Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion

In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.     

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.     

Pseudo almost periodic in distribution solutions and optimal solutions to impulsive partial stochastic differential equations with infinite delay

In this paper, we study a general class of impulsive partial stochastic differential equations with infinite delay and pseudo almost periodic coefficients in Hilbert spaces. Firstly, a more appropriate concept of pseudo almost periodic in distribution for stochastic processes of infinite class is introduced. Secondly, the existence of pseudo almost periodic in distribution mild solutions is investigated by utilizing the interpolation theory, the stochastic analysis techniques and fixed point theorem. The existence of optimal mild solutions of the systems is also proved. Finally, an example is provided to show the effectiveness of the theoretical results.     

EXISTENCE OF ALMOST PERIODIC SOLUTIONS FOR DIFFERENCE SYSTEMS

1IntroductionAsisknown,therehavebeenquiteafewresu1tsontheexistenceofalrilostperiodicsolutionsforordinaryanddelaydifferentialstystems.However,tothebestofourknowledge,therehavenotappearedanycorrespondingresu1tsfOrevenordinarydifferencesystemssofar.Todealwithsuchproblems,wewilldefinethenotionoftheso-calleduni-formlyalmostperiodicsequencesbesidesthealmostsequencesandasymptoti-callyalmostsequences.Thenweprovidethecriteriaandthebasicpropertiesofuniformlyalmostperiodicsequenceswhichareneededinestabl…     

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

In this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.     

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.     

A new approach to the existence,nonexistence and uniqueness of positive almost periodic solution for a model of Hematopoiesis

Since it is very difficult to obtain the compactness of an almost periodic function set, many classical methods controlled by compact conditions such as Schauder’s fixed point theorem and the coincidence degree cannot be applied to solve almost periodic cases. Therefore, it becomes more complicated to investigate the existence, nonexistence and uniqueness of positive almost periodic solution for a certain model by the traditional methods. In this paper, the authors establish a new fixed point theorem without the compact conditions. As its application, some sufficient conditions of the existence, nonexistence and uniqueness of positive almost periodic solution for a model of Hematopoiesis are obtained. Also, the technique used here is different from the usual methods employed to solve almost periodic cases such as the contraction mapping principle and the Lyapunov functional.     

Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences

The existence of almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions of differential equations with piecewise constant argument is characterized in terms of almost periodic, asymptotically, and pseudo almost periodic sequences. Thus Meisters's and Opial's theorems are extended.     

Lienard方程周期解、概周期解的存在性

本文考虑Ｌｉｅｎａｒｄ方程ｘ”十ｆ（ｘ）ｘ’＋ｇ（ｘ）＝ｅ（ｔ），我们得到：当且时，对于任意周期或概周期。数ｅ（ｔ），它有周期或概周期解．而对于Ｌｉｅｎａｒｄ方程ｘ”＋ｆ（ｘ）ｘ’＋ｃｘ＝ｅ（ｔ），我们得到：当ｃ＞０且时，对于任意周期、或概周期函数ｅ（ｔ），它有周期或概周期解．     

Weighted pseudo almost periodic solutions of neutral functional differential equations

In this paper, we study weighted pseudo almost periodic solutions of neutral functional differential equations. By applying the properties of weighted pseudo almost periodic functions and the exponential dichotomy of linear systems as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of weighted pseudo-almost periodic solution of the equations.     

Almost periodicity,finite automata mappings,and related effectiveness issues

We introduce a class of eventually almost periodic sequences where some suffix is almost periodic (i.e., uniformly recurrent). The class of generalized almost periodic sequences includes the class of eventually almost periodic sequences, and we prove this inclusion to be strict. We also prove that the class of eventually almost periodic sequences is closed under finite automata mappings and finite transductions. Moreover, we obtain an effective form of this result. In conclusion we consider some algorithmic questions related to the almost periodicity.     

ALMOST PERIODIC SOLUTIONS OF LASOTA-WAZEWSKA-TYPE DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT DELAYS AND ALMOST PERIODIC TIME DEPENDENCE

By using the concept of almost periodic "sequence", that is, a real valued function on Z satisfying the Bohr almost periodic condition, sufficient conditions are obtained for the existence of almost periodic solutions of Lasota-Wazewska-type differential equations with almost periodic time dependence.     

非单调时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数,结合上、下解方法及不动点理论,研究了一类反应项不具任何单调性的时滞反应扩散方程,证明了此方程对应的边值问题存在唯一的周期(或概周期)解.并通过一个经典的化学模型说明了所得结果的意义.     

Bochner’s theorem and Stepanov almost periodic functions

Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.      

Stepanov-like pseudo almost periodic solutions for impulsive perturbed partial stochastic differential equations and its optimal control

This paper is mainly concerned with the Stepanov-like pseudo almost periodicity to a class of impulsive perturbed partial stochastic differential equations. Firstly, we prove the existence of $p$-mean piecewise Stepanov-like pseudo almost periodic mild solutions for the impulsive stochastic dynamical system in a Hilbert space under non-Lipschitz conditions. The results are obtained by using the fixed point techniques with fractional power arguments. Then the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, an example is provided to illustrate the developed theory.     

一个A（f）≠M（f）的紧致系统（X，f）

本文证明，存在紧致系统，其几乎周期点集闭包不等于其测度中心．藉此，我们否定地回答了两个未解决的问题．     

Weighted pseudo‐almost periodic functions on time scales with applications to cellular neural networks with discrete delays

In this paper, we first propose a concept of weighted pseudo‐almost periodic functions on time scales and study some basic properties of weighted pseudo‐almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo‐almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo‐almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

Almost periodic upper and lower solutions

The method of upper and lower solutions is a classical tool in the theory of periodic differential equations of the second order. We show that this method does not have a direct extension to almost periodic equations. To do this we construct equations of this type without almost periodic solutions but having two constants as ordered upper and lower solutions.     

Two unsolved problems on almost periodic type functions

We first show that like a scalar-valued case, a vector-valued weakly almost periodic function also satisfies the Parseval’s equality. Then we show that the space of vector-valued weakly almost periodic functions is a proper subspace of pseudo almost periodic functions. These solve two open questions on almost periodic type functions.