<?xml version="1.0" encoding="utf-8"?> <feed xmlns="http://www.w3.org/2005/Atom"> <title type="text">Recent zbMATH articles in MSC 34K</title> <id>https://zbmath.org/atom/cc/34K</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/" /> <link href="https://zbmath.org/atom/cc/34K" rel="self" /> <generator>Werkzeug</generator> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Admissibility via induced delay equations</title> <id>https://zbmath.org/1553.34052</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34052" /> <author> <name>&quot;Barreira, Lu铆s&quot;</name> <uri>https://zbmath.org/authors/?q=ai:barreira.luis-m</uri> </author> <author> <name>&quot;Valls, Claudia&quot;</name> <uri>https://zbmath.org/authors/?q=ai:valls.claudia</uri> </author> <content type="text">The authors associate to any linear time-dependent finite-memory differential delay system \[ \dot{x}(t) = L(t) x_t \quad \text{with} \quad x_{\theta}(t) := x(t+\theta) \] an \textit{induced} system which shares many of its structural properties but is time-invariant: given the original evolution operator \(U\), they define a strongly continuous semigroup \(S\) that operates on ``stacking'' functions of two variables \(\psi\) such that \(\psi(\cdot, p)\) is the state of the system at time \(p\) and \(S(t)\) characterizes the evolution of the system during \(t\) seconds, for all initial times \(p\) simultaneously: \[ (S(t) \psi)(\theta, p) := (U(p, p-t) \psi(\cdot, p-t))(\theta). \] From the infinitesimal generator \(A\) of \(S\), they derive the induced system as \[ \frac{\partial u}{\partial t}(t, p) = (T u_t)(p) \quad \text{with} \quad T \psi := (A \psi)(0, \cdot), \; u_{\theta}(t, p) := x(t+\theta, p+\theta). \] It is known that the existence of an exponential dichotomy for the original system is equivalent to the existence of an exponential dichotomy for its induced system. In this paper, the authors additionally prove that the admissibility of a pair of Banach spaces satisfies a similar equivalence and that for the induced system, existence of an exponential dichotomy and admissibility of the pair are equivalent, establishing the same result for the original, time-dependent system. Reviewer: S茅bastien Boisg茅rault (Paris)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">An iterative method for the qualitative analysis of nonlinear neutral delay differential equations</title> <id>https://zbmath.org/1553.34053</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34053" /> <author> <name>&quot;Basu, R.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:basu.rupsa|basu.ronni|basu.ritam|basu.rudranil|basu.ranojoy|basu.rahul|basu.ruma|basu.riddhipratim|basu.rabeya|basu.rakhee|basu.raunak|basu.rajib|basu.reshmi</uri> </author> <author> <name>&quot;Lather, J.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lather.j-s</uri> </author> <content type="text">This paper concerns (i) oscillatory behavior of solutions of nonlinear neutral delay (and advanced) differential equation \[ (x(t) - r(t)x(\alpha(t)))' + \sum_{i=1}^n s_i(t)H_i(x(\beta_i(t))) = 0, \tag{E} \] with deviating arguments; (ii) existence of bounded positive solutions of (E); (iii) the effect of delay terms on the behavior of solutions of (E). The oscillatory nature of solutions of (E) is studied by an iterative method. The existence and uniqueness of bounded positive solutions of (E) is given using Banach fixed point theorem. The effect of delay terms on the behavior of solutions of (E) is shown by means of employing MATLAB algorithms. These results extend the theory of (E), and provide a basic for further research on (E) and their applications. Authors give three examples to illustrate main results. Other problems remain open for solutions of (E) and nonlinear fractional delay differential equations with time-varying arguments. Reviewer: Yige Zhao (Jinan)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Quasi-periodic response solution to scalar state-dependent delay differential equation with degenerate equilibrium</title> <id>https://zbmath.org/1553.34054</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34054" /> <author> <name>&quot;He, Xiaolong&quot;</name> <uri>https://zbmath.org/authors/?q=ai:he.xiaolong</uri> </author> <author> <name>&quot;Jin, Feng&quot;</name> <uri>https://zbmath.org/authors/?q=ai:jin.feng</uri> </author> <author> <name>&quot;Song, Yongli&quot;</name> <uri>https://zbmath.org/authors/?q=ai:song.yongli</uri> </author> <content type="text">The paper provides some sufficient conditions for the existence of response quasi-periodic solutions to a kind of state-dependent delay differential equation. The main difficulty of the problem is the lack of regularity arising from the state-dependent delay. To overcome this difficulty, the authors establish