators on an infinite dimensional real Banach space \(\mathbb{X},\) \(F: \mathbb{R}^+\times C([-r,0], \mbox{conv}\mathbb{D})\to 2^{\mathbb{X}}\setminus\emptyset\) has convex closed values, \(F(t,\cdot)\) is upper hemicontinuous for a.e. \(t\in \mathbb{R}^+,\) \(\mathbb{D}\) is a nonempty closed subset of \(\mathbb{X},\) \(\mbox{conv}\mathbb{D}\) is the convex hull of \(\mathbb{D}\) and \(u_t(\cdot)\in C([-r,0],\mathbb{D})\) is defined by \(u_t(s)=u(t+s), s\in [-r,0].\) It is proved that the solution map, having nonempty and compact values, is an \(R_{\delta}\)-map, which maps any connected set into a connected set. Several examples illustrating the applicability of the obtained abstract results are also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)Nonlinear equations of fourth-order with \(p\)-Laplacian like operators: oscillation, methods and applicationshttps://zbmath.org/1496.341022022-11-17T18:59:28.764376Z"Bazighifan, Omar"https://zbmath.org/authors/?q=ai:bazighifan.omar"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraThe paper is concerned with fourth order differential equations of the types found in fluid dynamics, electromagnetism and quantum theory. The main aim is the study of oscillatory behaviour of solutions of these equations when they are driven by a \(p\)-Laplace differential operator. The main results of the paper are two theorems that provide results that, under given hypotheses, the fourth order equation under consideration is oscillatory. As the authors point out, these results extend the existing known results. The paper concludes with an example.
Reviewer: Neville Ford (Chester)Rigorous verification of Hopf bifurcations in functional differential equations of mixed typehttps://zbmath.org/1496.341032022-11-17T18:59:28.764376Z"Church, Kevin E. M."https://zbmath.org/authors/?q=ai:church.kevin-e-m"Lessard, Jean-Philippe"https://zbmath.org/authors/?q=ai:lessard.jean-philippeThe paper is concerned with the development of a numerical method to prove the existence of Hopf bifurcations in simple functional differential equations of mixed type, otherwise known as advance-delay equations, or sometimes as forward-backward equations. The question of interest is to consider the properties of the eigenvalues, and use is made of the Newton-Kantorovich theorem. The authors prove the existence of Hopf bifurcations in the Lasota-Wazewska-Czyzewska model and the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. The overall objective of the work is to `develop numerical methods which can lead to computer-assisted proofs of existence of different type of dynamical objects arising in the study of differential equations.' Consequently, there is a section that discusses computer-assisted proofs of some of the theorems presented and links are provided to the relevant code.
Reviewer: Neville Ford (Chester)Periodic solutions to certain classes of third order delay differential equationshttps://zbmath.org/1496.341042022-11-17T18:59:28.764376Z"Olayemi, S. A."https://zbmath.org/authors/?q=ai:olayemi.s-a"Ogundare, B. S."https://zbmath.org/authors/?q=ai:ogundare.babatunde-sundaySummary: In this paper, existence of unique periodic solutions for two classes of third order delay differential equations are considered. The two equations are expressed in their integral equivalents with which suitable Green's function is constructed and its associated properties stated. The main tool adopted to establish the existence of unique periodic solutions to the classes of delay differential equations is the Krasnoselskii's fixed point theorem due to its applicability to the combination of a compact and contraction mappings which occur in dealing with perturbed differential operators.Bifurcation in car-following models with time delays and driver and mechanic sensitivitieshttps://zbmath.org/1496.341052022-11-17T18:59:28.764376Z"Padial, Juan Francisco"https://zbmath.org/authors/?q=ai:padial.juan-francisco"Casal, Alfonso"https://zbmath.org/authors/?q=ai:casal.alfonso-cSummary: In this work, we study a model of traffic flow along a one-way, one lane, road or street, the so-called car-following problem. We first present a historical evolution of models of this type corresponding to a successive improvement of requirements, to explain some real traffic phenomena. For both mathematical reasons and a better explanation of some of those phenomena, we consider more convenient and accurate requirements which lead to a better non-linear model with reaction delays, from several sources. The model can be written as an ordinary nonlinear delay differential equation. It has equilibrium solutions, which correspond to steady traffic. The mentioned reaction delays introduce perturbation terms in the equation, leading to of instabilities of equilibria and changes of the structure of the solutions. For some of the values of the delays, they may become oscillatory. We make a number of simulations to show these changes for different values of delays. We also show that, for certain values of the delays the above mentioned change of structure (representing regimes of real traffic) corresponds to a Hopf bifurcation.Discrete traveling waves in a relay system of Mackey-Glass equations with two delayshttps://zbmath.org/1496.341062022-11-17T18:59:28.764376Z"Preobrazhenskaya, M. M."https://zbmath.org/authors/?q=ai:preobrazhenskaya.m-mConsider a ring circuit of \(m\) identical Mackey-Glass generators of the form \[\dot{u}_{j}=-\beta u_{j}+\frac{\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)}{1+\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)^{\gamma}}, \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m\tag{1} \] with positive parameters \(\alpha,\beta,\gamma,\tau\). Letting \(\gamma \to \infty\) one obtains the limiting equation \[\dot{u}_{j}=-\beta u_{j}+\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right) F\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right), \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m \tag{2}\] with \[ F(u) := \lim _{\gamma \rightarrow+\infty} \frac{1}{1+u^{\gamma}}= \begin{cases}1, & 0<u<1, \\
1 / 2, & u=1, \\
0, & u>1.\end{cases} \] The motivation to study system (2) is twofold: on the one hand, \(\gamma\gg 1\) is a realistic assumption in applications, and on the other hand, equation (2) can be regarded as a relay circuit analogue of system (1).
