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Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions

Weiyuan Zhang1*, Junmin Li2 and Minglai Chen2

Author Affiliations

1 Institute of Mathematics and Applied Mathematics, Xianyang Normal University, Xianyang, 712000, China

2 School of Science, Xidian University, Xi’an, Shaanxi, 710071, China

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Boundary Value Problems 2013, 2013:105  doi:10.1186/1687-2770-2013-105


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/105


Received:18 October 2012
Accepted:12 April 2013
Published:26 April 2013

© 2013 Zhang et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, both global exponential stability and periodic solutions are investigated for a class of delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. By employing suitable Lyapunov functionals, sufficient conditions of the global exponential stability and the existence of periodic solutions are established for reaction-diffusion BAM neural networks with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature. A numerical example is given to show the effectiveness of the obtained results.

Keywords:
neural networks; reaction-diffusion; mixed time delays; global exponential stability; Poincaré mapping; Lyapunov functional

1 Introduction

Neural networks (NNs) have been extensively studied in the past few years and have found many applications in different areas such as pattern recognition, associative memory, combinatorial optimization, etc. Delayed versions of NNs were also proved to be important for solving certain classes of motion-related optimization problems. Various results concerning the dynamical behavior of NNs with delays have been reported during the last decade (see, e.g., [1-7]). Recently, the authors in [1] and [2] considered the problem of exponential passivity analysis for uncertain NNs with time-varying delays and passivity-based controller design for Hopfield NNs, respectively.

Since NNs related to bidirectional associative memory (BAM) were proposed by Kosko [8], the BAM NNs have been one of the most interesting research topics and have attracted the attention of researchers. In the design and applications of networks, the stability of the designed BAM NNs is one of the most important issues (see, e.g., [9-12]). Many important results concerning mainly the existence and stability of equilibrium of BAM NNs have been obtained (see, e.g., [9-15]).

However, strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. So, we must consider that the activations vary in space as well as in time. In [16-34], the authors considered the stability of NNs with diffusion terms which were expressed by partial differential equations. In particular, the existence and attractivity of periodic solutions for non-autonomous reaction-diffusion Cohen-Grossberg NNs with discrete time delays were investigated in [20]. The authors derived sufficient conditions on the stability and periodic solutions of delayed reaction-diffusion NNs (RDNNs) with Neumann boundary conditions in [21-25]. In these works, due to the divergence theorem employed, a negative integral term with gradient was removed in their deduction. Therefore, the stability criteria acquired by them do not contain diffusion terms; that is to say, the diffusion terms do not have any effect on their deduction and results. Meanwhile, some conditions dependent on the diffusion coefficients were given in [30,32-34] to ensure the global exponential stability and periodicity of RDNNs with Dirichlet boundary conditions based on 2-norm.

To the best of our knowledge, there are few reports about global exponential stability and periodicity of RDNNs with mixed time delays and Dirichlet boundary conditions, which are very important in theories and applications and also are a very challenging problem. In this paper, by employing suitable Lyapunov functionals, we shall apply inequality techniques to establish global exponential stability criteria of the equilibrium and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature [22,29].

Throughout this paper, we need the following notations. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M1">View MathML</a> denotes the n-dimensional Euclidean space. We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M2">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M3">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M5">View MathML</a>.

The remainder of this paper is organized as follows. In Section 2, the basic notations, model description and assumptions are introduced. In Sections 3 and 4, criteria are proposed to determine global exponential stability, and periodic solutions are considered for reaction-diffusion recurrent neural networks with mixed time delays, respectively. An illustrative example is given to illustrate the effectiveness of the obtained results in Section 5. We also conclude this paper in Section 6.

2 Model description and preliminaries

In this paper, the RDNNs with mixed time delays are described as follows:

(1)

The RDNNs model given in (1) can be regarded as RDNNs with two layers, where m is the number of neurons in the first layer and n is the number of neurons in the second layer. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M7">View MathML</a>, Ω is a compact set with smooth boundary Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M8">View MathML</a> in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M9">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M11">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M13">View MathML</a> represent the state of the ith neuron in the first layer and the jth neuron in the second layer at time t and in the space x, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M19">View MathML</a> are known constants denoting the synaptic connection strengths between the neurons in the two layers, respectively; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M25">View MathML</a> denote the activation functions of the neurons and the signal propagation functions, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M27">View MathML</a> denote the external inputs on the ith neuron and jth neuron, respectively; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M29">View MathML</a> are differentiable real functions with positive derivatives defining the neuron charging time; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M31">View MathML</a> represent continuous time-varying delay and discrete delay, respectively; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>, stand for the transmission diffusion coefficient along the ith neuron and jth neuron, respectively.

