Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations

In this paper, we first introduce the concepts and properties of the square-mean weighted pseudo almost automorphy and the square-mean bi-almost automorphy for a stochastic process. With these preliminary settings and by virtue of the theory of the semigroups of the operators, the Banach fixed point theorem and the stochastic analysis techniques, we investigate the well-posedness of the square-mean weighted pseudo almost automorphic solutions for a general class of non-autonomous stochastic evolution equations that satisfy either global or only local Lipschitz condition. Moreover, we estimate the boundedness of attractive domain for the case where the only local Lipschitz condition is taken into account. Finally, we provide two illustrative examples to show the practical usefulness of the analytical results that we establish in the paper.     

Square-mean pseudo almost automorphic process and its application to stochastic evolution equations

This paper introduces the concept of the square-mean pseudo almost automorphy for a stochastic process. Also it introduces the properties on the completeness and the composition of the space that consists of such processes. With appropriate settings and by virtue of the theory of the semigroups of the operators to an evolution family, the Banach fixed point theorem and the stochastic analysis techniques, this paper investigates the existence, the uniqueness and the global stability of the square-mean pseudo almost automorphic solutions for a general class of stochastic evolution equations. Finally, this paper provides an illustrative example to justify the practical usefulness of the established theoretical results.     

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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.     

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.     

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.     

Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u^′(t)=A(t)u(t)+f(t,u(t)) in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if A(t) is time independent.     

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.     

Existence and global attractiveness of a square‐mean ‐pseudo almost automorphic solution for some stochastic evolution equation driven by Lévy noise

In this work, we introduce the concept of μ‐pseudo almost automorphic processes in distribution. We use the μ‐ergodic process to define the spaces of μ‐pseudo almost automorphic processes in the square mean sense. We establish many interesting results on the functional space of such processes like a composition theorem. Under some appropriate assumptions, we establish the existence, the uniqueness and the stability of the square‐mean μ‐pseudo almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise. We provide an example to illustrate our results.     

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.     

Almost automorphic solutions for stochastic differential equations driven by Lévy noise

We introduce the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution. Under suitable conditions on the coefficients, we establish the existence of solutions which are almost automorphic in distribution for some semilinear stochastic differential equations with infinite dimensional Lévy noise. We further discuss the global asymptotic stability of these solutions. Finally, to illustrate the theoretical results obtained in this paper, we give several examples.     

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.     

Existence of positive almost automorphic solutions to nonlinear delay integral equations

This paper is concerned with the existence of positive almost automorphic solutions to some nonlinear delay integral equations. We first establish a new fixed point theorem for mixed monotone operator in a cone, and then, with its help, we obtain existence theorems of positive almost automorphic solutions. Some examples are given to illustrate our results. As one will see, even in the case of almost periodicity, our theorems extend some earlier results, and moreover, the approach dealing with the integral equation arising in an epidemic problem in this paper is also new.     

一类非自治随机微分方程的均方渐近概周期解

本文研究了一类在可分Hilbert空间中的非自治随机微分方程的均方渐近概周期解.利用"Acquistapace-Terreni"条件,开方族和Banach不动点原理讨论了该类随机微分方程的均方渐近概周期解的存在唯一性,推广了该类随机微分方程的均方概周期解的存在唯一性问题.     

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.     

Measure theory and pseudo almost automorphic functions: New developments and applications

In this work, we establish a new concept of weighted pseudo almost automorphic functions using the measure theory. We present new results on weighted ergodic functions like completeness and composition theorems. The theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions. For illustration, we provide some applications for evolution equations which include reaction-diffusion systems and partial functional differential equations.     

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.     

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.     

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.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Existence of positive almost automorphic solutions to neutral nonlinear integral equations

This paper is concerned with neutral nonlinear delay integral equations. We establish an existence theorem for positive almost automorphic solutions to the equations, which extend some existing results even in the case of almost periodicity. Some examples are given to illustrate our results.