Recent zbMATH articles in MSC 34Khttps://zbmath.org/atom/cc/34K2024-07-25T18:28:20.333415ZWerkzeugAn elementary inequality for dissipative Caputo fractional differential equationshttps://zbmath.org/1537.340132024-07-25T18:28:20.333415Z"Kloeden, Peter E."https://zbmath.org/authors/?q=ai:kloeden.peter-eris(no abstract)Delay differential equations with differentiable solution operators on open domains in \(C((- \infty, 0], \mathbb{R}^n)\) and processes for Volterra integro-differential equationshttps://zbmath.org/1537.340762024-07-25T18:28:20.333415Z"Walther, H.-O."https://zbmath.org/authors/?q=ai:walther.hans-ottoThe paper first considers autonomous differential delay equations of the form
\[
\dot x(t) = f(x_t), \; x(t) \in \mathbb{R}^n,\tag{1}
\]
where in this case \(x_t\) denotes the restriction of \( x(t + \cdot) \) to \((-\infty, 0]\), i.e., an infinite past history. The state space is the Fréchet space \(C^0((-\infty, 0], \mathbb{R}^n)\) with the topology of locally uniform convergence. In this context, two possible notions of differentiability (in the sense of Fréchet, and in the sense of Michal and Bastiani) are considered. In Section 3, a Picard-Lindelöf type ansatz for the construction of solutions is taken, rephrasing the differential equation as integral equation, which yields a contracting mapping for sufficiently short time intervals. Roughly speaking, each of the two mentioned smoothness types of the nonlinearity \(f\) is shown to yield the corresponding smoothness of the associated substitution operator (Section 2), and then of the time-\(t\)-maps defined by solutions (Section 4). Differentiability here is obtained from a parametrized contraction theorem (Theorem 8.7 in the appendix) which gives differentiability of the fixed point w.r. to the parameter. The results necessary for this approach are quoted (in the appendix) from an earlier paper of the author on differentiability in Fréchet spaces. In Section 5, the derivatives thus obtained are shown to solve a (differential) variational equation, which is achieved by first showing that they satisfy a corresponding integral equation.
Section 6 extends the results to non-autonomous systems for of the type
\[
\dot x(t) = g(t, x_t) \tag{2}
\]
with \( n\) real variables, relating these to autonomous systems for \( n + 1\) variables. Section 7 shows how the results apply, in particular, to Volterra integral equations of the type \(\displaystyle \dot x(t) = \int_0^t k(t,s) h(x(s))\, ds. \) These are shown to be equivalent to equation (2), as considered in Section 6, with appropriate choice of \(g\).
Reviewer: Bernhard Lani-Wayda (Gießen)On the oscillation and asymptotic behavior of solutions of third order nonlinear differential equations with mixed nonlinear neutral termshttps://zbmath.org/1537.340772024-07-25T18:28:20.333415Z"Salem, Shaimaa"https://zbmath.org/authors/?q=ai:salem.shaimaa"El-Sheikh, Mohamed M. A."https://zbmath.org/authors/?q=ai:el-sheikh.mohamed-m-a"Hassan, Ahmed Mohamed"https://zbmath.org/authors/?q=ai:hassan.ahmed-mohamedSummary: This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order nonlinear neutral differential equations with a middle term and mixed nonlinear neutral terms in the case of the canonical operator. We establish several oscillation criteria that guarantee that all solutions are oscillatory or converge to zero. The given results are obtained by applying the comparison method, the Riccati transformation and the integral averaging technique. The results improve significantly and extend existing ones in the literature. Finally, illustrative examples are given.Periodic solutions to reversible second order autonomous systems with commensurate delayshttps://zbmath.org/1537.340782024-07-25T18:28:20.333415Z"Balanov, Zalman"https://zbmath.org/authors/?q=ai:balanov.zalman-i"Chen, Fulai"https://zbmath.org/authors/?q=ai:chen.fulai"Guo, Jing"https://zbmath.org/authors/?q=ai:guo.jing"Krawcewicz, Wieslaw"https://zbmath.org/authors/?q=ai:krawcewicz.wieslaw-zSummary: Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer \(O(2) \times \Gamma \times \mathbb{Z}_2\)-equivariant degree theory, where \(O(2)\) is related to the reversing symmetry, \( \Gamma\) reflects the symmetric character of the coupling in the corresponding network and \(\mathbb{Z}_2\) is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with \(\Gamma = D_6\) - the dihedral group of order 12.Positive periodic solutions of a leukopoiesis model with iterative termshttps://zbmath.org/1537.340792024-07-25T18:28:20.333415Z"Khemis, Marwa"https://zbmath.org/authors/?q=ai:khemis.marwa"Bouakkaz, Ahlème"https://zbmath.org/authors/?q=ai:bouakkaz.ahleme"Khemis, Rabah"https://zbmath.org/authors/?q=ai:khemis.rabahIn this paper, a first-order differential equation as a leukopoiesis model with a harvesting strategy that the production and harvesting terms include iterative terms is investigated:
\[
\varphi'(t) = -a(t)\varphi(t) + \sum^N_{i=1}b_i(t)\frac{(\varphi^{[i]}(t))^m}{1+(\varphi^{[i]}(t))^n} - h(t, \varphi(t), \varphi^{[2]}(t), \ldots, \varphi^{[N]}(t)),
\]
where \(\varphi^{[2]}(t)= \varphi(\varphi(t))\) and \(\varphi^{[i]}(t)\) denotes the compound of \(\varphi\) with itself \(i\) times, \(a\) and \(b_i\) are continuous \(\omega\)-periodic functions, and \(h\) is the harvesting that is \(\omega\)-periodic with respect to the first argument and globally Lipschitz with respect to the other arguments. Applying the Krasnoselskii fixed point theorem with the aid of some Green's function properties the authors prove the existence of at least one positive periodic solution. Under another condition, the Banach fixed point theorem is used to prove the existence of a unique positive periodic solution. Moreover, the continuous dependence on parameters is also guaranteed.