System (1) is supplemented with the following boundary conditions and initial values:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M37">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M38">View MathML</a>

(3)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M41">View MathML</a> is the outer normal vector of Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M42">View MathML</a> are bounded and continuous, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M43">View MathML</a>. It is the Banach space of continuous functions which maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M44">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M45">View MathML</a> with the topology of uniform convergence for the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M46">View MathML</a>

Remark 1 Some famous NN models became a special case of system (1). For example, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M50">View MathML</a>), the special case of model (1) is the model which has been studied in [13-15]. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>, system (1) became NNs with distributed delays and reaction-diffusion terms [18,22,29].

Throughout this paper, we assume that the following conditions are made.

(A1) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M31">View MathML</a> are piecewise-continuous of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M57">View MathML</a> on the closure of each continuity subinterval and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M58">View MathML</a>

with some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M62">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>.

(A2) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M65">View MathML</a> are piecewise-continuous of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M57">View MathML</a> on the closure of each continuity subinterval and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M67">View MathML</a>

(A3) The activation functions and the signal propagation functions are bounded and Lipschitz continuous, i.e., there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M75">View MathML</a>

(A4) The delay kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M76">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>) are real-valued non-negative continuous functions that satisfy the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M80">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M82">View MathML</a>;

(iii) There exist a positive μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M83">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M84">View MathML</a> be the equilibrium point of system (1).

Definition 1 The equilibrium point of system (1) is said to be globally exponentially stable if we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85">View MathML</a> such that there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M88">View MathML</a>

(4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>.

Remark 2 It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1).

Lemma 1[33]

Let Ω be a cube<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M90">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M91">View MathML</a>), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M92">View MathML</a>be a real-valued function belonging to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M93">View MathML</a>which vanishes on the boundaryΩ of Ω, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M94">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M95">View MathML</a>

(5)

3 Global exponential stability

Now we are in a position to investigate the global exponential stability of system (1). By constructing a suitable Lyapunov functional, we arrive at the following conclusion.

Theorem 1Let (A1)-(A4) be in force. If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M97">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M102">View MathML</a>

(6)

in which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73">View MathML</a>are Lipschitz constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M112">View MathML</a>, then the equilibrium point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M113">View MathML</a>of system (1) is unique and globally exponentially stable.

Proof If (6) holds, we can always choose a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M114">View MathML</a> (may be very small) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M115">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M116">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>.

Let us consider the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M119">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M120">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>.

From (8) and (A4), we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M125">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M127">View MathML</a> are continuous for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M128">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M129">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M131">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M132">View MathML</a>. Thus there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M133">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M134">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M135">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>.

By using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M138">View MathML</a>, obviously, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M140">View MathML</a>

(10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M143">View MathML</a> is any solution of model (1). Rewrite model (1) as

(11)

(12)

Multiplying (11) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M146">View MathML</a> and integrating over Ω yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M147">View MathML</a>

(13)

According to Green’s formula and the Dirichlet boundary condition, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M148">View MathML</a>

(14)

Moreover from Lemma 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M149">View MathML</a>

(15)

From (11)-(15), (A2), (A3) and the Holder integral inequality, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M150">View MathML</a>

(16)

Multiplying both sides of (12) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M151">View MathML</a>, similarly, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M152">View MathML</a>

(17)

Choose a Lyapunov functional as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M153">View MathML</a>

Its upper Dini-derivative along the solution to system (1) can be calculated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M154">View MathML</a>

(18)

From (18) and the Young inequality, we can conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M155">View MathML</a>

(19)

From (6), we can conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M156">View MathML</a>

(20)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M157">View MathML</a>

(21)

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M158">View MathML</a>

(22)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M159">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87">View MathML</a>.

