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.     

Composition of Stepanov-like pseudo almost automorphic functions and applications to nonautonomous evolution equations

In this paper, we establish a new composition theorem about Stepanov-like pseudo almost automorphic functions under the local Lipschitz condition. Using this composition theorem, we also study the existence and uniqueness of pseudo almost automorphic solutions for nonautonomous evolution equations. Our results extend many recent known ones on these topics.     

Composition of pseudo almost automorphic and asymptotically almost automorphic functions

This paper is concerned with pseudo almost automorphic functions, which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions. New results, concerning the composition of pseudo almost automorphic functions, are established.     

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.     

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.     

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

EXISTENCE AND UNIQUENESS OF WEIGHTED PSEUDO ALMOST AUTOMORPHIC SEQUENCE SOLUTIONS TO SOME SEMILINEAR DIFFERENCE EQUATIONS

In this paper, we consider a semilinear difference equation in a Banach space. Under some suitable conditions on f, we prove the existence and uniqueness of a weighted pseudo almost automorphic sequence solution to the equation.     

On the integro-differential equations with reflection

In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed-point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general-type reflection integro-differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the Lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe.     

Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.     

加权Stepanov伪概自守函数的一些基本性质及其对Volterra积分方程的应用

本文主要研究加权Stepanov伪概自守函数的一些基本性质.首先,本文研究一个加权Stepanov伪概自守函数与它的Stepanov概自守部分的关系.利用这些关系,本文将这类函数的复合定理进行改进.其次,本文研究加权Stepanov伪概自守函数空间中的卷积算子,这里的卷积算子是由绝对可积函数所生成.最后,应用压缩映射原理,本文得到两类Volterra积分方程的加权Stepanov伪概自守解的存在唯一性.本文的结果推广了部分已知结果.     

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S ^p -a.a.) functions. We establish a composition theorem about S ^p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S ^p -a.a. functions are discussed. Some theorems extend earlier results.     

Weighted Stepanov-like pseudo almost automorphy and applications

In this paper, we propose a new class of functions called weighted Stepanov-like pseudo almost automorphic functions, which generalize in a natural fashion the concept of almost automorphy and its various extensions. We systematically explore the properties of the weighted Stepanov-like pseudo almost automorphic functions in general Banach space including a composition result. As an application, we establish some sufficient criteria for the existence, uniqueness of the weighted pseudo almost automorphic solution to a class of partial neutral functional differential equations and also to a class of nonlinear Volterra integral equations of convolution type with infinite delay in Banach space. Some interesting examples are presented to illustrate the main findings.     

Square-mean pseudo almost automorphic mild solutions for stochastic evolution equations driven by G-Brownian motion

In this article, we prove the existence and uniqueness of the square-mean pseudo almost automorphic mild solutions for a class of stochastic evolution equations driven by G-Brownian motion (G-SSEs). Our results are established by means of the fixed point theorem. An example is given to illustrate the theory.     

半线性微分方程的概自守与伪概自守解

在Banach空间中,利用发展系统的算子半群理论和Banach压缩原理,在半线性微分方程x′（t）=A（t）x（t）＋f（t,x（t））满足一定的条件下,证明了其概自守与伪概自守mild解的存在性与唯一性．     

Vector-valued pseudo almost periodic functions

Vector-valued pseudo almost periodic functions are defined and their properties are investigated. The vector-valued functions contain many known functions as special cases. A unique decomposition theorem is given to show that a vector-valued pseudo almost periodic function is a sum of an almost periodic function and an ergodic perturbation.     

On Stepanov-like (pseudo) almost automorphic functions

We prove new composition theorems for Stepanov-like almost automorphic and Stepanov-like pseudo-almost automorphic functions. An application is given to a class of evolution equations with Stepanov-like pseudo-almost automorphic perturbations.     

μ‐Pseudo almost automorphic solutions of abstract fractional integro‐differential neutral equations with an infinite delay

The aim of this work is to study μ‐pseudo almost automorphic solutions of abstract fractional integro‐differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Almost Automorphic Dynamics of Generalized Li\'enard Equation

In this study, we focus on the solutions of the Li\''enard equation being bounded in the future and characterize the almost automorphic, asymptotically almost automorphic, and weighted pseudo almost automorphic dynamics. An example is presented to illustrate the main findings.     

Weyl almost automorphic solutions for a class of Clifford-valued dynamic equations with delays on time scales

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford-valued shunt inhibitory cellular neural networks with time-varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results.