the normal form of the equation and apply the parameterization method, Lanford's Lemma and KAM method. This is also the novelty of this work. Reviewer: Hong-Xu Li (Chengdu)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">On asymptotic stability of second order differential equations with two commensurate delays</title> <id>https://zbmath.org/1553.34055</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34055" /> <author> <name>&quot;Cahlon, Baruch&quot;</name> <uri>https://zbmath.org/authors/?q=ai:cahlon.baruch</uri> </author> <author> <name>&quot;Schmidt, Darrell&quot;</name> <uri>https://zbmath.org/authors/?q=ai:schmidt.darrell</uri> </author> <content type="text">In this paper, the stability problem for the following second order delay differential equation is considered \[ y''(t) = b_0y'(t) + b_1y'(t-\tau) + a_0y(t) + a_1y(t-\tau) + a_2y(t-2\tau), \] where \(b_0, b_1, a_0, a_1, a_2\) are real constants and \(\tau&gt;0\) is the constant delay. By Pontryagin's theory for quasi-polynomials, the authors find practical necessary conditions for the zero solution to be asymptotically stable. Then sufficient conditions for stability are provided and stability regions for the coefficients are obtained. Reviewer: Zhanyuan Hou (London)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Floquet theory and stability for a class of first order differential equations with delays</title> <id>https://zbmath.org/1553.34056</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34056" /> <author> <name>&quot;Domoshnitsky, Alexander&quot;</name> <uri>https://zbmath.org/authors/?q=ai:domoshnitsky.alexander-i</uri> </author> <author> <name>&quot;Berenson, Elnatan&quot;</name> <uri>https://zbmath.org/authors/?q=ai:berenson.elnatan</uri> </author> <author> <name>&quot;Levi, Shai&quot;</name> <uri>https://zbmath.org/authors/?q=ai:levi.shai</uri> </author> <author> <name>&quot;Litsyn, Elena&quot;</name> <uri>https://zbmath.org/authors/?q=ai:litsyn.elena</uri> </author> <content type="text">In this work, a version of the Floquet theory for first order delay differential equations is proposed and the formula of solutions representation is obtained. Moreover, the stability of first order delay differential equations is studied. In addition, an analogue of the classical integral Lyapunov-Zhukovskii test of stability is proved and tests of the exponential stability are obtained on the basis of the Floquet theory. Reviewer: Genghong Lin (Guangzhou)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Exact stability criteria for linear differential equations with discrete and distributed delays</title> <id>https://zbmath.org/1553.34057</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34057" /> <author> <name>&quot;Qu, Mingzhu&quot;</name> <uri>https://zbmath.org/authors/?q=ai:qu.mingzhu</uri> </author> <author> <name>&quot;Matsunaga, Hideaki&quot;</name> <uri>https://zbmath.org/authors/?q=ai:matsunaga.hideaki</uri> </author> <content type="text">The authors study the asymptotic stability of a retarded scalar linear differential equation of the form \[ x'(t)=-ax(t)-bx(t-\tau)-c\int_{t-\tau}^tx(s)\,ds,\ \ \ t\ge0, \] where \(a\), \(b\), \(c\) and \(\tau&gt;0\) are real numbers. Two new delaydependent criteria on asymptotic stability of the zero solution are established. In particular, it is claimed that inequality \(a+b+c\tau&gt;0\) is necessary. Detailed proofs are given. It is shown that the new results are in good agreement with those known in the literature. The article will be useful to anyone interested in this type of delay equations. Reviewer: Oleg Anashkin (Simferopol)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">On the balanced pantograph equation of mixed type</title> <id>https://zbmath.org/1553.34058</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34058" /> <author> <name>&quot;Derfel, G.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:derfel.g-a|derfel.gregory</uri> </author> <author> <name>&quot;van Brunt, B.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:van-brunt.bruce|van-brunt.b</uri> </author> <content type="text">Summary: We consider the balanced pantograph equation \[ y'\left(x\right)+y\left(x\right)={\sum }_{k=1}^m{p}_ky\left({a}_kx\right), \tag{BPE} \] where \(a_k , p_k &gt; 0\) and \({\sum }_{k=1}^m{p}_k=1\). It is known that if \(K={\sum }_{k=1}^m{p}_k\ln{a}_k\le 0\) then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for \(K &gt; 0\) these solutions exist. In the present paper, we deal with a BPE of \textit{mixed type}, i.e., \(a_1&lt; 1 &lt; a_m,\) and prove that, in this case, the BPE has a nonconstant solution \(y\) and that \(y(x) \sim cx^\sigma\) as \(x \rightarrow \infty ,\) where \(c &gt; 0\) and \(\sigma\) is the unique positive root of the characteristic equation \(P\left(s\right)=1-\sum_{k=1}^m {p}_k{a}_k^{-s}=0.