Abstract

In this manuscript, we investigate the stability problems of neutral-type neural networks with operator and mixed delays. Some sufficient conditions are obtained for guaranteeing the existence, uniqueness, and global asymptotical stability of periodic solutions to the considered neural networks. Finally, a numerical example is performed to illustrate the theoretical results.

1. Introduction

In this article, we consider the following neutral neural networks with operator and mixed delays:where corresponds to the state of the ith unit at time t and , which is defined bywhere is a constant delay, with ; denotes the behaved function; the functions , and denote the neuron activations; and are the discrete time-varying delay and the distributed time-varying delay, respectively, which satisfy , where τ and μ are positive constants; and , for , and the norm is defined by . The matrix is a constant matrix, and are some unknown constant matrices. denotes the exogenous inputs at time t.

Remark 1. The neutral-type operator A in (2) represents the operator form which was put forward by Hale [1]. Then, we give the following properties for the difference operator . Define the operator on :where is a constant.

Lemma 1 (see [2, 3]). When , then has a unique continuous bounded inverse satisfyingWhen the neutral-type operator is defined on continuous function space , we have the following lemma:

Lemma 2 (see [3]). If then the inverse of difference operator denoted by exists, and

Proof. The proof of Lemma 2 is similar to the proof in [3]. For the convenience of the reader, we provide a detailed proof. Let , then . Thus, exists, and

Remark 2. Since the matrices , and in (1) are uncertain matrices, system (1) is interval neural networks. In the real world, the dynamic systems are often destroyed by various inevitable factors, such as parameter fluctuation, external disturbance, and random disturbance. Hence, the research for interval neural networks has important practical application value. There are lots of results for interval neural networks, e.g., [4, 5].
Let us recall the research for the stability problems of neural networks. Zhang [6] studied the stability problem of periodic solutions for a time-varying recurrent cellular neural network with time delays and impulses. Then, Zhang and Wang [7] further investigated dynamic properties of periodic solutions for high-order Hopfield neural networks with time delays and impulses. The models in [6, 7] are nonneutral-type models which are different from the model in the present paper. In [8], periodic oscillation problems of discrete-time bidirectional associative memory neural networks have been studied by employing the theory of coincidence degree and Halanay-type inequality technique. In the present paper, we obtain the existence, uniqueness, and global asymptotical stability of periodic solutions by using LMI approach, Lyapunov function, and a blend of matrix theory which are different from the methods in [8]. Zhu and Cao [9, 10] studied two different types of stochastic neural networks which are not neutral-type neural networks and are different from the model in the present paper. In [11, 12], the authors considered the following two kinds of neutral-type neural system:Liu et al. [13] studied a class of Markovian jumping neutral-type neural networks with mode-dependent mixed time delays: By the above work, we note that the neutral character in neural networks shows by the nonlinear term or differential operator . In this paper, we will study the neutral-type neural networks when the neutral term has the operator form which is shown by . For more references about neutral-type neural networks with mixed delays, see [14, 15].
To best knowledge of the authors, few authors have studied the global asymptotical stability problems of periodic solutions to neutral-type neural networks with mixed delays. The main areas of challenge are as follows: (1) since system (1) contains neutral-type operator A, constructing a viable Lyapunov–Krasovskii functional seems to be very difficult; (2) as the mixed delays exist in system (1), the corresponding LMI approach becomes more complicated since the LMI approach is required to reflect mixed delay’s influence; and (3) it is extraordinary to establish a unified framework to dispose of the uncertain matrices, the neutral-type terms, and mixed delays’ influence. Therefore, the main purpose of this article is to try for the first time to address the challenges listed.
In this paper, we will study global asymptotical stability of interval neural networks for a neutral-type neural network with mixed delays. Note that system (1) contains uncertain matrices, the neutral-type terms, and mixed delays that are all dependent on the properties of the neutral operator.

Remark 3. The main purpose of this paper is to obtain some sufficient conditions for guaranteeing asymptotic stability of system (1). We develop LMI approach to answer the addressed challenges. A simulation example is given to show the usefulness of the main results in the present paper. There are three aspects to the contribution of this paper: (1) For using the LMI method, we have to take into account the properties of the neutral-type operator. (2) Unlike the most existing results, we develop a new unified framework to deal with global asymptotical stability of interval neural networks by LMI approach, Lyapunov function, and a blend of matrix theory which may be of independent interest. It is worth pointing out that our main methods are also important for the case of nonneutral system with constant delays. (3) This article uses some new inequality techniques. In particular, using the properties of the neutral-type operator, we construct an appropriate Lyapunov–Krasovskii functional to handle the considered system.
The following sections are organized as follows: Section 2 gives some preliminaries including some useful lemmas and definitions. In Section 3, we obtain some sufficient conditions for existence, uniqueness, and global asymptotic stability of periodic solution to system (1). In Section 4, a numerical example verifies the accuracy of the results in the present paper.

2. Some Preliminaries

Throughout the paper, and denote the n-dimensional Euclidean space and the set of all real matrices, respectively. The superscript “T” represents the matrix transposition. (or ) denotes that A is a symmetric and positive definite (or negative definite) matrix. denotes the Euclidean norm of a vector z and denotes the induced norm of the matrix A, that is, , where means the largest eigenvalue of A. If their dimensions are not explicitly stated, they are assumed to be compatible for algebraic operations.

Lemma 3 (see [16]). For any vectors with , the inequalityholds, where is any matrix with .

Lemma 4 (see [17]). Let , and σ be the positive constants, and the function satisfies the scalar impulsive differential inequality:where . Assume that , thenwhere , and satisfies .

Remark 4. When , then is a continuous function on , and the results of Lemma 4 also hold.
Throughout this paper, the following assumptions are needed: (H₁) For the neuron activation functions in (1) satisfy where are the real constants and they may be positive, zero, or negative. (H₂) , and are all continuously periodic functions on .

3. Main Results

Assume that the unknown constant matrices satisfywherewith

Letwhere denotes the column vector with th element to be 1 and others to be 0. Then, system (1) can be rewritten asi.e.,where ,

Now, we present the main results of the present paper.

Theorem 1. Assume that the assumptions () and () hold. Then, system (1) has a global asymptotically stable periodic solution if there exist three constants , diagonal matrices , and , and three diagonal matrices , such that the following inequalities hold:where

Proof. Let be an arbitrary solution of system (1) with initial value . Define where is a constant and . Then is a solution of system (1) with initial value . Let . In view of system (18), we havewhereConstruct the following Lyapunov function:Derivation of the above Lyapunov function along the solution of (22) givesWe calculate every term of . It follows from the assumption (), Lemma 2, and Lemma 3 thatUsing the definition of , assumption (), and Lemma 2, we haveBy (29), we haveAdding the terms on the right-hand side of (26)–(28) and (30) to (25), and using condition (20), we getwhich together with Remark 4 yieldswhere and satisfies . Thus,By (33) and Lemma 2, we havei.e.,LetWe show that is uniformly convergent. For , by (35), we haveIt is obvious that the function sequence is uniformly convergent by the Cauchy convergence criterion. In addition,Hence, is also uniformly convergent, which implies thatHence, we can deduce that is a periodic solution of system (1) since is a solution of system (1). Now, we prove that is a unique periodic solution of system (1). Let and be different periodic solutions of system (1), where . It follows by (35) thatThus, for .