Reviewer: Zhanyuan Hou (London)Asymptotics of self-oscillations in chains of systems of nonlinear equationshttps://zbmath.org/1537.340802024-07-25T18:28:20.333415Z"Kashchenko, Sergey A."https://zbmath.org/authors/?q=ai:kashchenko.sergey-aleksandrovichSummary: We study the local dynamics of chains of coupled nonlinear systems of second-order ordinary differential equations of diffusion-difference type. The main assumption is that the number of elements of chains is large enough. This condition allows us to pass to the problem with a continuous spatial variable. Critical cases have been considered while studying the stability of the equilibrum state. It is shown that all these cases have infinite dimension. The research technique is based on the development and application of special methods for construction of normal forms. Among the main results of the paper, we include the creation of new nonlinear boundary value problems of parabolic type, whose nonlocal dynamics describes the local behavior of solutions of the original system.On the qualitative analysis of nonlinear \(q\)-fractional delay descriptor systemshttps://zbmath.org/1537.340812024-07-25T18:28:20.333415Z"Yiğit, Abdullah"https://zbmath.org/authors/?q=ai:yigit.abdullahSummary: In this manuscript, we obtain some sufficient conditions for a nonlinear \(q\) fractional integro singular system with constant delays to be asymptotically admissible and a nonlinear \(q\) fractional non-singular system to be asymptotically stable. We use Lyapunov-Krasovskii functionals and some inequalities to obtain these conditions. At the same time, we present some numerical examples that confirm the sufficient conditions we obtained theoretically, with their annotated solutions and graphs.Multistability analysis of octonion-valued neural networks with time-varying delayshttps://zbmath.org/1537.340822024-07-25T18:28:20.333415Z"Chouhan, Shiv Shankar"https://zbmath.org/authors/?q=ai:chouhan.shiv-shankar"Kumar, Rakesh"https://zbmath.org/authors/?q=ai:kumar.rakesh.7"Sarkar, Shreemoyee"https://zbmath.org/authors/?q=ai:sarkar.shreemoyee"Das, Subir"https://zbmath.org/authors/?q=ai:das.subirSummary: In this article, the multistability analysis is studied for \(n\)-dimensional octonion valued neural networks (OVNNs) with time-varying delays for a general class of activation functions. Firstly, OVNNs are decomposed into eight real-valued systems, and then based on geometrical properties of activation functions, \(3^{8n}\) disjoint regions are constructed in \(\mathbb{O}^n\). Then, by using the inequality technique, several sufficient conditions are obtained to ensure the existence of \(3^{8n}\) equilibrium points of the system, each of which is located in one of the regions, and \(2^{8n}\) of them are locally exponentially stable. Moreover positively invariant sets are also estimated in this scientific contribution. Two numerical examples are provided to illustrate the effectiveness of the obtained results. Especially, the numerical Example 2 demonstrates that the designed OVNNs work efficiently on storing and retrieving the truecolor images.Approximate controllability of impulsive integrodifferential equations with state-dependent delayhttps://zbmath.org/1537.340832024-07-25T18:28:20.333415Z"Fall, M."https://zbmath.org/authors/?q=ai:fall.mandiaye|fall.magatte|fall.moussa|fall.marfall-n|fall.mouhammed-moustapha|fall.m-l|fall.mbarack|fall.mouhamed-moustapha"Dehigbe, B."https://zbmath.org/authors/?q=ai:dehigbe.bertin"Ezzinbi, K."https://zbmath.org/authors/?q=ai:ezzinbi.khalil"Diop, M. A."https://zbmath.org/authors/?q=ai:diop.mamadou-abdul|diop.mamadou-abdoulSummary: This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer's resolvent operator theory and Schauder's fixed point theorem. We give an example at the end to ensure the compatibility of the results.A new approach to multi-delay matrix valued fractional linear differential equations with constant coefficientshttps://zbmath.org/1537.340842024-07-25T18:28:20.333415Z"Neto, Antônio Francisco"https://zbmath.org/authors/?q=ai:neto.antonio-francisco(no abstract)Stochastic stability of solutions for a fourth-order stochastic differential equation with constant delayhttps://zbmath.org/1537.340852024-07-25T18:28:20.333415Z"Mahmoud, Ayman M."https://zbmath.org/authors/?q=ai:mahmoud.ayman-mohammed"Adewumi, Adebayo O."https://zbmath.org/authors/?q=ai:adewumi.adebayo-olusegun"Ademola, Adeleke T."https://zbmath.org/authors/?q=ai:ademola.adeleke-timothy(no abstract)Bazykin's predator-prey model includes a dynamical analysis of a Caputo fractional order delay fear and the effect of the population-based mortality rate on the growth of predatorshttps://zbmath.org/1537.340862024-07-25T18:28:20.333415Z"Kumar, G. Ranjith"https://zbmath.org/authors/?q=ai:kumar.g-ranjith"Ramesh, K."https://zbmath.org/authors/?q=ai:ramesh.katta|ramesh.k-t|ramesh.kasilingam|ramesh.k-s|ramesh.k-v|ramesh.kiran"Khan, Aziz"https://zbmath.org/authors/?q=ai:khan.aziz-ullah"Lakshminarayan, K."https://zbmath.org/authors/?q=ai:lakshminarayan.kamakshi|lakshminarayan.k-j"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabetSummary: In this paper, we investigate a system of two differential equations of fractional order for the fear effect in prey-predator interactions, in which the density of predators controls the mortality pace of the prey population. The non-integer order differential equation is interpreted in terms of the Caputo derivative, and the development of the non-integer order scheme is described in terms of the influence of memory on population increase. The primary goal of existing research is to explore how the changing aspects of the current scheme are impacted by various types of parameters, including time delay, fear effect, and fractional order. The solutions' positivity, existence-uniqueness, and boundedness are established with precise mathematical conclusions. The requirements necessary for the local asymptotic stability of different equilibrium points and the global stability of coexistence equilibrium are established. Hopf bifurcation occurs in the system at various delay times. The model's fractional-order derivatives enhance the model behaviours and provide stability findings for the solutions. We have observed that fractional order plays an important role in population dynamics. Also, Hopf bifurcation for the proposed system have been observed for certain values of order of derivatives. Thus, the stability conditions of the equilibrium points may be changed by changing the order of the derivatives without changing other parametric values. Finally, a numerical simulation is run to verify our conclusions.On the Bari basis property for even-order differential operators with involutionhttps://zbmath.org/1537.340872024-07-25T18:28:20.333415Z"Polyakov, Dmitry M."https://zbmath.org/authors/?q=ai:polyakov.dmitrii-mikhailovichSummary: By using the method of similar operators we study even-order differential operators with involution. The domain of these operators are defined by periodic and antiperiodic boundary conditions. We obtain estimates for spectral projections and we prove the Bari basis property for the system of eigenfunctions and associated functions.Representations of abstract resolvent families on time scales via Laplace transformhttps://zbmath.org/1537.340922024-07-25T18:28:20.333415Z"Grau, Rogelio"https://zbmath.org/authors/?q=ai:grau.rogelio"Pereira, Aldo"https://zbmath.org/authors/?q=ai:pereira.aldo(no abstract)Abstract action spaces and their topological and dynamic propertieshttps://zbmath.org/1537.350152024-07-25T18:28:20.333415Z"Rossi, Riccarda"https://zbmath.org/authors/?q=ai:rossi.riccarda"Savaré, Giuseppe"https://zbmath.org/authors/?q=ai:savare.giuseppeSummary: We introduce the concept of \textit{action space}, a set \(\boldsymbol{X}\) endowed with an action cost \(\mathsf{a} : (0, +\infty) \times \boldsymbol{X} \times \boldsymbol{X} \to [0,+\infty)\) satisfying suitable axioms, which turn out to provide a 'dynamic' generalization of the classical notion of metric space. Action costs naturally arise as dissipation terms featuring in the Minimizing Movement scheme for gradient flows, which can then be settled in general action spaces.
As in the case of metric spaces, we will show that action costs induce an intrinsic topological and metric structure on \(\boldsymbol{X}\). Moreover, we introduce the related action functional on paths in \(\boldsymbol{X}\), investigate the properties of curves of finite action, and discuss their absolute continuity. Finally, under a condition akin to the \textit{approximate mid-point property} for metric spaces, we provide a dynamic interpretation of action costs.Hardy inequalities for magnetic \(p\)-Laplacianshttps://zbmath.org/1537.350172024-07-25T18:28:20.333415Z"Cazacu, Cristian"https://zbmath.org/authors/?q=ai:cazacu.cristian-m"Krejčiřík, David"https://zbmath.org/authors/?q=ai:krejcirik.david"Lam, Nguyen"https://zbmath.org/authors/?q=ai:lam.nguyen"Laptev, Ari"https://zbmath.org/authors/?q=ai:laptev.ari\noindent Let \(A:\mathbb{R}^d\to\mathbb{R}^d\) be a smooth magnetic potential, which can be interpreted as a \(1\)-differential form. The magnetic \(p\)-Laplacian is formally defined on \(C_c^{\infty}\left(\mathbb{R}^d\right)\) by
\[
\Delta_{A, p} u=\operatorname{div}_A\left(\left|\nabla_A u\right|^{p-2} \nabla_A u\right),
\]
where the magnetic gradient \(\nabla_A \) and magnetic divergence \( \operatorname{div}_A F\) are given by
\[
\nabla_A u=\nabla u+i A(x) u ; \quad \operatorname{div}_A F=\operatorname{div} F+i A \cdot F,
\]
for any smooth vector field \(F: \mathbb{R}^d \rightarrow \mathbb{C}^d\). The associated quadratic form \(h_{A, p}\) of the Dirichlet magnetic \(p\)-Laplacian \(\Delta_{A, p}\) with its form domain \(\mathcal{D}\left(h_{A, p}\right)\) is given by
\[
h_{A, p}[u]=\int_{\mathbb{R}^d}\left|\nabla_A u\right|^p \mathrm{dx}=\int_{\mathbb{R}^d}|\nabla u+i A(x) u|^p \mathrm{dx}, \quad \forall u \in \mathcal{D}\left(h_{A, p}\right)=\overline{C_c^{\infty}\left(\mathbb{R}^d\right)}^{\|\cdot\|},
\]
where the norm \(\|\cdot\|\) is given by
\[
\|u\|=\left(h_{A, p}[u]+\|u\|_{L^p(\mathbb{R}^d)}^p\right)^{\frac{1}{p}} .