\) We also show that \(y\) is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as \(x \rightarrow \infty .\)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Asymptotic estimations of the solution of a singularly perturbed equation with piecewise constant argument</title> <id>https://zbmath.org/1553.34059</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34059" /> <author> <name>&quot;Mirzakulova, A. E.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:mirzakulova.a-e</uri> </author> <author> <name>&quot;Dauylbayev, M. K.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:dauylbaev.m-k</uri> </author> <author> <name>&quot;Konisbayeva, K. T.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:konisbayeva.k-t</uri> </author> <content type="text">(no abstract)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Approximate controllability for nonautonomous integrodifferential equations with state-dependent delay</title> <id>https://zbmath.org/1553.34060</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34060" /> <author> <name>&quot;Diop, Mamadou Abdoul&quot;</name> <uri>https://zbmath.org/authors/?q=ai:diop.mamadou-abdoul</uri> </author> <author> <name>&quot;Elghandouri, Mohammed&quot;</name> <uri>https://zbmath.org/authors/?q=ai:elghandouri.mohammed</uri> </author> <author> <name>&quot;Ezzinbi, Khalil&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ezzinbi.khalil</uri> </author> <content type="text">Summary: We study the existence of mild solutions and the approximate controllability for nonautonomous integro-differential equations with state-dependent delay. We assume the approximate controllability of the linear part, and then we use resolvent operator theory to prove the approximate controllability of the nonlinear case. An example of the one-dimensional heat equation with memory is given to illustrate the basic idea our results.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Controllability of mild solutions for second-order neutral evolution equations with state-dependent delay</title> <id>https://zbmath.org/1553.34061</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34061" /> <author> <name>&quot;Boudefla, Chahrazed&quot;</name> <uri>https://zbmath.org/authors/?q=ai:boudefla.chahrazed</uri> </author> <author> <name>&quot;Sahraoui, Fatiha&quot;</name> <uri>https://zbmath.org/authors/?q=ai:sahraoui.fatiha</uri> </author> <author> <name>&quot;Baghli-Bendimerad, Selma&quot;</name> <uri>https://zbmath.org/authors/?q=ai:baghli-bendimerad.selma</uri> </author> <content type="text">Summary: The objective of our research is to demonstrate the controllability of mild solutions for a specific class of second-order neutral functional evolution equations that involve state-dependent delay. To achieve this, we rely on Avramescu's nonlinear alternative theorem and leverage cosine function theory.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Terminal value problem for implicit Katugampola fractional differential equations in \(b\)-metric spaces</title> <id>https://zbmath.org/1553.34062</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34062" /> <author> <name>&quot;Jamal, Benhadjeba&quot;</name> <uri>https://zbmath.org/authors/?q=ai:jamal.benhadjeba</uri> </author> <author> <name>&quot;Ali, Hakem&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ali.hakem</uri> </author> <content type="text">Summary: This paper mainly concerns a class of Caputo-Hadamard implicit fractional differential equations in \(b\)-metric spaces. The existence results are derived using the \(\alpha-\phi\)-Geraghty type contraction and the fixed point theory. An application is also considered to illustrate the novelty of the main result in the last section. The results obtained in this paper extend several contributions in this field.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Single and system of fractional neutral functional \(q\)-differential equations with application to particles in the plane</title> <id>https://zbmath.org/1553.34063</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.34063" /> <author> <name>&quot;Samei, Mohammad Esmael&quot;</name> <uri>https://zbmath.org/authors/?q=ai:samei.mohammad-esmael</uri> </author> <author> <name>&quot;Ahmadi, Ahmad&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ahmadi.ahmad</uri> </author> <author> <name>&quot;Kaabar, Mohammed K. A.