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M161">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87">View MathML</a> is a constant. This implies that the solution of (1) is globally exponentially stable. This completes the proof of Theorem 1. □

Remark 3 In this paper, the derived sufficient condition includes diffusion terms. Unfortunately, in the proof in the previous papers [21-24], a negative integral term with gradient is left out in their deduction. This leads to the fact that those criteria are irrelevant to the diffusion term. Obviously, Lyapunov functional to construct is more general and our results expand the model in [22,29].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M52">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M167">View MathML</a>), system (1) becomes the following BAM NNs with distributed delays and reaction-diffusion terms:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M168">View MathML</a>

(23)

For (23), we get the following result.

Corollary 1Let (A1)-(A4) be in force. If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M97">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M174">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M175">View MathML</a>

(24)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73">View MathML</a>are Lipschitz constants. Then the equilibrium point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M113">View MathML</a>of system (1) is unique and globally exponentially stable.

4 Periodic solutions

In this section, we consider the stability criterion for periodic oscillatory solutions of system (1), in which external input <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M104">View MathML</a>, are continuously periodic functions with period ω, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M189">View MathML</a>

By constructing a Poincaré mapping, the existence of a unique ω-periodic solution and its stability are readily established.

Theorem 2Let (A1)-(A4) be in force. There exists only oneω-periodic solution of system (1), and all other solutions converge exponentially to it as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M190">View MathML</a>if there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M192">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M100">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M167">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M198">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M199">View MathML</a>

(25)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M34">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M73">View MathML</a>are Lipschitz constants in (A3).

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M208">View MathML</a>, we denote the solutions of system (1) through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M212">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M213">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M214">View MathML</a>

respectively. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M215">View MathML</a>

Clearly, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M217">View MathML</a>. Now, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M218">View MathML</a>

Thus, we can obtain from system (1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M219">View MathML</a>

We consider the following Lyapunov functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M220">View MathML</a>

By a minor modification of the proof of Theorem 1, we can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M221">View MathML</a>

(26)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>,in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M87">View MathML</a> is a constant. Now, we can choose a positive integer N such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M224">View MathML</a>

(27)

Defining a Poincaré mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M225">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M226">View MathML</a>

(28)

due to the periodicity of system, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M227">View MathML</a>

(29)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M228">View MathML</a>, then from (26)-(29) we can derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M229">View MathML</a>

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230">View MathML</a> is a contraction mapping. Therefore, there exists a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M231">View MathML</a>, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M232">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M233">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M234">View MathML</a> is also a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230">View MathML</a>. Because of the uniqueness of a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M230">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M237">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M238">View MathML</a> be the solution of system (1) through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M239">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M240">View MathML</a> is also a solution of system (1). Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M241">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M243">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M63">View MathML</a>.

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M245">View MathML</a> is exactly one ω-periodic solution of system (1), and it is easy to see that all other solutions of system (1) converge exponentially to it as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M190">View MathML</a>. The proof is completed. □

5 Illustration example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

Example 1 Consider the following system on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M247">View MathML</a>:

(30)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M250">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M256">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M288">View MathML</a>. By simple calculation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M291">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M292">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M293">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M294">View MathML</a>

(33)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M295">View MathML</a>

(34)

that is, (6) holds.

The simulation results are shown in Figures 1-8. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M296">View MathML</a>, the states surfaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M297">View MathML</a> are shown in Figures 1-2, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M298">View MathML</a>, the states surfaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M299">View MathML</a> are shown in Figures 3-4. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M300">View MathML</a>, the states surfaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M301">View MathML</a> are shown in Figures 5-6, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M298">View MathML</a>, the states surfaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M303">View MathML</a> are shown in Figures 7-8, which illustrates that the system states in (30) converge to equilibrium solution. Therefore, it follows from Theorem 1 and the simulation study that (30) has one unique equilibrium solution which is globally exponentially stable.

thumbnailFigure 1. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M304">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305">View MathML</a>.

thumbnailFigure 2. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M306">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305">View MathML</a>.

thumbnailFigure 3. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M308">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309">View MathML</a>.

thumbnailFigure 4. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M310">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309">View MathML</a>.

thumbnailFigure 5. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M312">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305">View MathML</a>.

thumbnailFigure 6. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M314">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M305">View MathML</a>.

thumbnailFigure 7. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M316">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309">View MathML</a>.

thumbnailFigure 8. The surface of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M318">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M309">View MathML</a>.

