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Journal of Applied Mathematics
Volume 2012, Article ID 580482, 9 pages
http://dx.doi.org/10.1155/2012/580482
Research Article

Type-K Exponential Ordering with Application to Delayed Hopfield-Type Neural Networks

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 10 December 2011; Accepted 9 April 2012

Academic Editor: Chuanhou Gao

Copyright © 2012 Bin-Guo Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Order-preserving and convergent results of delay functional differential equations without quasimonotone condition are established under type-K exponential ordering. As an application, the model of delayed Hopfield-type neural networks with a type-K monotone interconnection matrix is considered, and the attractor result is obtained.

1. Introduction

Since monotone methods have been initiated by Kamke [1] and Müler [2], and developed further by Krasnoselskii [3, 4], Matano [5], and Smith [6], the theory and application of monotone dynamics have become increasingly important (see [718]).

It is well known that the quasimonotone condition is very important in studying the asymptotic behaviors of dynamical systems. If this condition is satisfied, the solution semiflows will admit order-preserving property. There are many interesting results, for example, [6, 812, 1417] for competitive (cooperative) or type-K competitive (cooperative) systems and [6, 7, 13] for delayed systems. In particular, for the scalar delay differential equations of the form 𝑥(𝑡)=𝑔(𝑥(𝑡),𝑥(𝑡𝑟)),(1.1) if the quasimonotone condition (𝜕𝑔(𝑥,𝑦))/𝜕𝑦>0 holds, then (1.1) generates an eventually strongly monotone semiflow on the space 𝐶([𝑟,0],), which is one of sufficient conditions for obtaining convergent results. In other words, the right hand side of (1.1) must be strictly increasing in the delayed argument. This is a severe restriction, and so the quasimonotone conditions are not always satisfied in applications. Recently, many researchers have tried to relax the quasimonotone condition by introducing a new cone or partial ordering, for example, the exponential ordering [6, 18, 19]. In particular, Smith [6] and Wu and Zhao [18] considered a new cone parameterized by a nonnegative constant, which is applicable to a single equation. Replacing the previous constant by a quasipositive matrix, the exponential ordering is generalized to some delay differential systems by Smith [6] and Y. Wang and Y. Wang [19]. However, the above results are not suitable to the type-K systems (see [6] for its definition). A typical example is a Hopfield-type neural network model with a type-K monotone interconnection matrix, which implies that the interaction among neurons is not only excitatory but also inhibitory. For this purpose, we introduce a type-K exponential ordering and establish order-preserving and convergent results under the weak quasimonotone condition (WQM) (see Section 2) and then apply the result to a network model with a type-K monotone interconnection matrix.

This paper is arranged as follows. In next section, the type-K exponential ordering parameterized by a type-K monotone matrix is introduced, and convergent result is established. In Section 3, we apply our results to a delayed Hopfield-type neural network.

2. Type-K Exponential Ordering

In this section, we establish a new cone and introduce some order-preserving and convergent results.