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:kaabar.mohammed</uri> </author> <author> <name>&quot;Siri, Zailan&quot;</name> <uri>https://zbmath.org/authors/?q=ai:siri.zailan</uri> </author> <author> <name>&quot;Alzabut, Jehad&quot;</name> <uri>https://zbmath.org/authors/?q=ai:alzabut.jehad-o</uri> </author> <author> <name>&quot;Akbulut, Arzu&quot;</name> <uri>https://zbmath.org/authors/?q=ai:akbulut.arzu</uri> </author> <author> <name>&quot;Kaplan, Melike&quot;</name> <uri>https://zbmath.org/authors/?q=ai:kaplan.melike</uri> </author> <content type="text">Summary: We obtain the existence of solutions of single and multidimensional fractional neutral functional \(q\)-differential equations with bounded delay based on operator equations by using Krasnoselskii's fixed point theorem. At the end, examples, which contain some tables, figures and related algorithms with numerical effect, are presented to show applications of our results.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Almost periodic and almost automorphic solutions for some nondensely nonautonomous linear periodic partial functional differential equations</title> <id>https://zbmath.org/1553.35021</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.35021" /> <author> <name>&quot;Afoukal, Abdallah&quot;</name> <uri>https://zbmath.org/authors/?q=ai:afoukal.abdallah</uri> </author> <author> <name>&quot;Ezzinbi, Khalil&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ezzinbi.khalil</uri> </author> <author> <name>&quot;Hilal, Khalid&quot;</name> <uri>https://zbmath.org/authors/?q=ai:hilal.khalid</uri> </author> <content type="text">Summary: In this work, we establish a new variation of constants formula for nonautonomous linear partial functional differential equations with linear part \(A\) satisfying the Hille-Yosida condition on a Banach space and is not necessarily densely defined. As well, we use this formula and the spectrum of the monodromy operator when the semigroup generated by the part of \(A\) in \(\overline{D(A)}\) is compact, to study the existence of almost periodic and almost automorphic solutions for a nonautonomous linear periodic partial functional differential equations.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Almost automorphic solutions for Lotka-Volterra systems with diffusion and time-dependent parameters</title> <id>https://zbmath.org/1553.35022</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.35022" /> <author> <name>&quot;Kpoumi茅, M. E.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil|kpoumie.moussa-el-khalill</uri> </author> <author> <name>&quot;Nsangou, A. H. G.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:nsangou.abdel-hamid-gamal</uri> </author> <author> <name>&quot;Zouine, A.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zouine.aziz</uri> </author> <content type="text">The article presents a thorough investigation into the existence and uniqueness of \(p\)-pseudo almost automorphic solutions for non-autonomous Lotka-Volterra systems with diffusion. The authors employ an exponential dichotomy approach and fixed-point arguments, demonstrating significant results applicable to predator-prey models with time-dependent parameters. Overall, the research contributes valuable insights into the dynamics of ecological systems, highlighting the complexity of interactions influenced by temporal changes. Reviewer: Mohamed Zitane (Mekn猫s)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Propagation dynamics for a time-periodic reaction-diffusion two group SIR epidemic model</title> <id>https://zbmath.org/1553.35067</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.35067" /> <author> <name>&quot;Zhao, Lin&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zhao.lin.2|zhao.lin.1</uri> </author> <content type="text">Summary: The paper was devoted to studying the spreading speed and traveling wave solutions for a time-periodic reaction-diffusion two group susceptible-infective-recovered (SIR) epidemic model. With regard to the basic reproduction number \(R_0\) of the corresponding periodic ordinary differential system and the minimal wave speed \(c^{\ast}\), spreading properties of the corresponding solution of the model when \(R_0 &gt; 1\) were established, which implied that the minimal wave speed \(c^{\ast}\) was equal to spreading speed of system (2). On the basis of it, the full information about the existence and nonexistence of traveling wave solutions of the system related with \(R_0\) nd \(c^{\ast}\) can be studied. More specifically, we proved that when \(R_0 &gt; 1\) and \(c \geq c^{\ast}\), the system admitted a nontrivial time-periodic traveling wave solution with wave speed \(c\), and for \(c &lt; c^{\ast}\) there were no such traveling waves satisfying the system. In addition, when \(R_0 &lt; 1\), the system admitted no nontrivial traveling waves.