The main result of the paper shows that for each positive integer \(m\geq 2\), there exists a range of values of the parameters such that for fixed \(\alpha, \beta,\tau\) and \(m\), there exists a discrete traveling wave solution of (2), that is, there is a \(\Delta>0\) for which system (2) has a periodic solution of the form \(u_j(t) = u_\ast (t + j\Delta)\).
The key observation is that discrete traveling wave solutions correspond to periodic solutions of the scalar version (i.e.\ \(m=1\)) of equation (2).
The proof is constructive: the exact formula for \(u_\ast\) is obtained.
Reviewer: Ábel Garab (Klagenfurt)Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive typehttps://zbmath.org/1496.341072022-11-17T18:59:28.764376Z"Zhu, Yu"https://zbmath.org/authors/?q=ai:zhu.yuSummary: In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type
\[
\bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t),
\] where \(f:(0,+\infty)\rightarrow \mathbb{R}\), \(\varphi(t)>0\) and \(\alpha(t)>0\) are continuous functions with \(T\)-periodicity in the \(t\) variable, \(c, \gamma\) are constants with \(|c|<1, \gamma\geq1\). Many authors obtained the existence of periodic solutions under the condition \(0<\mu\leq1\), and we extend the result to \(\mu>1\) by using Mawhin's continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.Globally exponential stability of piecewise pseudo almost periodic solutions for neutral differential equations with impulses and delayshttps://zbmath.org/1496.341082022-11-17T18:59:28.764376Z"He, Jianxin"https://zbmath.org/authors/?q=ai:he.jianxin"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchao"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Qiu, Hongjun"https://zbmath.org/authors/?q=ai:qiu.hongjunImpulsive differential equations are very important class of differential equations whose dynamics is very rich. In this work, authors consider a delayed impulsive neutral differential equations. The coefficients are assumed to be bounded. The main objective is to establish the existence of piecewise pseudo almost periodic solution. The techniques used are contraction mapping principle and generalized Gronwall-Bellmain inequality. Moreover, the stability of such solution is also shown. The stability is globally exponential. At the end, an example with numerical illustration is provided by the authors.
Reviewer: Syed Abbas (Mandi)Further results on delay-dependent stability for neutral singular systems via state decomposition methodhttps://zbmath.org/1496.341092022-11-17T18:59:28.764376Z"Chen, Wenbin"https://zbmath.org/authors/?q=ai:chen.wenbin"Gao, Fang"https://zbmath.org/authors/?q=ai:gao.fang"She, Jinhua"https://zbmath.org/authors/?q=ai:she.jinhua"Xia, Weifeng"https://zbmath.org/authors/?q=ai:xia.weifengSummary: This paper studies the delay-dependent stability for neutral singular systems. In the light of state decomposition method, a novel augmented Lyapunov-Krasovskii functional including less decision variables is developed. Then by means of zero-value equations technology, some sufficient stability conditions in the form of linear matrix inequalities are acquired, which guarantees the non-impulsiveness, regularity and stability for the proposed neutral singular systems. The obtained stability criterion takes the sizes of both the discrete- and neutral- delays into account. They are less conservative than those presented by previous analytical approaches. Numerical examples are given to show the feasibility of our method and the interrelation between the discrete- and neutral-delays.A phase model with large time delayed couplinghttps://zbmath.org/1496.341102022-11-17T18:59:28.764376Z"Al-Darabsah, Isam"https://zbmath.org/authors/?q=ai:al-darabsah.isam"Campbell, Sue Ann"https://zbmath.org/authors/?q=ai:campbell.sue-annSummary: We consider two identical oscillators with weak, time delayed coupling. We start with a general system of delay differential equations then reduce it to a phase model. With the assumption of large time delay, the resulting phase model has an explicit delay and phase shift in the argument of the phases and connection function, respectively. Using the phase model, we prove that for any type of oscillators and any coupling, the in-phase and anti-phase phase-locked solutions always exist and give conditions for their stability. We show that for small delay these solutions are unique, but with large enough delay multiple solutions of each type with different frequencies may occur. We give conditions for the existence and stability of other types of phase-locked solutions. We discuss the various bifurcations that can occur in the phase model as the time delay is varied. The results of the phase model analysis are applied to Morris-Lecar oscillators with diffusive coupling and compared with numerical studies of the full system of delay differential equations. We also consider the case of small time delay and compare the results with the existing ones in the literature.A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systemshttps://zbmath.org/1496.341112022-11-17T18:59:28.764376Z"Dineshkumar, C."https://zbmath.org/authors/?q=ai:dineshkumar.c"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion.Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaceshttps://zbmath.org/1496.341122022-11-17T18:59:28.764376Z"Guo, Shangjiang"https://zbmath.org/authors/?q=ai:guo.shangjiangThe paper extends existing methods for the analysis of autonomous delay differential equations on the existence of invariant manifolds to semilinear functional differential equations. The author summarises the paper in a succinct and comprehensive way: `We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.' Sections of the paper on decomposition of the phase space, equivariant normal form, and Hopf bifurcation with symmetry provide detail and useful discussion.
