Behavior of bounded solutions for some almost periodic neutral partial functional differential equations

We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.     

Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions for fractional differential and integro-differential equations in the case when the forcing term belongs to the class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions.     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

In this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.     

非自治中立型无穷时滞泛函微分方程的加权伪概自守解

首先引入h型Stepanov 加权伪概自守函数和∞型Stepanov加权伪概自守函数的概念, 接着建立了其函数空间的完备性以及相应组合定理, 最后证明了一类非自治无穷时滞偏中立型泛函微分方程在S^p-加权伪概自守系数下加权伪概自守解的存在唯一性.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t]

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences     

STABILITY AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Itiswellknownthattheexistenceofalmostperiodicsolutionsiscloselyrelatedtothestabilityofsolutions.Forfunctionaldifferentialequationswithinfinitedelay,Y.Hin.[5'6]studiedtheproblemsontheexistenceofalmostperiodicsolutionsandthestability.However,therearefewpapersll2]dealingwithneutralfunctionaldifferentialequationswithinfinitedelay.Inthepresentpaper,forneutralfunctionaldifferentialequationswithinfinitedelay,weprovetheinherencetheoremfortheuniformlystableoperatorD(t),definethestabilitywithrespecttot…     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Existence of $\mu$-pseudo almost automorphic solutions to abstract partial neutral functional differential equations with infinite delay

In this article, we introduce and investigate the concept of $\mu$-Stepanov-like pseudo almost automorphic functions of class $h$ and class infinity via measure theory. We present new results on completeness and composition theorems for the space of such functions. To illustrate our main results, we provide some applications to an abstract partial neutral functional differential equation with infinite delay.     

Existence of solutions for impulsive partial neutral functional differential equations

This paper is concerned with partial neutral functional differential equations of first and second order with impulses. We establish some results of existence of mild solutions for these classes of equations.     

On almost periodic mild solutions for stochastic functional differential equations

In this paper, a class of stochastic functional differential equations given by

     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of almost periodic solutions of second order neutral delay differential equations with piecewise constant argument

By constructing almost periodic sequence solutions to difference equations, the existence of almost periodic solutions of neutral delay differential equations with piecewise constant argument

Pseudo almost periodic solutions of differential equations with piecewise constant arguments

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.     

Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces

This work aims to investigate the existence and uniqueness of pseudo-almost-automorphic solutions for some neutral partial functional differential equations in Banach spaces. Recall that the new concept of pseudo-almost-automorphy generalizes the one of the pseudo-almost-periodicity and it has been recently introduced in the literature. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed parts are assumed to be pseudo-almost-automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.     

Pseudo almost periodic in distribution solutions and optimal solutions to impulsive partial stochastic differential equations with infinite delay

In this paper, we study a general class of impulsive partial stochastic differential equations with infinite delay and pseudo almost periodic coefficients in Hilbert spaces. Firstly, a more appropriate concept of pseudo almost periodic in distribution for stochastic processes of infinite class is introduced. Secondly, the existence of pseudo almost periodic in distribution mild solutions is investigated by utilizing the interpolation theory, the stochastic analysis techniques and fixed point theorem. The existence of optimal mild solutions of the systems is also proved. Finally, an example is provided to show the effectiveness of the theoretical results.     

Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

The aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.