Existence and Exponential Stability of Almost Periodic Solutions to General BAM Neural Networks with Leakage Delays on Time Scales

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(School of Mathematics and Statistics, Zhaotong University, Zhaotong 657000, China)

Abstract: In this paper,the existence of almost periodic solutions to general BAM neural networks with leakage delays on time scales is first studied, by using the exponential dichotomy method of linear differential equations and fixed point theorem. Then, the exponential stability of almost periodic solutions to such BAM neural networks on time scales is discussed by utilizing differential inequality. Finally, an example is given to support our results in this paper and the results are up-to-date.

Keywords: Almost periodic solution; Neural network; Time scale; Leakage delay;Existence and exponential stability

§1. Introduction

Neural networks are a class of mathematical models which derive from the simulation of biological neural systems. It can be used to perform functions similar to those of the biological neural systems to store and process information. BAM neural networks are a class of associative memory neural networks with the functions of memory, storage and association. This neural networks consist of two layers (X-layer, Y-layer) nonlinear feedback networks in the form of a bipartite graph, that is, the neurons in layer X and layer Y are fully connected to each other.And there is no connection between neurons in the same layer. BAM neural networks were first proposed by Kosko in [3], which have a widespread application in pattern recognition, image handling, associative memory, combinatorial optimization and so on. Hence, many scholars have seriously considered the related properties and achieved many meaningful results (for examples, [2,4–6,12,17]).

It is well known that the research about BAM neural networks on time scales is more and more active and interesting. The advantage of studying BAM neural networks on time scales is the unification of continuous and discrete processes (for example, [1,10,11,18]). In particular,there were many relevant results about periodic solutions of BAM neural networks on time scales (for example, [7,8,13,19]).

Furthermore, there is a delay in information transmission between various parts of the system. The switching speed of electronic and mechanical devices is limited. Delays must exist in neural networks. Sequentially, the presence of delays often bring about oscillations, dispersion or instability of the neural networks. In[5],authors obtained the conditions for the existence and stability of the anti-periodic solutions to the general BAM neural networks with multiple delays,by fundamental solution matrix of coefficients, inequality and Lyapunov methods. In [19], the authors discussed similar conditions for the existence and stability of periodic solutions to such BAM neural networks on time scales.

In addition, the self-decay process of neurons is not instantaneous in a real neural networks circuit. Then, there is an unavoidable delay in the leakage term. Since leakage delays have an unsteady impact on the dynamic behavior of the neural networks. It is necessary and important to consider the influence of leakage delays on neural networks (for example, [9,14–16]).

Due to the above reasons, the study about the existence and exponential stability of almost periodic solutions to general BAM neural networks with leakage delays on time scales is meaningful. In this paper, we consider the existence and exponential stability of almost periodic solutions to the following general BAM neural networks with leakage delays on time scales.

§2. Existence of almost periodic solutions

In this section, we will discuss the adequate conditions for the existence of the almost periodic solutions to the system (1.1). Let

Remark 2.1.Since the system (1.1) has leakage delays, in order to obtain the existence of the solution of system (1.1), we first obtain the equivalent form (2.1) of system (1.1) through integral transformation in this part. Then, the existence of almost periodic solutions for system(2.1) is studied by using exponential dichotomy and fixed point theory.

§3. Exponential stability of almost periodic solution

In this section, we will obtain the adequate conditions for exponential stability of almost periodic solutions to the system (1.1).

§4. Example

In this section, we will demonstrate the practicability of the results with an example.Consider the following general BAM neural networks on almost periodic time scales:

Fig. 1 The behavior of the system (4.1) on R.

Fig. 2 The behavior of the system (4.1) on Z.

§5. Conclusion

In this paper, we study the existence and exponential stability of almost periodic solutions for general BAM neural networks with leakage delays on time scales. Comparing with the existing studies, the system(1.1)we study is not only with multiple delays, but also with leakage delays. As far as we know, in the published papers, the existence and exponential stability of almost periodic solutions for general BAM neural networks with leakage delays have not been studied. Our research in this paper will fill the blank of almost periodic solutions for general BAM neural networks. Our results are new and meaningful.