Reviewer: Neville Ford (Chester)Existence and uniqueness of solutions for abstract integro-differential equations with state-dependent delay and applicationshttps://zbmath.org/1496.341132022-11-17T18:59:28.764376Z"Hernandez, Eduardo"https://zbmath.org/authors/?q=ai:hernandez.eduardo-m"Rolnik, Vanessa"https://zbmath.org/authors/?q=ai:rolnik.vanessa"Ferrari, Thauana M."https://zbmath.org/authors/?q=ai:ferrari.thauana-mIn this paper, the authors study the existence and uniqueness of solutions for a general class of abstract ordinary integro-differential equation with state dependent delay. The results are obtained by using a fixed point theorem. Some examples arising in the population dynamics and in the Solow's theory of economic growth are presented.
Reviewer: Krishnan Balachandran (Coimbatore)Stepanov-like pseudo almost periodic solutions of class \(r\) in \(\alpha \)-norm under the light of measure theoryhttps://zbmath.org/1496.341142022-11-17T18:59:28.764376Z"Zabsonre, Issa"https://zbmath.org/authors/?q=ai:zabsonre.issa"Nsangou, Abdel Hamid Gamal"https://zbmath.org/authors/?q=ai:nsangou.abdel-hamid-gamal"Kpoumiè, Moussa El-Khalil"https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil"Mboutngam, Salifou"https://zbmath.org/authors/?q=ai:mboutngam.salifouSummary: The aim of this work is to present some interesting results on weighted ergodic functions. We also study the existence and uniqueness of \((\mu,\nu)\)-weighted Stepanov-like pseudo almost periodic solutions class \(r\) for some partial differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and his co-authors.Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operatorhttps://zbmath.org/1496.341152022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Sun, Pan"https://zbmath.org/authors/?q=ai:sun.pan(no abstract)A study on controllability of impulsive fractional evolution equations via resolvent operatorshttps://zbmath.org/1496.341162022-11-17T18:59:28.764376Z"Gou, Haide"https://zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the \((\alpha ,\beta)\)-resolvent operator, we concern with the term \(u^\prime(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u^\prime b)=u^\prime_b\). Finally, we present an application to support the validity study.Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional differential equations with impulsive conditionshttps://zbmath.org/1496.341172022-11-17T18:59:28.764376Z"Ali, Arshad"https://zbmath.org/authors/?q=ai:ali.arshad"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Mahariq, Ibrahim"https://zbmath.org/authors/?q=ai:mahariq.ibrahim"Rashdan, Mostafa"https://zbmath.org/authors/?q=ai:rashdan.mostafaSummary: The current study is devoted to deriving some results about existence and stability analysis for a nonlinear problem of implicit fractional differential equations (FODEs) with impulsive and integral boundary conditions. The concerned problem involves proportional type delay term. By using Schaefer's fixed point theorem and Banach's contraction principle, the required conditions are developed. Also, different kinds of Ulam stability results are derived by using nonlinear analysis. Providing a pertinent example, we demonstrate our main results.Qualitative analysis of nonlinear coupled pantograph differential equations of fractional order with integral boundary conditionshttps://zbmath.org/1496.341182022-11-17T18:59:28.764376Z"Alrabaiah, Hussam"https://zbmath.org/authors/?q=ai:alrabaiah.hussam"Ahmad, Israr"https://zbmath.org/authors/?q=ai:ahmad.israr"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Rahman, Ghaus Ur"https://zbmath.org/authors/?q=ai:rahman.ghaus-urSummary: In this research article, we develop a qualitative analysis to a class of nonlinear coupled system of fractional delay differential equations (FDDEs). Under the integral boundary conditions, existence and uniqueness for the solution of this system are carried out. With the help of Leray-Schauder and Banach fixed point theorem, we establish indispensable results. Also, some results affiliated to Ulam-Hyers (UH) stability for the system under investigation are presented. To validate the results, illustrative examples are given at the end of the manuscript.Solvability of nonlinear functional differential equations of fractional order in reflexive Banach spacehttps://zbmath.org/1496.341192022-11-17T18:59:28.764376Z"Hashem, H. H. G."https://zbmath.org/authors/?q=ai:hashem.hind-h-g"El-Sayed, A. M. A."https://zbmath.org/authors/?q=ai:el-sayed.ahmed-mohamed-ahmed"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2In this paper, the authors discuss the solvability of nonlinear functional differential equations of fractional order in reflexive Banach spaces. There are two methods given in this paper. One is the coupled system approach, the other is the functional equation approach. By O'Regan's fixed point theorem in Banach spaces, the authors obtain weak and pseudo solutions for some initial value problems, which generalizes some related results in this topic.