\]
The authors prove the following improved Hardy inequalities for the magnetic \(p\)-Laplacian: Let \(B:\mathbb{R}^d\to\mathbb{R}^d\times\mathbb{R}^d\) be a smooth and closed magnetic field with \(B\not=0\).
\begin{itemize}
\item[1.] If \(p \geqslant d\), then there exists a constant \(C_{B, p, d}>0\) such that for any magnetic potential \(A\) with \(\mathrm{d} A=B\), where \(\mathrm{d}\) is the exterior derivative, we have
\[
\int_{\mathbb{R}^d}\left|\nabla_A u\right|^p \mathrm{dx} \geqslant C_{B, p, d} \int_{\mathbb{R}^d} \rho(x)|u|^p \mathrm{dx}, \quad \forall u \in \mathcal{D}\left(h_{A, p}\right),
\]
where
\[
\rho(x)=\frac{1}{|x|^d\left(|\log | x||^p+|x|^{p-d}\right)}.
\]
\item[2.] Assume \(2 \leqslant p<d\). Then, there are positive constants \(c(p)\) and \(C_{B, p, d}\) such that for any vector field \(A\) with \(\mathrm{d}A=B\) we have following:
\begin{itemize}
\item [(a)] for any \(u \in \mathcal{D}\left(h_{A, p}\right)\)
\[
\int_{\mathbb{R}^d}\left|\nabla_A u\right|^p \mathrm{dx}-\mu_{p, d} \int_{\mathbb{R}^d} \frac{|u|^p}{|x|^p} \mathrm{dx} \geqslant c(p) \int_{\mathbb{R}^d}\left|\nabla_A\left(u|x|^{\frac{d-p}{p}}\right)\right|^p|x|^{p-d} \mathrm{dx},
\]
where \(\mu_{p, d} =(\frac{d-p}{p})^{p}\) is the classical optimal constant for the \(L^{p}\)-Hardy inequality, and \(c(p)\) is explicitly given by
\[
c(p)=\inf _{(s, t) \in \mathbb{R}^2 \backslash\{(0,0)\}} \frac{\left[t^2+s^2+2 s+1\right]^{\frac{p}{2}}-1-p s}{\left[t^2+s^2\right]^{\frac{p}{2}}} \in(0,1].
\]
\item [(b)] for any \(u \in \mathcal{D}\left(h_{A, p}\right)\)
\[
\int_{\mathbb{R}^d}\left|\nabla_A u\right|^p \mathrm{dx}-\mu_{p, d} \int_{\mathbb{R}^d} \frac{|u|^p}{|x|^p} \mathrm{dx} \geqslant C_{B, p, d} \int_{\mathbb{R}^d} \rho(x)|u|^p \mathrm{dx}
\]
where
\[
\rho(x)=\frac{1}{|x|^p\left(1+\left.|\log | x|\right|^p\right)}.
\]
\end{itemize}
\item[3.] If \(d=2\), \(1<p<2\), and \(\beta\notin\mathbb{Z}\) then there exists a constant
\[
\lambda_\beta(p)>\left(\frac{2-p}{p}\right)^p
\]
such that
\[
\int_{\mathbb{R}^2}\left|\nabla_{A_\beta} u\right|^p \mathrm{dx} \geqslant \lambda_\beta(p) \int_{\mathbb{R}^2} \frac{|u|^p}{|x|^p} \mathrm{dx}, \quad \forall u \in C_c^{\infty}\left(\mathbb{R}^2\right),
\]
where \(A_\beta\) is the Aharonov-Bohm (AB) potential given by
\[
A_\beta(x)=\beta \frac{\left(x_2,-x_1\right)}{|x|^2}, \quad \beta \in \mathbb{R}.
\]
\end{itemize}
Reviewer: José Francisco De Oliveira (Teresina)Threshold dynamics for diffusive age-structured model over unbounded domains: age-dependent death and diffusion rateshttps://zbmath.org/1537.350602024-07-25T18:28:20.333415Z"AlJararha, Mohammadkheer"https://zbmath.org/authors/?q=ai:al-jararha.mohammadkheer-mSummary: The global dynamics of the typical age-structured model with age-dependent mortality and diffusion rates on unbounded domains have been established. On the one hand, we showed that a positive and constant state solution of the mature population is globally asymptotically stable with respect to the compact-open topology; on the other hand, we showed that the trivial solution is globally asymptotically stable with respect to the usual supremum norm. As an application of our result, we applied the result to birth functions appearing in biology. In addition to the theoretical results, we also present a numerical simulation.Long-time asymptotics for Toda shock waves in the modulation regionhttps://zbmath.org/1537.370632024-07-25T18:28:20.333415Z"Egorova, Iryna"https://zbmath.org/authors/?q=ai:egorova.iryna"Michor, Johanna"https://zbmath.org/authors/?q=ai:michor.johanna"Pryimak, Anton"https://zbmath.org/authors/?q=ai:pryimak.anton"Teschl, Gerald"https://zbmath.org/authors/?q=ai:teschl.geraldThe long-time asymptotics of Toda shock waves is studied. Proofs of asymptotic expansions of solutions are given, and the influence of resonances and eigenvalues on the leading terms of asymptotics is analyzed.