Remark 4 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M320">View MathML</a>, the conditions of Corollary 3.2 in [22] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/105/mathml/M321">View MathML</a>, under the conditions of Example 1, the conditions of Theorem 1 in [29] are not satisfied. However, by (31)-(34) and Theorem 1, we can derive that (30) has one unique equilibrium solution which is globally exponentially stable.

6 Conclusions

In this paper, by employing suitable Lyapunov functionals, Young’s inequality and Hölder’s inequality techniques, global exponential stability criteria of the equilibrium point and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions have been derived, respectively. The derived criteria contain and extend some previous NNs in the literature. Hence, our results have an important significance in design as well as in applications of periodic oscillatory NNs with mixed time delays. An example has been given to show the effectiveness of the obtained results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

WZ designed and performed all the steps of proof in this research and also wrote the paper. JL and MC participated in the design of the study and suggested many good ideas that made this paper possible and helped to draft the first manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work is partially supported by the National Natural Science Foundation of China under Grant No. 60974139, the Special Research Project in Shaanxi Province Department of Education (2013JK0578) and Doctor Introduced project of Xianyang Normal University under Grant No. 12XSYK008.

References

  1. Kwon, OM, Park, JH, Lee, SM, Cha, EJ: A new augmented Lyapunov-Krasovskii functional approach to exponential passivity for neural networks with time-varying delays. Appl. Math. Comput.. 217(24), 10231–10238 (2011). Publisher Full Text OpenURL

  2. Ji, DH, Koo, JH, Won, SC, Lee, SM, Park, JH: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput.. 217(13), 6168–6175 (2011). Publisher Full Text OpenURL

  3. Lee, SM, Kwon, OM, Park, JH: A novel delay-dependent criterion for delayed neural networks of neutral type. Phys. Lett. A. 374(17-18), 1843–1848 (2010). Publisher Full Text OpenURL

  4. Cao, J, Wang, J: Global exponential stability and periodicity of recurrent neural networks with time delays. IEEE Trans. Circuits Syst. I, Regul. Pap.. 52(5), 920–931 (2005)

  5. Huang, C, Cao, J: Convergence dynamics of stochastic Cohen-Grossberg neural networks with unbounded distributed delays. IEEE Trans. Neural Netw.. 22(4), 561–572 (2011). PubMed Abstract | Publisher Full Text OpenURL

  6. Ensari, T, Arik, S: Global stability of a class of neural networks with time varying delay. IEEE Trans. Circuits Syst. II, Express Briefs. 52(3), 126–130 (2005)

  7. Rakkiyappan, R, Balasubramaniam, P: Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays. Appl. Math. Comput.. 198(2), 526–533 (2008). Publisher Full Text OpenURL

  8. Kosko, B: Bi-directional associative memories. IEEE Trans. Syst. Man Cybern.. 18(1), 49–60 (1988). Publisher Full Text OpenURL

  9. Park, JH, Kwon, OM: Delay-dependent stability criterion for bidirectional associative memory neural networks with interval time-varying delays. Mod. Phys. Lett. B. 23(1), 35–46 (2009). Publisher Full Text OpenURL

  10. Park, JH, Park, CH, Kwon, OM, Lee, SM: New stability criterion for bidirectional associative memory neural networks of neutral-type. Appl. Math. Comput.. 199(2), 716–722 (2008). Publisher Full Text OpenURL

  11. Park, JH, Kwon, OM: On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with time-varying delays. Appl. Math. Comput.. 199(2), 435–446 (2008). Publisher Full Text OpenURL

  12. Cao, J, Wang, L: Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans. Neural Netw.. 13(2), 457–463 (2002). PubMed Abstract | Publisher Full Text OpenURL

  13. Park, J, Lee, SM, Kwon, OM: On exponential stability of bidirectional associative memory neural networks with time-varying delays. Chaos Solitons Fractals. 39(3), 1083–1091 (2009). Publisher Full Text OpenURL

  14. Wu, R: Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays. Nonlinear Anal., Real World Appl.. 11(1), 562–573 (2010). Publisher Full Text OpenURL

