A composition theorem for weighted pseudo-almost automorphic functions and applications

In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.     

Weighted pseudo-almost periodic solutions to some differential equations

The paper considers some new classes of functions called weighted pseudo-almost periodic functions, which implement in a natural fashion the classical pseudo-almost periodic functions due to Zhang. Properties of these weighted pseudo-almost periodic functions are discussed, including a composition result for weighted pseudo-almost periodic functions. The results obtained are subsequently utilized to study the existence and uniqueness of a weighted pseudo-almost periodic solution to the heat equation with Dirichlet conditions.     

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.     

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 for some Differential Equations with Reflection of the Argument

From the fixed-point methods and the properties of the μ-pseudo-almost automorphic functions, the existence and uniqueness of measure pseudo-almost automorphic solution are obtained for differential equations involving reflection of the argument.     

Almost automorphic solutions to second-order semilinear evolution equations

In this paper we give some sufficient conditions for ensuring the existence and uniqueness of a mild almost automorphic solution to a second-order semilinear evolution equation in a Banach space. We also present some properties of the (new) notion of a uniform spectrum of bounded functions.     

Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications

In this paper, we introduce a new concept of bi-almost automorphic functions and obtain new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to several nonautonomous differential equations. Moreover, two examples are given to illustrate the general theorems.     

Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations

Some sufficient conditions for the existence and uniqueness of pseudo-almost periodic (mild) solutions to some classes of partial evolution equations are given. We then make use of our abstract results to discuss the existence of pseudo-almost periodic solutions to some partial differential equations.     

伪概自首随机过程分解及应用

在较弱Lipschitz条件下,本文建立伪概自守随机过程分解定理,该定理推广了已知结果.利用所建立理论的性质,本文获得了一类随机微分方程的伪概自守弱解的存在唯一性条件.     

Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy~ and the results also hold for some nonlinear equations with the form x＇（t） = f（t,x（t）） ＋ λg（t,x（t））, where f,g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.     

Weighted pseudo-almost automorphic solutions for some partial functional differential equations

In this paper, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some nonhomogeneous partial functional differential equations. We use the variation of constants formula developed in Ezzinbi and N’Guérékata (2007) [11] and the spectral decomposition of the phase space to show the main result of this work. To illustrate our main result, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some diffusion equations with delay.     

Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces

In this paper we prove some existence results for almost automorphic and pseudo-almost automorphic mild solutions to a class of abstract differential equations in Banach spaces. The main technique is based on some composition theorems combined with the contraction mapping theorem. Finally, we present an application to a semilinear partial differential equation with Dirichlet conditions.     

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.     

Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations

The paper revisits the concept of S^p$S^p$-pseudo-almost periodicity recently introduced by the author. In particular, we study the existence of pseudo-almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and S^p$S^p$-pseudo-almost periodic for p>1$p>1$. Applications include the existence of pseudo-almost periodic solutions to the heat equation with a forcing term, which is S^p$S^p$-pseudo-almost periodic and jointly continuous.     

Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations

In this paper, under Acquistapace-Terreni conditions, we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations. Applications include the existence of weighted pseudo-almost periodic solutions to a nonautonomous heat equation with gradient coefficients.     

On Favard's theorems

In this paper, we improve and extend the classical Favard's theorems, i.e. Favard's theorem of the module containment, Favard's theorem of linear differential equations. We study Favard's theory of linear differential equations with piecewise constant argument. An example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument. The equivalence between almost automorphic functions and N-almost periodic ones is studied.     

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.     

Pseudo-almost periodic viscosity solutions of second-order nonlinear parabolic equations

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.     

Bounded mild solutions of perturbed Volterra equations with infinite delay

We study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions (in the Bohr sense), the space of compact almost automorphic functions, the space of almost automorphic functions and the space of asymptotically almost automorphic functions.