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Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales  [PDF]
Yongkun Li,Chao Wang
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/341520
Abstract: Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scale or , our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales. 1. Introduction In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [1, 2] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [3] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his fundamental work. It has been created in order to unify the study of differential and difference equations. Many papers have been published on the theory of dynamic equations on time scales [4–10]. Also, the existence of almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is among the most attractive topics in qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [11–29]. However, there are no concepts of almost periodic functions on time scales so that it is impossible for us to study almost periodic solutions for dynamic equations on time scales. Motivated by the above, our main purpose of this paper is firstly to propose a concept of uniformly almost periodic functions on time scales and investigate some basic properties of them. Then we study the existence and uniqueness of almost periodic solutions to linear dynamic equations on almost time scales. Finally, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales. The organization of this paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in the later sections. In Section 3, we propose the concept of uniformly almost periodic functions on almost periodic time scales and investigate the basic properties of uniformly almost
Some remarks on almost periodic time scales and almost periodic functions on time scales  [PDF]
Yongkun Li
Mathematics , 2015,
Abstract: In this note we communicate some important remarks about the concepts of almost periodic time scales and almost periodic functions on time scales that are proposed by Wang and Agarwal in their recent papers (Adv. Difference Equ. (2015) 2015:312; Adv. Difference Equ. (2015) 2015:296; Math. Meth. Appl. Sci. 2015, DOI: 10.1002/mma.3590).
Almost Periodic Solutions to Dynamic Equations on Time Scales and Applications
Yongkun Li,Chao Wang
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/463913
Abstract: We first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts and results, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples and numerical simulations given to illustrate our results are plausible and meaningful.
Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales  [PDF]
Yan Wang,Lan Li
Discrete Dynamics in Nature and Society , 2013, DOI: 10.1155/2013/784198
Abstract: We firstly introduce the concept and the properties of almost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept of -almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales. 1. Introduction The theory of dynamic equations on time scales was first introduced by Hilger [1]. The study of dynamic equations on time scales helps to avoid studying results twice, once for differential equations and once for difference equations. In recent years, the theory of first order and second order dynamic equations on time scales has been studied, and some important results have been presented in [2–6]. However, to the best of our knowledge, there are no results on the existence of almost periodic solutions for the second order dynamic equations on time scales. The aim of this paper is to consider the existence of almost periodic solutions for second order dynamic equations on time scales. The concept of almost periodicity was first introduced by Bohr [7] and later generalized by Bochner, Fink, N’Guérékata, and Shen and Yi and others (see [8–11]). Recently, Guan and Wang [12] and Li and Wang [13, 14] developed the theory of almost periodic functions on time scales, which do not only unify the almost periodic functions on and the almost periodic sequences on but also extend to nontrivial time scales, for example, -difference equations. The existence and uniqueness solutions for second order dynamic equations have become important in recent years in mathematical models and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The existence of oscillatory and nonoscillatory solutions for second order equations has been studied in [15–18] (and the references therein). This paper is concerned with the second order dynamic equation as follows: where is almost periodic in uniformly. Such a type of equation appears in many problems of applications, such as Brillouin focusing systems [19, 20], nonlinear elasticity [21], and Ermakov-Pinney equations [22, 23]. In this paper, we consider the existence of almost periodic solutions (1) and present alternative theorems for second order dynamic equations. In order to do this, we introduce a new concept called almost periodicity, which generalizes
Weighted Pseudo Almost-Periodic Functions and Applications to Semilinear Evolution Equations
Jun Zhang,Ti-Jun Xiao,Jin Liang
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/179525
Abstract: We first give a solution to a key problem concerning the completenessof the space of weighted pseudo almost-periodic functions and then establish a newcomposition theorem with respect to these functions. Some important remarks withconcrete examples are also presented. Moreover, we prove an existence theorem for theweighted pseudo almost-periodic mild solution to the semilinear evolution equation:()=()
Algebras of Almost Periodic Functions with Bohr-Fourier Spectrum in a Semigroup: Hermite Property and its Applications  [PDF]
L. Rodman,Ilya M. Spitkovsky
Mathematics , 2008,
Abstract: It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener--Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.
Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations
Junwei Liu and Chuanyi Zhang
Advances in Difference Equations , 2013, DOI: 10.1186/1687-1847-2013-11
Abstract: In this paper, we introduce the concept of piecewise pseudo almost periodic functions on a Banach space and establish some composition theorems of piecewise pseudo almost periodic functions. We apply these composition theorems to investigate the existence of piecewise pseudo almost periodic (mild) solutions to abstract impulsive differential equations. In addition, the stability of piecewise pseudo almost periodic solutions is considered.
A new composition theorem for $S^p$-weighted pseudo almost periodic functions and applications to semilinear differential equations
Zhi-Han Zhao,Yong-Kui Chang,G.M. N’Guérékata
Opuscula Mathematica , 2011,
Abstract: In this paper, we establish a new composition theorem for $S^p$-weighted pseudo almost periodic functions under weaker conditions than the Lipschitz ones currently encountered in the literatures. We apply this new composition theorem along with the Schauder's fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semilinear differential equation in a Banach space.
Almost periodic generalized functions  [PDF]
Chikh Bouzar,Mohammed Taha Khalladi
Mathematics , 2011,
Abstract: The aim of this paper is to introduce and to study an algebra of almost periodic generalized functions containing the classical Bohr almost periodic functions as well as almost periodic Schwartz distributions
Meromorphic almost periodic functions  [PDF]
S. Favorov,N. Parfyonova
Mathematics , 2007,
Abstract: We study meromorphic functions in a strip almost periodic with respect to the spherical metric. Then we get a complete description of zeros and poles for this class of functions, find a condition for a meromorphic almost periodic function to be a quotient of two holomorphic almost periodic function, find a necessary and suficient condition for sum and product of almost periodic meromorphic functions to be almost periodic.
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