  15. Liu, X, Martin, R, Wu, M: Global exponential stability of bidirectional associative memory neural networks with time delays. IEEE Trans. Neural Netw.. 19(2), 397–407 (2008). PubMed Abstract | Publisher Full Text OpenURL

  16. Zhang, W, Li, J: Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions. Bound. Value Probl.. 2012, Article ID 2. doi:10.1186/1687-2770-2012-2 (2012)

  17. Zhang, W, Li, J, Shi, N: Stability analysis for stochastic Markovian jump reaction-diffusion neural networks with partially known transition probabilities and mixed time delays. Discrete Dyn. Nat. Soc.. 2012, Article ID 524187. doi:10.1155/2012/524187 (2012)

  18. Song, Q, Zhao, Z, Li, YM: Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms. Phys. Lett. A. 335(2-3), 213–225 (2005). Publisher Full Text OpenURL

  19. Zhang, W, Li, J: Global exponential stability of reaction-diffusion neural networks with discrete and distributed time-varying delays. Chin. Phys. B. 20(3), Article ID 030701 (2011)

  20. Pan, J, Zhan, Y: On periodic solutions to a class of non-autonomously delayed reaction-diffusion neural networks. Commun. Nonlinear Sci. Numer. Simul.. 16(1), 414–422 (2011). Publisher Full Text OpenURL

  21. Song, Q, Cao, J: Global exponential stability and existence of periodic colutions in BAM with delays and reaction-diffusion terms. Chaos Solitons Fractals. 23(2), 421–430 (2005). Publisher Full Text OpenURL

  22. Cui, B, Lou, X: Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solitons Fractals. 27(5), 1347–1354 (2006). Publisher Full Text OpenURL

  23. Zhao, H, Wang, G: Existence of periodic oscillatory solution of reaction-diffusion neural networks with delays. Phys. Lett. A. 343(5), 372–383 (2005). Publisher Full Text OpenURL

  24. Song, Q, Cao, J, Zhao, Z: Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal., Real World Appl.. 7(1), 65–80 (2006). Publisher Full Text OpenURL

  25. Wang, Z, Zhang, H: Global asymptotic stability of reaction-diffusion Cohen-Grossberg neural network with continuously distributed delays. IEEE Trans. Neural Netw.. 21(1), 39–49 (2010). PubMed Abstract | Publisher Full Text OpenURL

  26. Zhang, X, Wu, S, Li, K: Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Commun. Nonlinear Sci. Numer. Simul.. 16(3), 1524–1532 (2011). Publisher Full Text OpenURL

  27. Wang, L, Zhang, R, Wang, Y: Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays. Nonlinear Anal., Real World Appl.. 10(2), 1101–1113 (2009). Publisher Full Text OpenURL

  28. Zhu, Q, Li, X, Yang, X: Exponential stability for stochastic reaction-diffusion BAM neural networks with time-varying and distributed delays. Appl. Math. Comput.. 217(13), 6078–6091 (2011). Publisher Full Text OpenURL

  29. Lou, X, Cui, B, Wu, W: On global exponential stability and existence of periodic solutions for BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solitons Fractals. 36(4), 1044–1054 (2008). Publisher Full Text OpenURL

  30. Zhang, W, Li, J, Chen, M: Dynamical behaviors of impulsive stochastic reaction-diffusion neural networks with mixed time delays. Abstr. Appl. Anal.. 2012, Article ID 236562. doi:10.1155/2012/236562 (2012)

  31. Wang, Z, Zhang, H, Li, P: An LMI approach to stability analysis of reaction-diffusion Cohen-Grossberg neural networks concerning Dirichlet boundary conditions and distributed delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern.. 40(6), 1596–1606 (2010)

  32. Lu, J: Robust global exponential stability for interval reaction-diffusion Hopfield neural networks with distributed delays. IEEE Trans. Circuits Syst. II, Express Briefs. 54(12), 1115–1119 (2007)

  33. Lu, J, Lu, L: Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos Solitons Fractals. 39(4), 1538–1549 (2009). Publisher Full Text OpenURL

  34. Wang, J, Lu, J: Global exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms. Chaos Solitons Fractals. 38(3), 878–885 (2008). Publisher Full Text OpenURL