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     

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.     

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.     

Asymptotic Behavior of Quasi-Autonomous Expansive Type Evolution Systems in a Hilbert Space

We introduce the notion of almost expansive sequences and curves and study their ergodic and asymptotic properties in a Hilbert space H. We apply our results to study the asymptotic behavior of solutions to the quasi-autonomous expansive type evolution system (du/dt)(t) + f(t) ∈ Au(t) on [0, ∞).     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

Positive pseudo almost periodic solutions for a class of differential iterative equations with biological background

The article considers a class of differential iterative equations with biological background. We establish some sufficient conditions for the existence of positive pseudo almost periodic solutions by applying exponential dichotomy and contraction mapping principle. The obtained results extend some known ones.     

Sequences of harmonic maps in the 3‐sphere

We define two transforms of non‐conformal harmonic maps from a surface into the 3‐sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3‐sphere, H‐surfaces in Euclidean 3‐space and almost complex surfaces in the nearly Kähler manifold . As a consequence we can construct sequences of H‐surfaces and almost complex surfaces.     

Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions for fractional differential and integro-differential equations in the case when the forcing term belongs to the class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions.     

Some generalizations of locally and weakly locally uniformly convex space

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.