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For the nonlinear functional differential equation with several constant delays, the variation formulas for its solution are proved, in which the effects of perturbations of delays and the initial moment are detected.

In this paper, a split-step collocation method is proposed for solving linear stochastic Volterra integral equations (SVIEs) with smooth kernels. The H\"older condition and the conditional expectations of the exact solutions are investigated. The solvability and mean-square boundedness of numerical solutions are proved and the strong convergence orders of collocation solutions and iterated collocation solutions are also shown. In addition, numerical experiments are provided to verify the conclusions.

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0$$\end{document}R0 as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0\le 1$$\end{document}R0≤1 then the disease will be eradicated from each region, while if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0> 1$$\end{document}R0>1 there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0$$\end{document}R0 may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

We investigate the existence of mild solutions on acompact interval to some classes of semilinear neutral functional differential inclusions. We will rely on a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler and on Schaefer's fixed-point theorem combined with lower semicontinuous multivalued operators with decomposable values.

We study evolution semigroups associated with nonautonomous functional differential equations. In fact, we convert a given functional differential equation to an abstract autonomous evolution equation and then derive a representation theorem for the solutions of the underlying functional differential equation. The representation theorem is then used to study the boundedness and almost periodicity of solutions of a class of nonautonomous functional differential equations.

We consider the second-order integrodifferential boundary value problem ¶ ν ( y ) g ( y ) − ∫ 0 + ∞ k ( x ) g ( x ) d x @ [ D ( y ) g ′ ( y ) @ ] @ ′ = p ( y ) for y ≥ 0 , g ′ ( 0 ) = 0 , g ( + ∞ ) = 0 , ¶ arising from the kinetic theory of dusty plasmas, and we provide information on the existence and other qualitative properties of the solution that have been essential in the numerical investigation.

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the $$C$$ -norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.

By using the alternative method and the topological degree theory, we obtain some sufficient conditions for the existence of 2π-periodic solutions of some semilinear equations at resonance where the kernel of the linear part has an arbitrary dimension

On demontre deux resultats d'existence locale concernant des equations differentielles fonctionnelles fortement non lineaires

On considere le systeme d'equations differentielles x'(t)=y(t)−F(x(t)), y'(t)=−g(t,x(t−r(t))), ou F:R→R, et g:[0,∞)×R→R, r:[0,∞)→(0,∞) sont continues. On donne une condition necessaire et suffisante pour la continuabilite des solutions

On considere la solution fondamentale au sens de S. Nakagiri de l'equation differentielle lineaire a retard: du(t)/dt=Au(t)+A 1 u(t−r)+∫ −2 0 a(τ)A 2 u(t+τ)dτ+f(t) dans un espace de Banach X

Soit L(t,Φ) continu et l'espace de phase pour Φ l'espace general pour des equations a retard infini introduit par Hale et Kato. On deduit la mesurabilite de Borel de η(t,θ). On ne peut plus faire l'hypothese de la propriete de constance de η(t,θ)