Reviewer: Piotr Biler (Wrocław)New approach on the study of operator matrixhttps://zbmath.org/1537.390152024-07-25T18:28:20.333415Z"Marzouk, Ines"https://zbmath.org/authors/?q=ai:marzouk.ines"Walha, Ines"https://zbmath.org/authors/?q=ai:walha.inesSummary: In the present paper, a new technique is presented to study the problem of invertibility of unbounded block \(3\times 3\) operator matrices defined with diagonal domain. Sufficient criteria are established to guarantee our interest and to prove some interaction between such a model of an operator matrix and its diagonal operator entries. The effectiveness of the proposed new technique is shown by a physical example of an integro differential equation named the neutron transport equation with partly elastic collision operators. In particular, the obtained results answer the question in [\textit{H. Zguitti}, Mediterr. J. Math. 10, No. 3, 1497--1507 (2013; Zbl 1304.47005)] and the conjecture in [\textit{A. Bahloul} and \textit{I. Walha}, Numer. Funct. Anal. Optim. 43, No. 16, 1836--1847 (2022; Zbl 07618159)].Master equations for finite state mean field games with nonlinear activationshttps://zbmath.org/1537.490352024-07-25T18:28:20.333415Z"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo"Li, Wuchen"https://zbmath.org/authors/?q=ai:li.wuchenSummary: We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on graphs induced by finite-state reversible continuous time Markov chains. In these graphs, each edge is weighted by the transition probability and invariant measure of the original process. Using these controlled dynamics on the graph and the dynamic programming principle for the value function, we derive several key components: the mean field game systems, the functional Hamilton-Jacobi equations, and the master equations on a finite probability space for potential mean field games. The existence and uniqueness of solutions to the potential mean field game system are ensured through a convex optimization reformulation in terms of the density-flux pair. We also derive variational principles for the master equations of both non-potential games and mixed games on a continuous state space. Finally, we offer several concrete examples of discrete mean field game dynamics on a two-point space, complete with closed-formula solutions. These examples include discrete Wasserstein distances, mean field planning, and potential mean field games.Fractional SDEs with stochastic forcing: existence, uniqueness, and approximationhttps://zbmath.org/1537.600682024-07-25T18:28:20.333415Z"Kubilius, Kęstutis"https://zbmath.org/authors/?q=ai:kubilius.kestutisThe article delves into fractional stochastic differential equations (FSDEs) with stochastic forcing, which extend traditional FSDEs by integrating a stochastic forcing term. It establishes conditions for the existence and uniqueness of solutions for such equations, while also analyzing the convergence rate of the implicit Euler approximation scheme. These equations are particularly useful in modeling FSDEs with a permeable wall.
Additionally, the article highlights the importance of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) in various scientific domains, mainly because of their introduction of memory elements. Numerous authors have explored the existence and uniqueness of solutions for SDEs driven by fBm, with growing interest in numerical methods for solving such equations.
Interest in FSDEs with a stochastic forcing term was initially sparked by \textit{T. Vojta} et al. [Reflected fractional Brownian motion in one and higher dimensions, Phys. Rev. E, 102:032108 (2020; \url{doi:10.1103/PhysRevE.102.032108})], who introduced repulsive forces to the recursion relation, leading to the development of a ``soft wall'' model. The article further expands on this model by introducing FSDEs with stochastic forcing terms defined by specific equations.
The primary objective of the paper is to establish conditions for the unique solution of FSDEs with stochastic forcing and to scrutinize the convergence rate of the implicit Euler approximation method. The proof of existence and uniqueness relies on estimates from prior literature and the application of the implicit Picard iteration procedure.
Reviewer: Gerardo Hernandez-del-Valle (Ciudad de México)The implicit multistep block method with an off-step point for initial value problems of neutral delay Volterra integro-differential equationshttps://zbmath.org/1537.651992024-07-25T18:28:20.333415Z"Ismail, N. I. N."https://zbmath.org/authors/?q=ai:ismail.nur-inshirah-naqiah"Majid, Z. A."https://zbmath.org/authors/?q=ai:majid.zanariah-abdulSummary: The aim of this manuscript is to solve the initial-value problems of neutral delay Volterra integro-differential equations with constant or proportional delays. Hence, a proposed hybrid technique named as an implicit multistep block method with an off-step point (1OBM4) is formulated for the numerical solution of NDVIDE. A LMM associated with an off-point is known as hybrid LMM. The proposed technique, 1OBM4, attempts to solve the problem synchronously in a block manner. Moreover, a Taylor expansion is implemented to develop 1OBM4 in predictor-corrector mode. Two different approaches are presented in order to solve both integral and differential parts of the problem. Some analyses on 1OBM4 are considered in terms of order and convergence of the method. A stability polynomial is also obtained for the stability regions to be constructed. In the last section, some numerical results are demonstrated to show the applicability of 1OBM4 in solving NDVIDE with constant or proportional delays.Numerical solution on neutral delay Volterra integro-differential equationhttps://zbmath.org/1537.652002024-07-25T18:28:20.333415Z"Ismail, Nur Inshirah Naqiah"https://zbmath.org/authors/?q=ai:ismail.nur-inshirah-naqiah"Majid, Zanariah Abdul"https://zbmath.org/authors/?q=ai:majid.zanariah-abdulSummary: In this research, the constant type of neutral delay Volterra integro-differential equations (NDVIDEs) are currently being resolved by applying the proposed technique in numerical analysis namely, two-point two off-step point block multistep method (2OBM4). This new technique is being applied in solving NDVIDE, identified as a hybrid block multistep method, developed using Taylor series interpolating polynomials. To complete the algorithm, two alternative numerical approaches are introduced to resolve the integral and differential parts