ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONAUTONOMOUS OPERATOR

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脉冲时滞微分方程解的整体存在唯一性、振动性与非振动性

本文讨论脉冲时滞微分方程X’(t)=f(t,x(t-T_1(t)),…,x(t-T_n(t))),x(t_k)-x(t_k~-)=I_k(x(t_k~- )).获得了方程(E) 解的一个整体存在唯一性定理.当(E)是线性方程时,给出了由时滞微分方程解的振动性或非振动性刻划出相应的脉冲时滞微分方程的同样性质的一般性脉冲条件.     

Standing Wave Solutions in Nonhomogeneous Delayed Synaptically Coupled Neuronal Networks

The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed systemof integral differential equations and a nonlinear scalar integral differential equation. It will be shown that there exist six standing wave solutions (u(x,t),w(x,t))=(U(x),W(x)) to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave solutions u(x,t)=U(x) to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.     

具有转向点的三阶半线性奇摄动边值问题解的存在性

本文应用微分不等式技术证明一类具有转向点的三阶半线性奇摄动边值问题解的存在性.     

On the existence of almost periodic solutions of a singularly perturbed differential equation with piecewise constant argument

Based on the investigation of almost periodic solutions to difference equations, the existence of almost periodic solution for a nonautonomous, singularly perturbed differential equation with piecewise constant argument is considered. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data.     

Pseudo Almost Periodic Solutions of a Singularly Perturbed Differential Equation with Piecewise Constant Argument

Under suitable assumptions, the existence and the uniqueness of the pseudo-almost periodic solution for a singularly perturbed differential equation with piecewise constant argument are obtained. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data.     

Oscillation of second‐order perturbed differential equations

We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second‐order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Darboux problem for perturbed partial differential equations of fractional order with finite delay

In this paper we investigate the existence of solutions for functional partial perturbed hyperbolic differential equations with fractional order. These results are based upon a fixed point theorem for the sum of contraction and compact operators.     

Front motion in the parabolic reaction-diffusion problem

A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

具扰动项的时滞Logistic方程的周期解

本文首先讨论一类时滞微分方程的周期解的存在性，然后将本文结果应用到具扰动项的 时滞Ｌｏｇｉｓｔｉｃ方程得出其正周期解的存在性，本文结果推广和改进了「１」和「６」的相应结果。     

Existence of a solution to the inhomogeneous equation with the one-dimensional Schr?dinger operator in the space of quickly decreasing functions

We prove the theorem of existence of a solution to the inhomogeneous equation with the one-dimensional Schr?dinger operator in the space of quickly decreasing functions arising in the theory of asymptotic solutions to singularly perturbed partial differential equations.     

On a new almost periodic type solution of a class of singularly perturbed differential equations with piecewise constant argument

A new class of ergodic sequences, pseudo almost periodic sequence, is introduced, and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, we investigate the existence of pseudo almost periodic solutions for a nonautonomous, singularly perturbed differential equations with piecewise constant argument.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

On mild solutions of gradient systems in Hilbert spaces

We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.     

Partial differential equations with delayed random perturbations: existence uniqueness and stability of solutions

We consider a stochastic non–linear Partial Differential Equation with delay which may be regarded as a perturbed equation. First, we prove the existence and the uniqueness of solutions. Next, we obtain some stability results in order to prove the following: if the unperturbed equation is exponentially stable and the stochastic perturbation is small enough then, the perturbed equations remains exponentially stable. We impose standard assumptions on the differential operators and we use strong and mild solutions     

On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.     

Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection

For a singularly perturbed parabolic equation termed in applications as the reaction-diffusion-advection equation, stationary solutions with internal transition layers (contrast structures) are studied. An arbitrary-order asymptotic approximation of such solutions is constructed, and an existence theorem is proved. An efficient algorithm for constructing an asymptotic approximation of the transition point is proposed. The constructed asymptotic approximation is justified by applying the asymptotic method of differential inequalities, which is extended to the class of problems under study. This method is also used to establish the Lyapunov stability of such stationary solutions.     

On the almost periodic solutions of Lattices equations

By using the Ekeland variational principle and the calculus of variations in mean, we study the existence of almost periodic solutions of a class of advanced-retarded differential equation. We show that under some hypothesis, for any given almost periodic forcing term can be ‘perturbed’ so that the corresponding forced equation admits an almost periodic solution.     

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.