Let (𝑋𝑖,𝑋+𝑖),𝑖𝑁={1,2,,𝑛}, be ordered Banach spaces with Int𝑋+𝑖. For 𝑥𝑖, 𝑦𝑖𝑋𝑖, we write 𝑥𝑖𝑋𝑖𝑦𝑖 if 𝑦𝑖𝑥𝑖𝑋+𝑖; 𝑥𝑖<𝑋𝑖𝑦𝑖 if 𝑦𝑖𝑥𝑖𝑋+𝑖{0}; 𝑥𝑖𝑋𝑖𝑦𝑖 if 𝑦𝑖𝑥𝑖Int𝑋+𝑖. For 𝑘𝑁, we denote 𝐼={1,2,,𝜅} and 𝐽=𝑁𝐼={𝜅+1,,𝑛}. Thus, we can define the product space 𝑋=𝑖=𝑛𝑖=1𝑋𝑖 which generates two cones 𝑋+=𝑖=𝑛𝑖=1𝑋+𝑖 and 𝐾=𝑖=𝜅𝑖=1𝑋+𝑖×𝑖=𝑛𝑖=𝜅+1(𝑋+𝑖) with nonempty interiors Int𝑋+=𝑖=𝑛𝑖=1Int𝑋+𝑖 and Int𝐾=𝑖=𝜅𝑖=1Int𝑋+𝑖×𝑖=𝑛𝑖=𝜅+1(Int𝑋+𝑖). The ordering relation on 𝑋+ and 𝐾 is defined in the following way:𝑥𝑋𝑦𝑥𝑖𝑋𝑖𝑦𝑖,𝑖𝑁,𝑥<𝑋𝑦𝑥𝑦,𝑥𝑖<𝑋𝑖𝑦𝑖,forsome𝑖𝑁,thatis,𝑥𝑋𝑦,𝑥𝑦,𝑥𝑋𝑦𝑥𝑖𝑋𝑖𝑦𝑖,𝑖𝑁,𝑥𝐾𝑦𝑥𝑖𝑋𝑖𝑦𝑖,𝑖𝐼,x𝑖𝑋𝑖𝑦𝑖,𝑖𝐽,𝑥<𝐾𝑦𝑥𝐾𝑦,𝑥𝑖<𝑋𝑖𝑦𝑖,forsome𝑖𝐼or𝑥𝑖>𝑋𝑖𝑦𝑖,forsome𝑖𝐽,𝑥𝐾𝑦𝑥𝑖𝑋𝑖𝑦𝑖,𝑖𝐼,𝑥𝑖𝑋𝑖𝑦𝑖,𝑖𝐽.(2.1)

A semiflow on 𝑋 is a continuous mapping Φ: 𝑋×+𝑋, (𝑥,𝑡)Φ(𝑥,𝑡), which satisfies (i) Φ0=𝑖𝑑 and (ii) Φ𝑡Φ𝑠=Φ𝑡+𝑠 for 𝑡,𝑠+. Here, Φ𝑡(𝑥)Φ(𝑥,𝑡) for 𝑥𝑋 and 𝑡0. The orbit of 𝑥 is denoted by 𝑂(𝑥):𝑂Φ(𝑥)=𝑡(𝑥)𝑡0.(2.2) An equilibrium point is a point 𝑥 for which Φ𝑡(𝑥)=𝑥 for all 𝑡0. Let 𝐄 be the set of all equilibrium points for Φ. The omega limit set 𝜔(𝑥) of 𝑥 is defined in the usual way. A point 𝑥𝑋 is called a quasiconvergent point if 𝜔(𝑥)𝐄. The set of all such points is denoted by Q. A point 𝑥𝑋 is called a convergent point if 𝜔(𝑥) consists of a single point of 𝐄. The set of all convergent points is denoted by C.

The semiflow Φ is said to be type-K monotone providedΦ𝑡(𝑥)𝐾Φ𝑡(𝑦)whenever𝑥𝐾𝑦𝑡0.(2.3)Φ is called type-K strongly order preserving (for short type-K SOP), if it is type-K monotone, and whenever 𝑥<𝐾𝑦, there exist open subsets 𝑈, 𝑉 of 𝑋 with 𝑥𝑈, 𝑦𝑉 and 𝑡0>0, such thatΦ𝑡(𝑈)𝐾Φ𝑡(𝑉)𝑡𝑡0.(2.4) The semiflow Φ is said to be strongly type-K monotone on 𝑋 if Φ is type-K monotone, and whenever 𝑥<𝐾𝑦 and 𝑡>0, then Φ𝑡(𝑥)𝐾Φ𝑡(𝑦). We say that Φ is eventually strongly type-K monotone if it is type-K monotone, and whenever 𝑥<𝐾𝑦, there exists 𝑡0>0 such that Φ𝑡0(𝑥)𝐾Φ𝑡0(𝑦). Clearly, strongly type-K monotonicity implies eventually strongly type-K monotonicity.

