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.     

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.     

Approximation of almost periodic functions by periodic ones

It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on =(+).     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

Population models in almost periodic environments

We establish the basic theory of almost periodic sequences on ?₊. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.     

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 multivalued maps

We consider multivalued maps with compact images in a complete metric spaceU. Conditions on the set of selections of a multivalued mapF are obtained which are necessary and sufficient for the multivalued mapF to be Stepanov almost periodic. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 82–90, July, 2000.     

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.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

Reducibility of a linear system of differential equations with odd almost periodic coefficients

A theorem on the Kolmogorov reducibility of a system of ordinary differential equations with odd almost periodic coefficients is proved. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 109–119, July, 1996.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.     

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].     

概周期系统概周期解的存在唯一性

运用Ｌｉａｐｕｎｏｖ函数，研究了概周期系统概周期解的存在唯一性，得到了一个方便应用的判定定理．     

Transformation semigroup compactifications and norm continuity of weakly almost periodic functions

We prove if there exists a separately continuous action of a topologically right simple semitopological semigroupS on a topological spaceX and ifS acts topologically surjective onX then each weakly almost periodic function onX, with respect toS, is left norm continuous.     

An additive property of almost periodic sets

We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method. This equivalence is used to give simple proofs of theorems of J. Brüdern.     

Almost periodic homogeneous linear difference systems without almost periodic solutions

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.     

A note on quasi-weakly almost periodic point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606(2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in[Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502(2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.