Reviewer: Zhenbin Fan (Jiangsu)A new approach on approximate controllability of fractional evolution inclusions of order \(1<r<2\) with infinite delayhttps://zbmath.org/1496.341202022-11-17T18:59:28.764376Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.rSummary: This manuscript is mainly focusing on the approximate controllability of fractional differential evolution inclusions of order \(1<r<2\) with infinite delay. We study our primary outcomes by using the theoretical concepts about fractional calculus, cosine, and sine function of operators and Dhage's fixed point theorem. Initially, we prove the approximate controllability for the fractional evolution system. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.On a new structure of the pantograph inclusion problem in the Caputo conformable settinghttps://zbmath.org/1496.341212022-11-17T18:59:28.764376Z"Thabet, Sabri T. M."https://zbmath.org/authors/?q=ai:thabet.sabri-t-m"Etemad, Sina"https://zbmath.org/authors/?q=ai:etemad.sina"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann-Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann-Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delayshttps://zbmath.org/1496.341222022-11-17T18:59:28.764376Z"Jin, Shoubo"https://zbmath.org/authors/?q=ai:jin.shouboSummary: The Picard iteration method is used to study the existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays. Several sufficient conditions are specified to ensure that the equation has a unique solution. First, the stochastic Volterra-Levin equation is transformed into an integral equation. Then, to obtain the solution of the integral equation, the successive approximation sequences are constructed, and the existence and uniqueness of solutions for the stochastic Volterra-Levin equation are derived by the convergence of the sequences. Finally, two examples are given to demonstrate the validity of the theoretical results.Asymptotic behavior of a predator-prey system with delayshttps://zbmath.org/1496.341232022-11-17T18:59:28.764376Z"El-Owaidy, H. M."https://zbmath.org/authors/?q=ai:el-owaidy.hassan-mostafa"Ismail, M."https://zbmath.org/authors/?q=ai:ismail.m-i|ismail.mohamed-m|ismail.mohammad|ismail.m-a-h|ismail.mohd-tahir|ismail.mourad-el-houssieny|ismail.mahamod|ismail.mohammad-vaseem|ismail.mohammed|ismail.mehmet-s|ismail.mardhiyah|ismail.moshira-a|ismail.mohd-vaseem|ismail.m-ghazie|ismail.mat-rofa-bin|ismail.mourad|ismail.mohd-azmi|ismail.mohammad-s|ismail.munira|ismail.mohamed-a|ismail.mahmoud-h|ismail.m-n|ismail.muhammad-faizal(no abstract)Stability and bifurcation analyses of p53 gene regulatory network with time delayhttps://zbmath.org/1496.341242022-11-17T18:59:28.764376Z"Hou, Jianmin"https://zbmath.org/authors/?q=ai:hou.jianmin"Liu, Quansheng"https://zbmath.org/authors/?q=ai:liu.quansheng|liu.quansheng.1"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei"Wang, Lixin"https://zbmath.org/authors/?q=ai:wang.lixin"Bi, Yuanhong"https://zbmath.org/authors/?q=ai:bi.yuanhongSummary: In this paper, based on a p53 gene regulatory network regulated by Programmed Cell Death 5(PDCD5), a time delay in transcription and translation of Mdm2 gene expression is introduced into the network, the effects of the time delay on oscillation dynamics of p53 are investigated through stability and bifurcation analyses. The local stability of the positive equilibrium in the network is proved through analyzing the characteristic values of the corresponding linearized systems, which give the conditions on undergoing Hopf bifurcation without and with time delay, respectively. The theoretical results are verified through numerical simulations of time series, characteristic values and potential landscapes. Furthermore, combined effect of time delay and several typical parameters in the network on oscillation dynamics of p53 are explored through two-parameter bifurcation diagrams. The results show p53 reaches a lower stable steady state under smaller PDCD5 level, the production rates of p53 and Mdm2 while reaches a higher stable steady state under these larger ones. But the case is the opposite for the degradation rate of p53. Specially, p53 oscillates at a smaller Mdm2 degradation rate, but a larger one makes p53 reach a low stable steady state. Besides, moderate time delay can make the