of the problems. Note that the differentiation is approximated by the divided difference formula while the integration is interpolated using composite Simpson's rule. The proposed method has been analysed thoroughly in terms of its order, consistency, zero stability and convergence. The suitable stability region for 2OBM4 in solving NDVIDE has been constructed and the stability region is built based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed 2OBM4.Dissipativity and contractivity of the second-order averaged \(L1\) method for fractional Volterra functional differential equationshttps://zbmath.org/1537.652022024-07-25T18:28:20.333415Z"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yin"Xiao, Aiguo"https://zbmath.org/authors/?q=ai:xiao.aiguoSummary: This paper focuses on the dissipativity and contractivity of a second-order numerical method for fractional Volterra functional differential equations (F-VFDEs). Firstly, an averaged \(L1\) method for the initial value problem of F-VFDEs is presented based on the averaged \(L1\) approximation for Caputo fractional derivative together with an appropriate piecewise interpolation operator for the functional term. Then the averaged \(L1\) method is proved to be dissipative with an absorbing set and contractive with an algebraic decay rate. Finally, the numerical experiments further confirm the theoretical results.Analyzing stability of equilibrium points in impulsive neural network models involving generalized piecewise alternately advanced and retarded argumenthttps://zbmath.org/1537.920062024-07-25T18:28:20.333415Z"Chiu, Kuo-Shou"https://zbmath.org/authors/?q=ai:chiu.kuo-shouSummary: In this paper, we investigate the models of the impulsive cellular neural network with piecewise alternately advanced and retarded argument of generalized argument (in short IDEPCAG). To ensure the existence, uniqueness and global exponential stability of the equilibrium state, several new sufficient conditions are obtained. The method is based on utilizing Banach's fixed point theorem and a new IDEPCAG's Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of proposed results.A new approach to stability analysis for stochastic Hopfield neural networks with time delayshttps://zbmath.org/1537.920122024-07-25T18:28:20.333415Z"Lv, Xiang"https://zbmath.org/authors/?q=ai:lv.xiang|lv.xiang.1Editorial remark: No review copy delivered.Stability and bifurcation analysis of Alzheimer's disease model with diffusion and three delayshttps://zbmath.org/1537.920282024-07-25T18:28:20.333415Z"Li, Huixia"https://zbmath.org/authors/?q=ai:li.huixia"Zhao, Hongyong"https://zbmath.org/authors/?q=ai:zhao.hongyong(no abstract)Kinetic analysis of p53 gene network with time delays and PIDDhttps://zbmath.org/1537.920452024-07-25T18:28:20.333415Z"Huo, Ruimin"https://zbmath.org/authors/?q=ai:huo.ruimin"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nan"Yang, Hongli"https://zbmath.org/authors/?q=ai:yang.hongli"Yang, Liangui"https://zbmath.org/authors/?q=ai:yang.lianguiSummary: p53 kinetics plays a key role in regulating cell fate. Based on the p53 gene regulatory network composed by the core regulatory factors ATM, Mdm2, Wip1, and PIDD, the effect of the delays in the process of transcription and translation of Mdm2 and Wip1 on the dynamics of p53 is studied theoretically and numerically. The results show that these two time delays can affect the stability of the positive equilibrium. With the increase of delays, the dynamics of p53 presents an oscillating state. Further, we also study the effects of PIDD and chemotherapeutic drug etoposide on the kinetics of p53. The model indicates that (i) PIDD low-level expression does not significantly affect p53 oscillatory behavior, but high-level expression could induce two-phase kinetics of p53; (ii) Too high and too low concentration of etoposide is not conducive to p53 oscillation. These results are in good agreement with experimental findings. Finally, we consider the influence of internal noise on the system through binomial \(\tau\)-leap algorithm. Stochastic simulations reveal that high-intensity noise completely destroys p53 dynamics in the deterministic model, whereas low-intensity noise does not alter p53 dynamics. Interestingly, for the stable focus, the internal noise with appropriate intensity can induce quasi-limit cycle oscillations of the system. Our work may provide the useful insights for the development of anticancer therapy.Analysis of an extended gene regulatory network model with time delayhttps://zbmath.org/1537.920472024-07-25T18:28:20.333415Z"Öztürk, Dilan"https://zbmath.org/authors/?q=ai:ozturk.dilan"Özbay, Hitay"https://zbmath.org/authors/?q=ai:ozbay.hitaySummary: This paper is concerned with the analysis of an extended gene regulatory network model with time delayed negative feedback. An extended model amounts to considering the unmodeled dynamics in the earlier gene regulatory network (GRN) designs. Stability analysis of the extended model is further investigated with different design parameters. A local stability condition is derived for a model of extended GRN with time delay. Several numerical examples are given to illustrate the stability characteristics of the delayed extended GRN model and the results are compared with the benchmark homogeneous gene regulatory network.Existence of positive periodic solutions for a predator-prey modelhttps://zbmath.org/1537.920762024-07-25T18:28:20.333415Z"Feng, Chunhua"https://zbmath.org/authors/?q=ai:feng.chunhuaSummary: In this paper, a class of nonlinear predator-prey models with three discrete delays is considered. By linearizing the system at the positive equilibrium point and analyzing the instability of the linearized system, two sufficient conditions to guarantee the existence of positive periodic solutions of the system are obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation which is very complex for such a predator-prey model with three discrete delays. Some numerical simulations are provided to illustrate our theoretical prediction.A class of natural \textit{Pinus koraiensis} population system with time delay and diffusion termhttps://zbmath.org/1537.920772024-07-25T18:28:20.333415Z"Feng, Guo-Feng"https://zbmath.org/authors/?q=ai:feng.guofeng"Chen, Jiaqi"https://zbmath.org/authors/?q=ai:chen.jiaqi"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.binSummary: In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given.Dynamics of a delay-induced