New results on anti-periodic solutions for SICNNs with oscillating coefficients in leakage terms
Introduction
It was well known that the following delayed differential system:has been used to describe shunting inhibitory cellular neural networks (SICNNs), which has been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, image processing, and population dynamics [1], [2], [3], [4], [5], [6]. Here is the cell at the position of the lattice, and the r neighborhood of is defined as follows: is similarly specified. represents the state of the cell , means the passive decay rate of the cell activity, and describe the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell , and . Moreover, the first term in each of the right sides of SICNNs (1.1) corresponds to a stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as forgetting or leakage terms (see for instance [7], [8], [9], [10], [11], [12], [13]).
On the other hand, the existence and stability of anti-periodic solutions play a key role in characterizing the behavior of nonlinear differential equations (see [14], [15], [16], [17], [18]). The signal transmission process of neural networks can often be described as an anti-periodic process. In particular, assume that the leakage term coefficient function is not oscillating, i.e., some sufficient conditions ensuring the existence and exponential stability of anti-periodic solutions for SICNNs (1.1) and its generalized models have been established in [19], [20] and [21], [22], [23], [24], [25], [26], respectively.
In the past few decades, people have paid much attention to the stability on biological systems with oscillating coefficients appear in linearizations of population dynamics models with seasonal fluctuations, where during some seasons the death or harvesting rates may be greater than the birth rate (see [27], [28], [29], [30], [31]). This motivates us to find the new criteria to guarantee the existence and exponential stability of anti-periodic solutions for (1.1) with oscillating coefficients in leakage terms.
The main purpose of this paper is to establish some sufficient conditions on the existence and exponential stability of anti-periodic solutions for (1.1) without the conditions (1.2). To the best of our knowledge, this has not been done before.
The remaining of this paper is organized as follows. In Section 2, we give some basic definitions and lemmas, which play an important role in Section 3 to establish the existence of anti-periodic solutions of (1.1). Here we also study the global exponential stability of anti-periodic solutions. The paper concludes with an example to illustrate the effectiveness of the obtained results by numerical simulation.
Section snippets
Preliminary results
In this section, we shall first recall some basic definitions and lemmas which are used in what follows.
The initial conditions associated with system (1.1) are of the formwhere BC denotes the set of all real-valued bounded continuous functions defined on . Set Let be continuous in t. is said to be T-anti-periodic on if where T is a positive constant. Also, we let
Main results
In this section, we establish sufficient conditions on the existence and exponential stability of anti-periodic solutions of (1.1). Theorem 3.1 Let , and hold. Then, there exists a T-anti-periodic solution of system (1.1). Proof First, we consider the mapping which is defined in Lemma 2.1 as It is easy to see that is a closed convex subset of . According to the definition of the norm of Banach space we get
An example
In this section, we give an example and its numerical simulations to demonstrate the results obtained in previous sections. Example 4.1 Consider the following SICNNs with oscillating coefficients in leakage terms:
Acknowledgments
The author would like to express the sincere appreciation to the editor and reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by National Natural Science Foundation of China (Grant no. 11171098).
Zhiwen Long was born in Hunan, China, in 1980. He received the B.S. degree in mathematics, from the Hunan University of Arts and Science, Hunan, China, in 2003, and the M.S. degree in applied mathematics from the Central South University, Changsha, China, in 2005, respectively. Currently he is a Lecturer in the Department of Mathematics and Econometrics, Hunan University of Humanities, Science and Technology, Loudi, Hunan, PR China. His research interests are in the areas of dynamics of neural
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Zhiwen Long was born in Hunan, China, in 1980. He received the B.S. degree in mathematics, from the Hunan University of Arts and Science, Hunan, China, in 2003, and the M.S. degree in applied mathematics from the Central South University, Changsha, China, in 2005, respectively. Currently he is a Lecturer in the Department of Mathematics and Econometrics, Hunan University of Humanities, Science and Technology, Loudi, Hunan, PR China. His research interests are in the areas of dynamics of neural networks and biological mathematics.