steady state switch from stable to unstable and induce p53 oscillation for moderate value of these parameters. Theses results reveal that time delay has a significant impact on p53 oscillation and may provide a useful insight into developing anti-cancer therapy.Qualitative analysis of equations of the regulatory mechanisms of a multicellular organismhttps://zbmath.org/1496.341252022-11-17T18:59:28.764376Z"Khidirov, B. N."https://zbmath.org/authors/?q=ai:khidirov.b-n(no abstract)Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noisehttps://zbmath.org/1496.370552022-11-17T18:59:28.764376Z"Lv, Xiang"https://zbmath.org/authors/?q=ai:lv.xiangSummary: This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \(f\) is monotone (or anti-monotone) and the global Lipschitz constant of \(f\) is smaller than the positive real part of the principal eigenvalue of the competitive matrix \(A\), the random dynamical system (RDS) generated by SDEs has an unstable \(\mathscr{F}_+\)-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \(\mathscr{F}_+ = \sigma \{ \omega \mapsto W_t (\omega):t\geq 0\}\) is the future \(\sigma\)-algebra. In addition, we get that the \(\alpha\)-limit set of all pull-back trajectories starting at the initial value \(x(0) = x\in\mathbb{R}^n\) is a single point for all \(\omega\in\Omega\), i.e., the unstable \(\mathscr{F}_+\)-measurable random equilibrium. Applications to stochastic neural network models are given.Qualitative analysis of integro-differential equations with variable retardationhttps://zbmath.org/1496.450072022-11-17T18:59:28.764376Z"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-j"Tunç, Osman"https://zbmath.org/authors/?q=ai:tunc.osmanThe paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs), which reads as
\[
\frac{\mathrm{d} x}{\mathrm{~d} t}=A(t) x+B F(x(t-\tau(t)))+\int_{t-\tau(t)}^{t} \Omega(t, s) F(x(s)) \mathrm{d} s+G(t, x),
\]
where \(x \in \mathbb{R}^{n},\ t \in \mathbb{R}^{+}:=[0, \infty), \ \tau \in \mathrm{C}^{1}\left(\mathbb{R}^{+}, \ \mathbb{R}^{+}\right), \ A=\left(a_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+},\ \mathbb{R}^{n \times n}\right)\), \(\Omega=\left(\Omega_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \ \mathbb{R}^{n \times n}\right)\), \(B=\left(b_{i j}\right) \in \mathbb{R}^{n \times n}, \ F \in \mathrm{C}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right), \ F(0)=0\), and \(G\in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \mathbb{R}^{n}\right)\). The authors focus on the uniform
stability and uniform asymptotic stability of the zero solution and integrability and boundedness of solutions in the case \(G(x,t)=0\). The boundedness of solutions at infinity is discussed for \(G(x,t)\neq 0\) too. Also, the authors provide two illustrative examples. Remarkably the given theorems are not only applicable to time-varying linear RIDEs, but also to nonlinear RIDEs depending on time.
Reviewer: Gaston Vergara-Hermosilla (Dublin)Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulseshttps://zbmath.org/1496.600642022-11-17T18:59:28.764376Z"Liu, Jiankang"https://zbmath.org/authors/?q=ai:liu.jiankang"Xu, Wei"https://zbmath.org/authors/?q=ai:xu.wei.1"Guo, Qin"https://zbmath.org/authors/?q=ai:guo.qin"Wang, Jinbin"https://zbmath.org/authors/?q=ai:wang.jinbinSummary: This paper aims at addressing the issue of a periodic averaging method for neutral stochastic functional differential equations with delayed impulses. Two periodic averaging theorems are presented and the approximate equivalence between the solutions to the original systems and those to the reduced averaged systems without impulses is proved. Further, we show a brief framework of extending our main results to Lévy case. At last, an example is given to demonstrate the procedure and validity of the proposed periodic averaging method.RETRACTED ARTICLE: Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motionshttps://zbmath.org/1496.600752022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo|sun.jianguo.1"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: By using new Schrödinger type inequalities appearing in [\textit{Z. Jiang} and \textit{F. M. Usó}, J. Inequal. Appl. 2016, Paper No. 233, 10 p. (2016; Zbl 1350.35070)], we study the existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions.