An 𝑛×𝑛 matrix 𝑀 is said to be type-K monotone if it has the following manner:𝑀=𝐴𝐵𝐶𝐷,(2.5) where 𝐴=(a𝑖𝑗)𝑘×𝑘 satisfies (𝑎𝑖𝑗)0 if 𝑖𝑗, similarly for the (𝑛𝑘)×(𝑛𝑘) matrix 𝐷 and 𝐵0, 𝐶0.

In this paper, the following lemma is necessary.

Lemma 2.1. If 𝑀 is a type-K monotone matrix, then 𝑒𝑀𝑡 remains type-K monotone with diagonal entries being strictly positive for all 𝑡>0.

Proof. The product of two type-K monotone matrices remains type-K monotone; the rest is obvious and we omit it here.

Let 𝑟>0 be fixed and let 𝐶=𝐶([𝑟,0],𝑋). The ordering relations on 𝐶 are understood to hold pointwise. Consider the family of sets parameterized by type-K monotone matrix 𝑀 given by𝐾𝑀=𝜙𝜙=1,𝜙2,,𝜙𝑛𝐶𝜙(𝑠)𝐾[]𝜙0,𝑠𝑟,0(𝑡)𝐾𝑒𝑀(𝑡𝑠)𝜙(𝑠),0𝑡𝑠𝑟.(2.6) It is easy to see that 𝐾𝑀 is a closed cone in 𝐶 and generates a partial ordering on 𝐶 which is written by 𝑀. Assume that 𝜙𝐶 is differentiable on (𝑟,0), a similar argument to [18, lemma 2.1] implies that 𝜙𝑀0 if and only if 𝜙(𝑟)𝐾0 and 𝑑𝜙(𝑠)/𝑑𝑠𝑀𝜙(𝑠)𝐾0 for all 𝑠(𝑟,0).

Consider the abstract functional differential equation𝑥𝑥(𝑡)=𝑓𝑡,(2.7) where 𝑓𝐷𝑋 is continuous and satisfies a local Lipschitz condition on each compact subset of 𝐷 and 𝐷 is an open subset of 𝐶. By the standard equation theory, the solution 𝑥(𝑡,𝜙) of (2.7) can be continued to the maximal interval of existence [0,𝜎𝜙). Moreover, if 𝜎𝜙>𝑟, then 𝑥(𝑡,𝜙) is a classical solution of (2.7) for 𝑡(𝑟,𝜎𝜙). In this section, for simplicity, we assume that, for each 𝜙𝐷, (2.7) admits a solution 𝑥(𝑡,𝜙) defined on [0,). Therefore, (2.7) generates a semiflow on 𝐶 by Φ𝑡(𝜙)𝑥𝑡(𝜙), where 𝑥𝑡(𝜙)(𝑠)=𝑥(𝑡+𝑠,𝜙) for 𝑡0 and 𝑟𝑠0.

In the following, we will seek a sufficient condition for the solution of (2.7) to preserve the ordering 𝑀.

(WQM) Whenever 𝜙,𝜓𝐷,𝜓𝑀𝜙, then𝑓(𝜓)𝑓(𝜙)𝐾𝑀(𝜓(0)𝜙(0)).(2.8)

Theorem 2.2. Suppose that (WQM) holds. If 𝜓𝑀𝜙, then 𝑥𝑡(𝜓)𝑀𝑥𝑡(𝜙) for all 𝑡0.

