Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ?‐problem converges in L² to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma‐convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.     

Pseudo–almost‐periodic solutions for delayed differential equations with integrable dichotomies and bi–almost‐periodic Green functions

Using the existence of integrable bi–almost‐periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost‐periodic mild solutions under quite general hypotheses for the differential equation with constant delay in a Banach space X, where τ>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.     

A note on Arzela–Ascoli's lemma in almost periodic problems

This paper is concerned with a generalized Arzela–Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela–Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of μ−pseudo almost periodic solutions to some evolution equations

In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace‐Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of μ‐pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Decoupled schemes for unsteady MHD equations. I. time discretization

In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional stability and convergence of this scheme are provided. The optimal error estimates for velocity and magnet are provided, and the optimal error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017     

一类具有收获率的脉冲Lotka-Volterra竞争合作系统的八个正概周期解

研究了一类具有收获率的脉冲Lotka-Volterra竞争合作系统的正概周期解.通过利用重合度理论延拓定理、概周期理论和不等式分析技巧,获得了系统至少存在8个正概周期解的充分条件,推广和改进了早期文献的相关结果.