Editorial remark. This article has been retracted. According to the retraction notice [\textit{J. Sun} et al., J. Inequal. Appl. 2021, Paper No. 109, 1 p. (2021; Zbl 1496.60076)], ``the Editors-in-Chief have retracted this article because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Retraction note: ``Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions''https://zbmath.org/1496.600762022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo.1|sun.jianguo"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: The Editors-in-Chief have retracted this article [\textit{J. Sun}, ibid. 2018, Paper No. 100, 15 p. (2018; Zbl 1496.60075)] because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Stochastic model of innovation diffusion that takes into account the changes in the total market volumehttps://zbmath.org/1496.910522022-11-17T18:59:28.764376Z"Parphenova, Alena Yu."https://zbmath.org/authors/?q=ai:parphenova.alena-yu"Saraev, Leonid A."https://zbmath.org/authors/?q=ai:saraev.leonid-aleksandrovichSummary: The article proposes a stochastic mathematical model of the diffusion of consumer innovations, which takes into account changes over time in the total number of potential buyers of an innovative product. A stochastic differential equation is constructed for a random value of the number of consumers of an innovative product. The interaction of random changes in the number of consumers with changes in the total market volume of the product under consideration is investigated. Following the Euler-Maruyama method, an algorithm for the numerical solution of the stochastic differential equation for the diffusion of innovations is constructed. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for a random function of the number of consumers of an innovative product. A variant of the method for calculating the mathematical expectation of a random function of the number of consumers of an innovative product is developed and the corresponding differential equation is obtained. It is shown that the numerical solution of this equation and the average value of the function of the number of consumers calculated for all the implemented implementations of stochastic trajectories give practically the same results. Numerical analysis of the developed model showed that taking into account an external random disturbing factor in the stochastic model leads to significant deviations from the classical deterministic model of smooth market development with innovative goods.The impact of distributed time delay in a tumor-immune interaction systemhttps://zbmath.org/1496.920212022-11-17T18:59:28.764376Z"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Biswas, Santosh"https://zbmath.org/authors/?q=ai:biswas.santosh"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhasSummary: The impact of continuously distributed delay has been investigated in this paper to describe the interaction among tumor cells, tumor-specific CD8+T cells, helper T cells and immuno-stimulatory cytokine interleukin-2 (IL-2) through a system of coupled nonlinear ordinary differential equations. We analyze the qualitative properties of the model such as positivity of the solutions and the existence of biologically feasible equilibrium points. Next, we discuss the local asymptotic stability for the delayed and non-delayed system. Our model system experiences Hopf bifurcation with respect to the activation rate \(\lambda_1\) of tumor-specific CD8+T cells. The effect of continuously distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Our study reveals that the activation rate of CD8+T cells can prevent the oscillation of tumor-presence equilibria as well as tumor-free equilibria of the system. Then we performed some numerical results and interpret their biological implications to validate our analytical findings.Effects of delays in mathematical models of cancer chemotherapyhttps://zbmath.org/1496.920312022-11-17T18:59:28.764376Z"Abdulrashid, Ismail"https://zbmath.org/authors/?q=ai:abdulrashid.ismail"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Han, Xiaoying"https://zbmath.org/authors/?q=ai:han.xiaoyingSummary: Two mathematical models of chemotherapy cancer treatment are studied and compared, one modeling the chemotherapy agent as the predator and the other modeling the chemotherapy agent as the prey. In both models constant delay parameters are introduced to incorporate the time lapsed from the instant the chemotherapy agent is injected to the moment it starts to be effective. For each model, the existence and uniqueness of non-negative bounded solutions are first established. Then both local and Lyapunov stability for all steady states are investigated. In particular, sufficient conditions dependent of the delay parameters under which each steady state is asymptotically stable are constructed. Numerical simulations are presented to illustrate the theoretical results.Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycleshttps://zbmath.org/1496.920432022-11-17T18:59:28.764376Z"Li, Michael Y."https://zbmath.org/authors/?q=ai:li.michael-yi"Lin, Xihui"https://zbmath.org/authors/?q=ai:lin.xihui"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4(no abstract)Delay induced nonlinear dynamics of oxygen-plankton interactionshttps://zbmath.org/1496.920902022-11-17T18:59:28.764376Z"Gökçe, Aytül"https://zbmath.org/authors/?q=ai:gokce.aytul"Yazar, Samire"https://zbmath.org/authors/?q=ai:yazar.samire"Sekerci, Yadigar"https://zbmath.org/authors/?q=ai:sekerci.yadigarSummary: The present investigation deals with a generic oxygen-plankton model with constant time delays using the combinations of analytical and numerical methods. First, a two-component delayed model: the interaction between the concentration of dissolved oxygen and the density of the phytoplankton is examined in terms of the local stability and Hopf bifurcation analysis around the positive steady state. Then, a three-component model (oxygen-phytoplankton-zooplankton system) is investigated. The prime objective of this trio model is to explore how a constant time delay in growth response of phytoplankton and in the gestation time of zooplankton affects the dynamics of interaction between the concentration of oxygen and the density of plankton. The analytical and numerical investigations reveal that the positive steady states for both models are stable in the absence of time delays for a given hypothetical parameter space. Analysing eigenvalues of the characteristic equation which depends on the delay parameters, the conditions for linear stability and the existence of delay-induced Hopf bifurcation threshold are studied for all possible cases. As the delay rate increases, stability of coexistence state switches from stable to unstable. To support the analytical results, detailed numerical simulations are performed. Our findings show that time delay has a significant impact on the dynamics and may provide useful insights into underlying ecological oxygen-plankton interactions.Dynamics of fractional-order delay differential model of prey-predator system with Holling-type III and infection among predatorshttps://zbmath.org/1496.920952022-11-17T18:59:28.764376Z"Rihan, F. A."https://zbmath.org/authors/?q=ai:rihan.fathalla-a"Rajivganthi, C."https://zbmath.org/authors/?q=ai:rajivganthi.chinnathambiSummary: In this work, we study the dynamics of a fractional-order delay differential model of prey-predator system with Holling-type III and predator population is infected by an infectious disease. We use Laplace transform, Lyapunov functional, and stability criterion to establish new sufficient conditions that ensure the asymptotic stability of the steady states of the system. Existence of Hopf bifurcation is investigated. The model undergoes Hopf bifurcation, when the feedback time-delays passes through the critical values \(\tau_1^*\) and \(\tau_2^*\). Fractional-order improves the dynamics of the model; while time-delays play a considerable influence on the creation of Hopf bifurcation and stability of the system. Some numerical simulations are provided to validate the theoretical results.A time-delay model for prey-predator growth with stage structurehttps://zbmath.org/1496.920972022-11-17T18:59:28.764376Z"Saito, Yasuhisa"https://zbmath.org/authors/?q=ai:saito.yasuhisa"Takeuchi, Yasuhiro"https://zbmath.org/authors/?q=ai:takeuchi.yasuhiroSummary: This paper studies a stage-structured prey-predator model based on the model proposed by \textit{W. G. Aiello} and \textit{H. I. Freedman} [Math. Biosci. 101, No. 2, 139--153 (1990; Zbl 0719.92017)], where the stage structure was introduced by considering a time to maturity for the predator as a time delay. We establish conditions for the local asymptotic stability and global attractivity of an interior equilibrium of the model.Stability in distribution of a stochastic predator-prey system with S-type distributed time delayshttps://zbmath.org/1496.920982022-11-17T18:59:28.764376Z"Wang, Sheng"https://zbmath.org/authors/?q=ai:wang.sheng"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixin"Wei, Tengda"https://zbmath.org/authors/?q=ai:wei.tengda"Wang, Linshan"https://zbmath.org/authors/?q=ai:wang.linshanThe paper studies stability of the following Lotka-Volterra stochastic system
\[
d x_1(t)=x_1(t)\left(r_1-a_{11}x_1(t)-\int_{-\tau_{11}}^0 x_1(t+\theta) d \mu_{11}(\theta)-a_{12}x_2(t)\right.
\]
\[
\left.-\int_{-\tau_{12}}^0 x_2(t+\theta) d \mu_{12}(\theta)\right)dt+\sigma_1x_1(t)d B_1(t)
\]
\[
d x_2(t)=x_2(t)\left(r_2-a_{22}x_2(t)-\int_{-\tau_{22}}^0 x_2(t+\theta) d \mu_{22}(\theta)-a_{21}x_1(t)\right.
\]
\[
\left.-\int_{-\tau_{21}}^0 x_1(t+\theta) d \mu_{21}(\theta)\right)dt+\sigma_2x_2(t)d B_2(t),
\]
where \(B_i(t)\) are mutually independent standard Wiener processes, \(\tau_{ij}>0\).
