Decomposition of weighted pseudo-almost periodic functions

First, we show by constructing two counterexamples that the decomposition of weighted pseudo-almost periodic functions is not unique in general. Then we prove that the decomposition of such functions is unique if PAP₀(X,ρ) is translation invariant, but not necessarily unique without the assumption. Moreover, we give an example to show that the mean value under a certain weight ρ may not exist for all almost periodic functions. With these results, we answer some fundamental questions on weighted pseudo-almost periodic functions.     

Asymptotically almost periodic,almost periodic and pseudo-almost periodic mild solutions for neutral differential equations

In this paper we establish the existence and uniqueness of almost periodic, asymptotically almost periodic and pseudo-almost periodic mild solutions for neutral differential equations in Banach spaces.     

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.     

The existence of pseudo-almost periodic solutions of third-order neutral differential equations with piecewise constant argument

In this paper, we present some existence theorems for pseudo-almost periodic solutions of differential equations with piecewise constant argument by means of pseudo-almost periodic solutions of relevant difference equations.     

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBV _loc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].     

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.     

Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u^′(t)=A(t)u(t)+h(t) and on , assuming that A(t) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A(t) has exponential dichotomy, that R(λ₀,A()) is almost periodic, that B,C(t,s)_t≥s are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument and that h, f, F are Stepanov-like pseudo-almost periodic for p>1 and continuous. To illustrate our abstract result, a concrete example is given.     

Behavior of bounded solutions for some almost periodic neutral partial functional differential equations

We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.     

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.      

On almost periodic solutions of logistic delay differential equations with almost periodic time dependence

In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458].     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

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.     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

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.     

逐段常变量微分方程组概周期型解存在唯一性

通过差分方程指数二分法,我们研究了一类具有变系数逐段常变量微分方程组概周期与伪概周期解的存在唯一性与渐近稳定性,改进了已有文献的结果,并且得到了差分方程满足指数二分性的一个充分条件．     

Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.     

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.     

PSEUDO-ALMOST PERIODIC SOLUTIONS TO SOME NON-AUTONOMOUS DIFFERENCE EQUATIONS WITH DELAY

This paper is concerned with the pseudo-almost periodic solutions to the nonautonomous difference equations with delay. Under some suitable assumptions, some results for the existence and uniqueness of pseudo-almost solutions are obtained.     

Cauchy matrix for linear almost periodic systems and some consequences

A novel method to construct the fundamental matrix for a linear almost periodic system is proposed, provided that the diagonal terms satisfy an average separation condition and the off-diagonal coefficients are L^∞-small. The idea is to transform the system in a set of Riccati type equations and use exponential dichotomy and its consequences. It is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients. Finally, our result enables us to obtain: (i) explicit almost periodic matrices Q(t), Q⁻¹(t) and Q^′(t), which diagonalize the original system and (ii) sufficient conditions for the stability. Two illustrative examples are shown.     

含参数泛函微分方程概周期正解的存在性

研究了一类含参数泛函微分方程概周期正解的存在性问题.结合有界性及渐近概周期性获得了系统存在概周期正解的几组充分条件,并将结果应用于几类种群动力学模型,分别获得了系统在概周期环境下存在概周期解的一组充分条件.