prey-predator system with interaction between immature prey and predatorshttps://zbmath.org/1537.920902024-07-25T18:28:20.333415Z"Pandey, Soumik"https://zbmath.org/authors/?q=ai:pandey.soumik"Sarkar, Abhijit"https://zbmath.org/authors/?q=ai:sarkar.abhijit"Das, Debashis"https://zbmath.org/authors/?q=ai:das.debashis"Chakraborty, Sarbani"https://zbmath.org/authors/?q=ai:chakraborty.sarbaniSummary: In biological pest control systems, several pests (including insects, mites, weeds, etc.) are controlled by biocontrol agents that rely primarily on predation. Following this biocontrol management ecology, we have created a three-tier prey-predator model with prey phase structure and predator gestation delay. Several studies have demonstrated that predators with Holling type-II functional responses sometimes consume immature prey. A study of the well-posedness and local bifurcation (such as saddle-node and transcritical) near the trivial and planer equilibrium points is carried out. Without any time lag, the prey development coefficient has a stabilizing impact, while increasing attack rate accelerates instability. Energy transformation rate and handling time are shown to cause multiple stability switches in the system. Numerical results demonstrate time delay is the key destabilizer that destroys stability. Our model can replicate more realistic events by including time-dependent factors and exploring the dynamic behavior of nonautonomous systems. In the presence of time delay, sufficient conditions of permanence and global attractivity of the nonautonomous system are derived. Finally, MATLAB simulations are performed to validate the analytical findings.Influence of toxic substances on dynamical behavior of a delayed diffusive predator-prey modelhttps://zbmath.org/1537.921002024-07-25T18:28:20.333415Z"Zhu, Honglan"https://zbmath.org/authors/?q=ai:zhu.honglan"Zhang, Xuebing"https://zbmath.org/authors/?q=ai:zhang.xuebing"Zhang, Hao"https://zbmath.org/authors/?q=ai:zhang.hao.13|zhang.hao.5|zhang.hao.15|zhang.hao.12|zhang.hao.3|zhang.hao-helen|zhang.hao|zhang.hao.4|zhang.hao.2Summary: In this paper, we propose and investigate a delayed diffusive predator-prey model affected by toxic substances. We first study the boundedness and persistence property of the model. By analyzing the associated characteristic equation, we obtain the conditions for the existence of steady state bifurcation, Hopf bifurcation and Turing bifurcation. Furthermore, we also study the Hopf bifurcation induced by the delay. Finally, our theoretical results are verified by numerical simulation. The numerical observation results are in good agreement with the theoretically predicted results. Theoretical and numerical simulations indicate that toxic substances have a great impact on the dynamics of the system.Delay epidemic models determined by latency, infection, and immunity durationhttps://zbmath.org/1537.921372024-07-25T18:28:20.333415Z"Saade, Masoud"https://zbmath.org/authors/?q=ai:saade.masoud"Ghosh, Samiran"https://zbmath.org/authors/?q=ai:ghosh.samiran.1|ghosh.samiran"Banerjee, Malay"https://zbmath.org/authors/?q=ai:banerjee.malay"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: We propose new single and two-strain epidemic models represented by systems of delay differential equations and based on the number of newly exposed individuals. Transitions between exposed, infectious, recovered, and back to susceptible compartments are determined by the corresponding time delays. Existence and positiveness of solutions are proved. Reduction of delay differential equations to integral equations allows the analysis of stationary solutions and their stability. In the case of two strains, they compete with each other, and the strain with a larger individual basic reproduction number dominates the other one. However, if the basic reproduction number exceeds some critical values, stationary solution loses its stability resulting in periodic time oscillations. In this case, both strains are present and their dynamics is not completely determined by the basic reproduction numbers but also by other parameters. The results of the work are illustrated by comparison with data on seasonal influenza.Severe acute respiratory syndrome-coronavirus-2 (SARS-COV-2) infection of pneumocytes with vaccination and drug therapy: mathematical analysis and optimal controlhttps://zbmath.org/1537.921442024-07-25T18:28:20.333415Z"Viriyapong, R."https://zbmath.org/authors/?q=ai:viriyapong.ratchada"Inkhao, P."https://zbmath.org/authors/?q=ai:inkhao.pSummary: We propose a mathematical model studying a within-host infection dynamics of SARS-CoV-2 in pneumocytes. This model incorporates immune response, vaccination and antiviral drugs. The crucial properties of the model -- the existence, positivity and boundary of solutions -- are established. Equilibrium points and the basic reproduction number are calculated. The stability of each equilibrium point is analyzed. Optimal control is applied to the model by adding three control variables: vaccination, treatment by favipiravir and treatment by molnupiravir. Numerical results show that each individual control could reduce SARS-CoV-2 infection in some aspects; however, with a combination of three controls, we obtain the best results in reducing SARS-CoV-2 infection. This study has emphasized the importance of prevention by vaccine and the antiviral treatments.Pattern dynamics and bifurcation in delayed SIR network with diffusion networkhttps://zbmath.org/1537.921482024-07-25T18:28:20.333415Z"Yang, Wenjie"https://zbmath.org/authors/?q=ai:yang.wenjie"Zheng, Qianqian"https://zbmath.org/authors/?q=ai:zheng.qianqian"Shen, Jianwei"https://zbmath.org/authors/?q=ai:shen.jianweiSummary: The spread of infectious diseases often presents the emergent properties, which leads to more difficulties in prevention and treatment. In this paper, the SIR model with both delay and network is investigated to show the emergent properties of the infectious diseases' spread. The stability of the SIR model with a delay and two delay is analyzed to illustrate the effect of delay on the periodic outbreak of the epidemic. Then the stability conditions of Hopf bifurcation are derived by using central manifold to obtain the direction of bifurcation, which is vital for the generation of emergent behavior. Also, numerical simulation shows that the connection probability can affect the types of the spatio-temporal patterns, further induces