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Dynamical behaviors of fractional-order complex dynamical networks</title> <id>https://zbmath.org/1553.37020</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.37020" /> <author> <name>&quot;Wang, Jin-Liang&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wang.jinliang.2|wang.jinliang|wang.jinliang.1</uri> </author> <content type="text">Publisher's description: This book benefits researchers, engineers, and graduate students in the field of fractional-order complex dynamical networks. Recently, the dynamical behaviors (e.g., passivity, finite-time passivity, synchronization, and finite-time synchronization, etc.) for fractional-order complex networks (FOCNs) have attracted considerable research attention in a wide range of fields, and a variety of valuable results have been reported. In particular, passivity has been extensively used to address the synchronization of FOCNs.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Optimal control, well-posedness and sensitivity analysis for a class of generalized evolutionary systems</title> <id>https://zbmath.org/1553.49017</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.49017" /> <author> <name>&quot;Li, Xiuwen&quot;</name> <uri>https://zbmath.org/authors/?q=ai:li.xiuwen</uri> </author> <author> <name>&quot;Luo, Zhi&quot;</name> <uri>https://zbmath.org/authors/?q=ai:luo.zhi</uri> </author> <author> <name>&quot;Liu, Zhenhai&quot;</name> <uri>https://zbmath.org/authors/?q=ai:liu.zhenhai</uri> </author> <content type="text">The authors study a system governed by the following fractional differential variational-hemivariational inequality in a reflexive Banach space \(E\): \[ \left\{ \begin{array}{l} ^CD_t^\alpha x(t) \in Ax(t) + F(t,x(t),u(t))\quad \mbox{a.e.}\,\,t \in I := [0,b], \\ u(t) \in SOL(K; g(t,x(t),\cdot0, J(x(t),\cdot), J(x(t),\cdot),\phi,h)\quad \mbox{a.e.}\,\,t \in I,\\ x(0) = x_0. \end{array} \right. \] Here \(^CD_t^\alpha,\) \(0 &lt; \alpha \leq 1\) denotes the Caputo fractional derivative, \(A\) is an infinitesimal generator of a compact and uniformly bounded \(C_0\)-semigroup in \(E,\) \(F \colon I \times E \times U \multimap E,\) where \(U\) is a reflexive Banach space, is a multimap with convex closed values. For a nonempty convex closed subset \(K \subset U\), the notation \(SOL(K;g(t,x(t),\cdot),J(x(t),\cdot),\phi,h)\) stands for the solution set of the mixed variational-hemivariational inequality of the form \[ \langle g(t,x(t),u(t)),v - u(t)\rangle_U + J^0(x(t),\gamma u(t);\gamma(v - u(t))) + \phi(v,u(t)) \geq \langle h,v - u(t)\rangle_U \] for all \(v \in K\), where \(J^0\) denotes the Clarke directional derivative of a locally Lipschitz function \(J\). The authors demonstrate the nonemptiness and compactness of the solution set to the above problem. The existence of an optimal control is proved and the well-posedness of the problem, including the existence, uniqueness and stability of solutions is studied and a sensitivity analysis of the problem related to multiparameters is explored. An application to a fractional heat equation is considered. Reviewer: Valerii V. Obukhovskij (Voronezh)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">A geometrical index for the group \(S^n\) and its applications in periodic solutions of differential and differential-difference equations</title> <id>https://zbmath.org/1553.58007</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.58007" /> <author> <name>&quot;Li, Lin&quot;</name> <uri>https://zbmath.org/authors/?q=ai:li.lin.10</uri> </author> <author> <name>&quot;Ge, Weigao&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ge.weigao</uri> </author> <content type="text">Summary: In this paper, we construct a new index theory, \(S^n\) index theory, which is an improvement of the \(S^1\) index theory given by \textit{V. Benci} [Trans. Am. Math. Soc. 274, 533--572 (1982; Zbl 0504.58014)]. Also \(S^1\) index theory is a powerful tool in the study of the multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional \(\Phi\) is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Graph limit of the consensus model with self-delay</title> <id>https://zbmath.org/1553.82029</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.82029" /> <author> <name>&quot;Haskovec, Jan&quot;</name> <uri>https://zbmath.org/authors/?q=ai:haskovec.jan</uri> </author> <content type="text">Summary: It is known that models of interacting agents with self-delay (reaction-type delay) do not admit, in general, the classical mean-field limit description in terms of a Fokker-Planck equation. In this paper we propose the graph limit of the nonlinear consensus model with self-delay as an alternative continuum description and study its mathematical properties. We establish the well-posedness of the resulting integro-differential equation in the Lebesgue \(L^p\) space. We present a rigorous derivation of the graph limit from the discrete consensus system and derive a sufficient condition for reaching global asymptotic consensus. We also