Abstract:
By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a periodic generalized food limited model with state dependent delays and distributed delays is studied, respectively.

Abstract:
A discrete periodic mutualism model with time delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, the existence of positive periodic solutions of the model is established.

Abstract:
We study the existence and global exponential stability of positive periodic solutions for a class of continuous-time generalized bidirectional neural networks with variable coefficients and delays. Discrete-time analogues of the continuous-time networks are formulated and the existence and global exponential stability of positive periodic solutions are studied using the continuation theorem of coincidence degree theory and Lyapunov functionals. It is shown that the existence and global exponential stability of positive periodic solutions of the continuous-time networks are preserved by the discrete-time analogues under some restriction on the discretization step-size. An example is given to illustrate the results obtained.

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).

Abstract:
By using an integral inequality, we establish some sufficient conditions for the existence and p-exponential stability of periodic solutions for a class of impulsive stochastic BAM neural networks with time-varying delays in leakage terms. Moreover, we present an example to illustrate the feasibility of our results. 1. Introduction Since it was proposed by Kosko (see [1]), the bidirectional associative memory (BAM) neural networks have attracted considerable attentions due to their extensive applications in classification of patterns, associative memories, image processing, and other areas. In the past few years, many scholars have obtained lots of good results on the dynamical behaviors analysis of BAM neural networks. The reader may see [2–8] and the references therein. But in a real nervous system, it is usually unavoidably affected by external perturbations which are in many cases of great uncertainty and hence may be treated as random. As pointed out by Haykin [9], in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. And the stability of neural networks could be stabilized or destabilized by some stochastic inputs [10]. Therefore, it is significant and of prime importance to consider the dynamics of stochastic neural networks. With respect to stochastic neural networks, there are many works on the stability. For example, in？？[11–17], the scholars studied the stability of different classes of stochastic neural networks. For other results on stochastic neural networks, the reader may see？？[18–23] and the references therein. However, the above results are mainly on the stability of considered stochastic neural networks. And it is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior. On the other hand, the neural networks are often subject to impulsive effects that in turn affect dynamical behaviors of the systems. Moreover, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term. Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks. Therefore, it is

Abstract:
In this paper, we first give a new definition of almost periodic time scales, two new definitions of almost periodic functions on time scales and investigate some basic properties of them. Then, as an application, by using the fixed point theorem in Banach space and the time scale calculus theory, we obtain some sufficient conditions for the existence and exponential stability of positive almost periodic solutions for a class of Nicholson's blowflies models on time scales. Finally, we present an illustrative example to show the effectiveness of obtained results. Our results show that under a simple condition the continuous-time Nicholson's blowflies models and their discrete-time analogue have the same dynamical behaviors.

Abstract:
By using critical point theory, some new sufficient conditions for the existence of solutions of impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions are obtained. Some examples are also given to illustrate our results. 1. Introduction Consider the following Duffing dynamic equations on time scales with impulsive effects where , is a regressive constant, , , , is continuous, and , are continuous. Obviously, system (1.1) covers Duffing equations (when ) The Duffing equation has been used to model the nonlinear dynamics of special types of mechanical and electrical systems. This differential equation has been named after the studies of Duffing in 1918 [1], has a cubic nonlinearity, and describes an oscillator. The main applications have been in electronics, but it can also have applications in mechanics and in biology. For example, the brain is full of oscillators at micro- and macrolevel [2]. There are applications in neurology, ecology, secure communications, chaotic synchronization, and so on. Due to the rich behaviour of these equations, the most general forced forms of the Duffing equation (1.2) have been studied by many researchers [3–12]. The study of dynamic equations on time scales goes back to its founder Stefan Hilger [13], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models [14–16]. Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [17–24]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory, and so on (see [25–32]). In addition, system (1.1) also includes which were studied by papers [33, 34], and some existence results were obtained by using some critical point theorems. Our purpose in this paper is to study the variational structure of problem (1.1) in an appropriate space of functions and the existence of solutions for problem (1.1) by means of some critical point theorems. The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, we will study the variational structure of

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

Abstract:
Sufficient conditions are obtained for the existence of at least one positive periodic solution of a periodic cooperative model with delays and impulses by using Mawhin's continuation theorem of coincidence degree theory.

Abstract:
We investigate local robust stability of fuzzy neural networks (FNNs) with time-varying and S-type distributed delays. We derive some sufficient conditions for local robust stability of equilibrium points and estimate attracting domains of equilibrium points except unstable equilibrium points. Our results not only show local robust stability of equilibrium points but also allow much broader application for fuzzy neural network with or without delays. An example is given to illustrate the effectiveness of our results.