Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

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.     

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.     

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In this paper, we introduce a concept of Poisson $p$-mean almost automorphy for stochastic processes and give the composition theorems for (Poisson) $p$-mean almost automorphic functions under non-Lipschitz conditions. Our abstract results are, subsequently, applied to study a class of neutral stochastic evolution equations driven by L\'evy noise, and we present sufficient conditions for the existence of square-mean almost automorphic mild solutions. An example is provided to illustrate the effectiveness of the proposed result.     

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.     

Pseudo Almost Automorphic Solutions for Non-autonomous Stochastic Differential Equations with Exponential Dichotomy

In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.     

Exis tence and st ability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.     

Rosenthal’s Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations

In this paper, the authors generalize the concept of asymptotically almost \linebreak negatively associated random variables from the classic probability space to the upper expectation space. Within the framework, the authors prove some different types of Rosenthal''s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal''s inequalities.     

Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

In this paper, a new concept of Poisson asymptotically almost automorphy for stochastic processes is introduced. And then, some fundamental properties including composition theorems for the space of such processes are proved. Subsequently, this concept is applied to investigate the existence and uniqueness of asymptotically almost automorphic solutions in distribution to some linear and semilinear stochastic differential equations driven by a Lévy process under some suitable conditions. Finally, an example is given to illustrate the main results.     

Pseudo almost automorphy of two-term fractional functional differential equations

In this paper, by measure theory, we introduce and investigate the concepts of (Stepanov-like) $(\mu,\nu)$-pseudo almost automorphic of class $r$ and class infinity, respectively. As applications, we establish some sufficient criteria for the existence, uniqueness of pseudo almost automorphic mild solutions to two-term fractional functional differential equations with finite or infinite delay. The working tools are based on the generalization of semigroup theory, Banach contraction mapping principle and Leray-Schauder alternative theorem. Finally, we explore the same topic for a fractional partial functional differential equation with delay.     

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.     

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.     

A spectral countability condition for almost automorphy of solutions of differential equations

We consider the almost automorphy of bounded mild solutions to equations of the form

with (generally unbounded) -periodic and almost automorphic in a Banach space . Under the assumption that does not contain , the part of the spectrum of the monodromy operator associated with the evolutionary process generated by on the unit circle is countable. We prove that every bounded mild solution of on the real line is almost automorphic.

     

Stepanov‐like doubly weighted pseudo almost automorphic processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion

This paper investigates the properties of the p‐mean Stepanov‐like doubly weighted pseudo almost automorphic (S^pDWPAA) processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion. We firstly prove the equivalent relation between the S^pDWPAA and Stepanov‐like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for S^pDWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the S^p DWPAA solution for a class of nonlinear Sobolev‐type stochastic differential equations driven by G‐Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.     

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.     

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.     

Asymptotic periodicity and almost automorphy for a class of Volterra integro‐differential equations

This work deals with the existence of asymptotically periodic (resp. asymptotically almost automorphic) solutions for a class of Volterra integro‐differential equations. As application, we examine sufficient conditions for the existence of asymptotically periodic (resp. asymptotically almost automorphic) mild solutions of an equation arising in the study of heat conduction in materials with memory. Copyright © 2012 John Wiley & Sons, Ltd.     

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.