consider a linear variant of the model with a given interaction kernel, which can be interpreted as a dynamical system over a graphon. Here we derive an optimal (i.e. sufficient and necessary) condition for reaching global asymptotic consensus. Finally, we give a detailed explanation of how the presence of the self-delay term rules out a description of the mean-field limit in terms of a particle density governed by a Fokker-Planck-type equation. In particular, we show that the indistinguishability-of-particles property does not hold, which is one of the main ingredients for deriving the classical mean-field description. {{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd}</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Estimates of solutions in a model of antiviral immune response</title> <id>https://zbmath.org/1553.92029</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92029" /> <author> <name>&quot;Skvortsova, Mariya Aleksandrovna&quot;</name> <uri>https://zbmath.org/authors/?q=ai:skvortsova.mariya-aleksandrovna</uri> </author> <content type="text">Summary: We consider a model of antiviral immune response suggested by \textit{G. I. Marchuk} [in: Current problems of applied mathematics and mathematical physics. 11--19 (1988; Zbl 0685.92002)]. The model is described by a system of differential equations with several delays. We study asymptotic stability for a stationary solution of the system that corresponds to a completely healthy organism. We estimate the attraction set of this stationary solution. We also find estimates of solutions characterizing the stabilization rate at infinity. A Lyapunov-Krasovskii functional is used in the proof.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Periodic solutions of an NPZ model with periodic delay and space heterogeneity</title> <id>https://zbmath.org/1553.92037</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92037" /> <author> <name>&quot;Cui, Mengran&quot;</name> <uri>https://zbmath.org/authors/?q=ai:cui.mengran</uri> </author> <author> <name>&quot;Lv, Yunfei&quot;</name> <uri>https://zbmath.org/authors/?q=ai:lv.yunfei</uri> </author> <author> <name>&quot;Zhang, Qianying&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zhang.qianying</uri> </author> <content type="text">Summary: In this paper, we investigate a nutrient-phytoplankton-zooplankton (NPZ) model with seasonality and spatial heterogeneity, in which the immature zooplankton with size structure is described by the quasi-linear first-order partial differential equation. A three-dimensional periodic delay food chain model with initial value is derived from the original model. To establish the threshold dynamics of NPZ model, we first obtain the global attractivity of the positive periodic solution of the nutrient-phytoplankton (NP) subsystem by using the method of sub-super solutions and associated iterations. We then give the integral expression of threshold for the corresponding system, which is convenient for numerical simulations. Based on this, the threshold dynamics is discussed by appealing to the comparison arguments and the persistence theory. Finally, the analytic results are in good consistence with our numerical simulations.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Effect of incubation delay in an SIRS model: Hopf bifurcation and stability switch</title> <id>https://zbmath.org/1553.92049</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92049" /> <author> <name>&quot;Das, Tanuja&quot;</name> <uri>https://zbmath.org/authors/?q=ai:das.tanuja</uri> </author> <author> <name>&quot;Srivastava, Prashant K.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:srivastava.prashant-kr</uri> </author> <content type="text">An delayed SIRS epidemic model is proposed to consider the impact of incubation period on the spread of infectious diseases, where the incidence is of the Monod-Haldane type. First the existence of equilibrium points is discussed. Then their local stability is investigated. Delay can lead to Hopf bifurcation. These theoretical results are illustrated with numerical simulations. It should be pointed out that the expression of \(\frac{dN}{dt}\) on page 369 is not correct. It should be \[ \frac {dN}{dt}=bS\left(1-\frac{S}{K}\right)-d(N-S)-\delta I(t+\tau)-\gamma R(t+\tau)+\gamma R. \] As the focus of the paper is only about local dynamics, this error does not affect the remaining discussion of the paper. For the entire collection see [Zbl 1515.92004]. Reviewer: Yuming Chen (Waterloo)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Analysis of a delayed malaria transmission model including vaccination with waning immunity and