Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces

In this paper, we prove that the space is complete. This not only gives an affirmative answer to a basic problem in this field, but also enables us to obtain an existence and uniqueness theorem of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. An example is given to illustrate our theorem. The work was supported partly by the National Natural Science Foundation of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.     

Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations

In this paper, we reveal several basic properties about nonlinear vector-valued weighted pseudo almost automorphic functions, including equivalence, completeness, translation invariance, composition theorem, and convolution theorem of these functions. We also give some concrete examples to illustrate our results. Finally, we obtain a new existence theorem of nonlinear weighted pseudo almost automorphic solutions for semilinear evolution equations in Banach spaces.     

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.     

Almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations

In this paper, we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations

where AA_m|kerL generates a hyperbolic analytic semigroup on a Banach space X. The functions h and are defined on some intermediate subspaces X_β,0<β<1, and take values in X and in a boundary space ∂X respectively.     

On analytic structure of solutions to higher order abstract Cauchy problems

We prove the existence of entire solutions to some abstract higher order Cauchy problem for a dense subset of initial values.

     

Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions

Of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions. Under some suitable hypotheses, we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations. Moreover, an example is given to illustrate our results.     

Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay

We consider the existence and uniqueness of a Stepanov-like almost automorphic solution to the nonautonomous semilinear evolution equations with a constant delay:

where , generates an exponentially stable evolution family {U(t,s)} and satisfies a Lipschitz condition with respect to the second argument.     

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.     

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.     

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.     

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 pseudo almost automorphic mild solutions to stochastic fractional differential equations

Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.     

Positive almost automorphic solutions for a class of non-linear delay integral equations

This article is concerned with a class of non-linear delay integral equations, which unify some extensively studied delay integral equations. We establish a new existence and uniqueness theorem about positive almost automorphic solutions of the delay integral equations. Our theorem can deal with some cases to which many known results are not applicable. Two examples are given to illustrate 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.     

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.     

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.