Reviewer: Leonid Berezanski (Be'er Sheva)Stability analysis of a fractional-order novel hepatitis B virus model with immune delay based on Caputo-Fabrizio derivativehttps://zbmath.org/1496.921112022-11-17T18:59:28.764376Z"Gao, Fei"https://zbmath.org/authors/?q=ai:gao.fei"Li, Xiling"https://zbmath.org/authors/?q=ai:li.xiling"Li, Wenqin"https://zbmath.org/authors/?q=ai:li.wenqin"Zhou, Xianjin"https://zbmath.org/authors/?q=ai:zhou.xianjinSummary: In mathematical epidemiology, mathematical models play a vital role in understanding the dynamics of infectious diseases. Therefore, in this paper, a novel mathematical model for the hepatitis B virus (HBV) based on the Caputo-Fabrizio fractional derivative with immune delay is introduced, while taking care of the dimensional consistency of the proposed model. Initially, the existence and uniqueness of the model solutions are proved by Laplace transform and the fixed point theorem. The positivity and boundedness of the solutions are also discussed. Sumudu transform and Picard iteration were used to analyze the stability and iterative solution of the fractional order model of HBV. Further, using the stability theory of fractional order system, the stability and bifurcation of equilibrium point are discussed. Finally, results are presented for different values of the fractional parameter.Global dynamics of a two-strain disease model with latency and saturating incidence ratehttps://zbmath.org/1496.921242022-11-17T18:59:28.764376Z"Rahman, S. M. Ashrafur"https://zbmath.org/authors/?q=ai:ashrafur-rahman.s-m"Zou, Xingfu"https://zbmath.org/authors/?q=ai:zou.xingfu(no abstract)Global dynamics of a time-delayed dengue transmission modelhttps://zbmath.org/1496.921252022-11-17T18:59:28.764376Z"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen|wang.zhen.2|wang.zhen.6|wang.zhen.1|wang.zhen.7|wang.zhen.3|wang.zhen.5"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiao-qiang|zhao.xiaoqiangSummary: We present a time-delayed dengue transmission model. We first introduce the basic reproduction number for this model and then show that the disease persists when \(\mathcal R_0>1\). It is also shown that the disease will die out if \(\mathcal R_0<1\), provided that the invasion intensity is not strong. We further establish a set of sufficient conditions for the global attractivity of the endemic equilibrium by the method of fluctuations. Numerical simulations are performed to illustrate our analytic results.An investigation of delay induced stability transition in nutrient-plankton systemshttps://zbmath.org/1496.921312022-11-17T18:59:28.764376Z"Thakur, Nilesh Kumar"https://zbmath.org/authors/?q=ai:thakur.nilesh-kumar"Ojha, Archana"https://zbmath.org/authors/?q=ai:ojha.archana"Tiwari, Pankaj Kumar"https://zbmath.org/authors/?q=ai:tiwari.pankaj-kumar"Upadhyay, Ranjit Kumar"https://zbmath.org/authors/?q=ai:kumar-upadhyay.ranjitSummary: In this paper, a nutrient-plankton interaction model is proposed to explore the characteristic of plankton system in the presence of toxic phytoplankton and discrete time delay. Anti-predator efforts of phytoplankton by toxin liberation act as a prominent role on plankton dynamics. Toxicity controls the system dynamics and reduces the grazing rate of zooplankton. The toxic substance released by phytoplankton is not an instantaneous process, it requires some time for maturity. So, a discrete time delay is incorporated in the toxin liberation by the phytoplankton. The choice of functional response is important to understand the toxin liberation and it depends on the nonlinearity of the system, which follows the Monod-Haldane type functional response. Theoretically, we have studied the boundedness condition along with all the feasible equilibria analysis and stability criteria of delay free system. We have explored the local stability conditions of delayed system. The existence criterion for stability and direction of Hopf-bifurcation are also derived by using the theory of normal form and center manifold arguments. The essential features of time delay are studied by time series, phase portrait and bifurcation diagram. We perform a global sensitivity analysis to identify the important parameters of the model having a significant impact on zooplankton. Our numerical investigation reveals that the toxin liberation delay switches the stability of the system from stable to limit cycle and after a certain interval chaotic dynamics is observed. High rate of toxic substances production shows extinction of zooplankton. Further, the negative and positive impacts of other control parameters are studied. Moreover, to support the occurrence of chaos, the Poincaré map is drawn and the maximum Lyapunov exponents are also computed.Finite-time \(H_\infty\) control of linear singular fractional differential equations with time-varying delayhttps://zbmath.org/1496.931052022-11-17T18:59:28.764376Z"Niamsup, Piyapong"https://zbmath.org/authors/?q=ai:niamsup.piyapong"Thanh, Nguyen T."https://zbmath.org/authors/?q=ai:thanh.nguyen-trung|thanh.nguyen-thi"Phat, Vu N."https://zbmath.org/authors/?q=ai:vu-ngoc-phat.Summary: In this paper, we propose an efficient analytical approach based on fractional calculus and singularity value theory to designing the finite-time \(H_\infty\) controller for linear singular fractional differential equations with time-varying delay. By introducing new fractional-order \(H_\infty\) norm, the state feedback controller is designed to guarantee that the closed-loop system is singular, impulse-free and finite-time stable with prescribed \(H_\infty\) performance. New sufficient conditions for designing the \(H_\infty\) finite-time controller are presented. The results of this paper improve the corresponding ones of integer-order singular systems with time-varying delay. Finally, a numerical example demonstrates the validity and effectiveness of the proposed theoretical results.