reinfection</title> <id>https://zbmath.org/1553.92053</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92053" /> <author> <name>&quot;Li, Jinhui&quot;</name> <uri>https://zbmath.org/authors/?q=ai:li.jinhui</uri> </author> <author> <name>&quot;Teng, Zhidong&quot;</name> <uri>https://zbmath.org/authors/?q=ai:teng.zhi-dong</uri> </author> <author> <name>&quot;Wang, Ning&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wang.ning.4</uri> </author> <author> <name>&quot;Chen, Wei&quot;</name> <uri>https://zbmath.org/authors/?q=ai:chen.wei.27</uri> </author> <content type="text">In this paper, a delayed malaria is proposed to study the effect of incubation delays, waning immunity of vaccine, and reinfection. It is shown that the existence of equilibria is completely determined by the basic reproduction number \(R_0\). There is always the disease-free equilibrium \(P^0\) and there is also a unique endemic equilibrium \(P^*\) when \(R_0&gt;1\). Moreover, delays do not affect the stability of \(P^0\), which is locally asymptotically stable if \(R_0&lt;1\). However, delays can destablize \(P^*\) and lead to Hopf bifurcation. The model is applied to simulate the reported cases in Nigeria. Moreover, sensitivity analysis was carried out to investigate the correlation between each parameter and \(R_0\). It indicated that vaccination is not very effective in reducing \(R_0\) but is efficient in controlling the disease. Reviewer: Yuming Chen (Waterloo)</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Dynamics of an epidemic model with relapse and delay</title> <id>https://zbmath.org/1553.92054</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92054" /> <author> <name>&quot;Liu, Jian&quot;</name> <uri>https://zbmath.org/authors/?q=ai:liu.jian.18</uri> </author> <author> <name>&quot;Ding, Qian&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ding.qian</uri> </author> <author> <name>&quot;Guo, Hongpeng&quot;</name> <uri>https://zbmath.org/authors/?q=ai:guo.hongpeng</uri> </author> <author> <name>&quot;Zheng, Bo&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zheng.bo|zheng.bo.1</uri> </author> <content type="text">Summary: In this paper, we consider a new epidemiological model with delay and relapse phenomena. Firstly, a basic reproduction number \(R_0\) is identified, which serves as a threshold parameter for the stability of the equilibria of the model. Then, beginning with the delay-free model, the global asymptotic stability of the equilibria is obtained through the construction of suitable Lyapunov functions. For the delay model, the stability of the positive equilibrium and the existence of the local Hopf bifurcation are discussed. Furthermore, the application of the normal form theory and center manifold theorem is used to determine the direction and stability of these Hopf bifurcations. Finally, we shed light on corresponding biological implications from a numerical perspective. It turns out that time delay affects the stability of the positive equilibrium, leading to the occurrence of periodic oscillations and disease recurrence.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">\textit{Wolbachia} invasion in mosquitoes with incomplete CI, imperfect maternal transmission and maturation delay</title> <id>https://zbmath.org/1553.92068</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.92068" /> <author> <name>&quot;Ma, Xiaoke&quot;</name> <uri>https://zbmath.org/authors/?q=ai:ma.xiaoke</uri> </author> <author> <name>&quot;Su, Ying&quot;</name> <uri>https://zbmath.org/authors/?q=ai:su.ying</uri> </author> <content type="text">Summary: The mechanism of cytoplasmic incompatibility (CI) is important in the study of \textit{Wolbachia} invasion in wild mosquitoes. \textit{Y. Su} et al. [Bull. Math. Biol. 84, No. 9, Paper No. 95, 21 p. (2022; Zbl 1497.92215)] proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for \textit{Wolbachia} successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way.</content> </entry> <entry xml:base="https://zbmath.org/atom/cc/34K"> <title type="text">Dupire It么's formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control</title> <id>https://zbmath.org/1553.93222</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.93222" /> <author> <name>&quot;Zhang, Ning&quot;</name> <uri>https://zbmath.org/authors/?q=ai:zhang.ning.6</uri> </author> <author> <name>&quot;Wang, Haodong&quot;</name> <uri>https://zbmath.org/authors/?q=ai:wang.haodong</uri> </author> <author> <name>&quot;Li, Wenxue&quot;</name> <uri>https://zbmath.org/authors/?q=ai:li.wenxue</uri> </author> <content type="text">Summary: This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain'' is used to qualify the impulse intensity. Our approach expands Dupire functional It么's formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua's circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.</content> </entry> </feed>