the emergent properties. Finally, the emergent properties of COVID-19 are explained by the above results.Dynamic analysis of a latent HIV infection model with CTL immune and antibody responseshttps://zbmath.org/1537.921522024-07-25T18:28:20.333415Z"Zhang, Zhiqi"https://zbmath.org/authors/?q=ai:zhang.zhiqi"Chen, Yuming"https://zbmath.org/authors/?q=ai:chen.yuming"Wang, Xia"https://zbmath.org/authors/?q=ai:wang.xia"Rong, Libin"https://zbmath.org/authors/?q=ai:rong.libinSummary: This paper develops a mathematical model to investigate the Human Immunodeficiency Virus (HIV) infection dynamics. The model includes two transmission modes (cell-to-cell and cell-free), two adaptive immune responses (cytotoxic T-lymphocyte (CTL) and antibody), a saturated CTL immune response, and latent HIV infection. The existence and local stability of equilibria are fully characterized by four reproduction numbers. Through sensitivity analyses, we assess the partial rank correlation coefficients of these reproduction numbers and identify that the infection rate via cell-to-cell transmission, the number of new viruses produced by each infected cell during its life cycle, the clearance rate of free virions, and immune parameters have the greatest impact on the reproduction numbers. Additionally, we compare the effects of immune stimulation and cell-to-cell spread on the model's dynamics. The findings highlight the significance of adaptive immune responses in increasing the population of uninfected cells and reducing the numbers of latent cells, infected cells, and viruses. Furthermore, cell-to-cell transmission is identified as a facilitator of HIV transmission. The analytical and numerical results presented in this study contribute to a better understanding of HIV dynamics and can potentially aid in improving HIV management strategies.Directional switches in network-organized swarming systems with delayhttps://zbmath.org/1537.921632024-07-25T18:28:20.333415Z"Xiao, Rui"https://zbmath.org/authors/?q=ai:xiao.rui"Li, Wang"https://zbmath.org/authors/?q=ai:li.wang.1"Zhao, Donghua"https://zbmath.org/authors/?q=ai:zhao.donghua"Sun, Yongzheng"https://zbmath.org/authors/?q=ai:sun.yongzheng(no abstract)Optimal feedback control problems for a semi-linear neutral retarded integro-differential systemhttps://zbmath.org/1537.932462024-07-25T18:28:20.333415Z"Huang, Hai"https://zbmath.org/authors/?q=ai:huang.hai"Feng, Tingting"https://zbmath.org/authors/?q=ai:feng.tingting"Fu, Xianlong"https://zbmath.org/authors/?q=ai:fu.xianlongSummary: This article considers the optimal and time optimal feedback control problems for a semi-linear neutral retarded integro-differential system. The existence of mild solutions and feasible pairs for the considered system is studied by applying theory of resolvent operators for linear neutral integro-differential evolution systems, fractional powers of operators and Schauder fixed point theorem. Then the Lagrange optimal feedback control problem for the system is investigated via limit arguments under some suitable assumptions. The time optimal feedback control problem is proposed and discussed deliberately here as well. An example is presented in the end to illustrate the obtained results.Almost periodic solutions of memristive multidirectional associative memory neural networks with mixed time delayshttps://zbmath.org/1537.933072024-07-25T18:28:20.333415Z"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.136"Qiao, Yuanhua"https://zbmath.org/authors/?q=ai:qiao.yuanhua"Duan, Lijuan"https://zbmath.org/authors/?q=ai:duan.lijuanSummary: Traditional biological neural networks cannot simulate the real situation of the abrupt synaptic connections between neurons while modeling associative memory of human brains. In this paper, the memristive multidirectional associative memory neural networks (MAMNNs) with mixed time-varying delays are investigated in the sense of Filippov solution. First, three steps are given to prove the existence of the almost periodic solution. Two new lemmas are proposed to prove the boundedness of the solution and the asymptotical almost periodicity of the solution by constructing Lyapunov function. Second, the uniqueness and global exponential stability of the almost periodic solution of memristive MAMNNs are investigated by a new Lyapunov function. The sufficient conditions guaranteeing the properties of almost periodic solution are derived based on the relevant definitions, Halanay inequality and Lyapunov function. The investigation is an extension of the research on the periodic solution and almost periodic solution of bidirectional associative memory neural networks. Finally, numerical examples with simulations are presented to show the validity of the main results.Vector extensions of Halanay's inequalityhttps://zbmath.org/1537.933542024-07-25T18:28:20.333415Z"Mazenc, Frédéric"https://zbmath.org/authors/?q=ai:mazenc.frederic"Malisoff, Michael"https://zbmath.org/authors/?q=ai:malisoff.michael"Krstic, Miroslav"https://zbmath.org/authors/?q=ai:krstic.miroslavEditorial remark: No review copy delivered.Asymptotic stability of fractional-order Hopfield neural networks with event-triggered delayed impulses and switching effectshttps://zbmath.org/1537.935822024-07-25T18:28:20.333415Z"Luo, Lingao"https://zbmath.org/authors/?q=ai:luo.lingao"Li, Lulu"https://zbmath.org/authors/?q=ai:li.lulu"Huang, Wei"https://zbmath.org/authors/?q=ai:huang.wei.6|huang.wei.1|huang.wei|huang.wei.2|huang.wei.5|huang.wei.3Summary: This paper examines the asymptotic stability of nonlinear fractional-order switched systems (FOSSs) under a mode-dependent event-triggered delayed impulsive mechanism (MDETDIM). The impulses and switched signals are asynchronous. A novel MDETDIM is proposed to determine the impulsive sequence, which can prevent the Zeno phenomenon. Lyapunov-based asymptotic stability conditions for general FOSSs are derived using the proposed MDETDIM. The theoretical results are then applied to a fractional-order Hopfield neural network (FOHNN) with event-based delayed impulses and switching effects. Two examples are provided to demonstrate the effectiveness of our proposed results.Time-varying Halanay inequalities with application to stability and control of delayed stochastic systemshttps://zbmath.org/1537.937812024-07-25T18:28:20.333415Z"Zhao, Xueyan"https://zbmath.org/authors/?q=ai:zhao.xueyan"Deng, Feiqi"https://zbmath.org/authors/?q=ai:deng.feiqiEditorial remark: No review copy delivered.