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

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.

Abstract:
In this paper, we first propose a single-species system with impulsive effects on time scales and by establishing some new comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then we prove a Massera type theorem for impulsive dynamic equations on time scales and based on this theorem, we establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Finally, we give an example to show the feasibility of our main results. Our example also shows that the continuous time system and its corresponding discrete time system have the same dynamics. Our results of this paper are completely new.

Abstract:
By using the theory of calculus on time scales and -matrix theory, the unique existence theorem of solution of almost periodic differential equations on almost periodic time scales is established. The result can be used to a large of dynamic systems.

Abstract:
In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new.

Abstract:
We first propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then by using the theory of calculus on time scales and some mathematical methods, some basic results about almost periodic differential equations on almost periodic time scales are established. Based on these results, a class of high-order Hopfield neural networks with variable delays are studied on almost periodic time scales, and some sufficient conditions are established for the existence and global asymptotic stability of the almost periodic solution. Finally, two examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results. 1. Introduction It is well known that in celestial mechanics, almost periodic solutions and stable solutions to differential equations or difference equations are intimately related. In the same way, stable electronic circuits, ecological systems, neural networks, and so forth exhibit almost periodic behavior. A vast amount of researches have been directed toward studying these phenomena (see [1–6]). Also, the theory of calculus on time scales (see [7] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [8] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. However, there are no concepts of almost periodic time scales and almost periodic functions on time scales, so that it is impossible for us to study almost periodic solutions to differential equations on time scales. Motivated by the above, the main purpose of this paper is to propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then establish some basic results about almost periodic differential equations on almost periodic time scales by using the theory of calculus on time scales and some mathematical methods. Furthermore, based on these results, as an application, we consider the following high-order Hopfield neural networks with variable delays on time scales: where corresponds to the number of units in a neural network, corresponds to the state vector of the th unit at the time , represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and

Abstract:
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate the main results.

Abstract:
We present a model with feedback controls based on ecology theory, which effectively describes the competition and cooperation of enterprise cluster in real economic environments. Applying the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and the existence of uniformly asymptotically stable almost periodic solution of the system are obtained. 1. Introduction In recent years, a few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1–9] and references cited therein. In [1], two models from biology were given and explained by economic view, and sufficient conditions were obtained to guarantee the coexistence and stability of enterprise clusters. In [3], the developing strategy of enterprise clusters was analyzed based on the logistic model, and the suggestions of constructing cooperative relation and choosing generalization or specialization tactics for commodity were put forward. In addition, based on the theoretical model of ecological population science, Wang and Pan [6] made a detailed analysis to the equilibrium mechanism of enterprise clusters, including net model and center halfback model and drew a conclusion that the relationship of pierce competition and beneficial cooperation among enterprise clusters was the crucial factor for them to keep stability. More related research about enterprises cluster one can refer to the literatures [10–13]. Recently, the literature [5] considered the competition and cooperation system of two enterprises based on ecosystem: where , represent the output of enterprises and , , are the intrinsic growth rate, denotes the carrying capacity of market under nature unlimited conditions, , are the competitive coefficients of two enterprises, , are the initial production of two enterprises. Accordingly, we consider now the equation with nonconstant coefficients, which can be obtained as a modified system (1) with variable coefficients (Letting , , , in system (1)): In real world, the situation of enterprises is often distributed by unpredictable forces which can result in changes in enterprises' parameters such as intrinsic growth rates. So it is necessary to study models with control variables which are so-called disturbance functions [14–17]. As well known, both continuous and discrete systems are very important in implementation and

Abstract:
We exhibit examples of almost periodic Verblunsky coefficients for which Herman's subharmonicity argument applies and yields that the associated Lyapunov exponents are uniformly bounded away from zero.

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).