Proof. Let 𝜂Int𝐾. For any 𝜀>0, define 𝑓𝜀(𝜙)=𝑓(𝜙)+𝜀𝜂 for 𝜙𝐷, and let 𝑥𝜀𝑡(𝜓) be a unique solution of the following equation: 𝑥(𝑡)=𝑓𝜀𝑥𝑡,𝑡0,𝑥(𝑠)=𝜓(𝑠),𝑟𝑠0.(2.9)
Let 𝑦𝜀(𝑡)=𝑥𝜖(𝑡,𝜓)𝑥(𝑡,𝜙) and define [𝑆=𝑡0,)𝑦𝜖𝑡𝑀0.(2.10) Since 𝜓𝑀𝜙, 𝑆 is closed and nonempty. We first prove the following two claims.
Claim 1. If 𝑡0𝑆, there exists 𝛿0>0 such that [𝑡0,𝑡0+𝛿0]𝑆.
According to the integral expression of (2.9) we have 𝑦𝜖(𝑡)=𝑒𝑀(𝑡𝑠)𝑦𝜖(𝑠)+𝑡𝑠𝑒𝑀(𝜏𝑠)𝑓𝑥𝜖𝜏(𝑥𝜓)𝑓𝜏(𝜙)𝑀(𝑥𝜖(𝜏,𝜓)𝑥(𝜏,𝜙))+𝜖𝜂𝑑𝜏.(2.11) Since 𝑡0𝑆 and (WQM) hold, we have 𝑓𝑥𝜖𝑡𝑥(𝜓)𝑓𝑡(𝜙)𝑀(𝑥𝜖(𝑡,𝜓)𝑥(𝑡,𝜙))+𝜖𝜂|𝑡=𝑡0𝐾𝜖𝜂𝐾0.(2.12) By the characteristic of a cone, there is 𝛿0>0 such that 𝑓𝑥𝜖𝑡𝑥(𝜓)𝑓𝑡(𝜙)𝑀(𝑥𝜖(𝑡,𝜓)𝑥(𝑡,𝜙))+𝜖𝜂𝐾𝑡0,𝑡0,𝑡0+𝛿0.(2.13) By Lemma 2.1, we have 𝑦𝜖(𝑡)𝐾𝑒𝑀(𝑡𝑠)𝑦𝜖(𝑠),𝑡0𝑠𝑡𝑡0+𝛿0,(2.14) which, together with the definition of 𝐾𝑀, implies that 𝑥𝜖𝑡(𝜓)𝑀𝑥𝑡𝑡(𝜙),𝑡0,𝑡0+𝛿0.(2.15)

Claim 2. Let 𝑆1={𝑡[0,𝑡]𝑆}. Then sup𝑆1=.
If 𝑡=sup𝑆1<, then there is a sequence {𝑡𝑛}𝑆1𝑆 such that 𝑡𝑛𝑡 as 𝑛. From the closeness of 𝑆 we have 𝑡𝑆. By Claim 1, [𝑡,𝑡+𝛿]𝑆 for some 𝛿>0, which contradicts the definition of 𝑡. Therefore, sup𝑆1=, which implies 𝑆=[0,).

Since 𝑓𝜖𝑓 uniformly on bounded subset of 𝐷 as 𝜖0+, then lim𝜖0+𝑥𝜖𝑡(𝜓)=𝑥𝑡(𝜓),𝑡0.(2.16) Letting 𝜖0+ in 𝑦𝜖𝑡=𝑥𝜖𝑡(𝜓)𝑥𝑡(𝜙)𝑀0, we have 𝑥𝑡(𝜓)𝑥𝑡(𝜙)𝑀0, which implies that 𝑥𝑡(𝜓)𝑀𝑥𝑡(𝜙).

By the definition of the semiflow Φ𝑡, it is easy to see from (WQM) that Φ𝑡 is monotone with respect to 𝑀 in the sense that Φ𝑡(𝜓)𝑀Φ𝑡(𝜙) whenever 𝜓𝑀𝜙 for all 𝑡0.

As we all know the strongly order-preserving property is necessary for obtaining some convergent results. However, it is easy to check that the cone 𝐾𝑀 has empty interior on 𝐶; we cannot, therefore, expect to show that the semiflow generated by (2.7) is eventually strongly type-K monotone in 𝐶. Let 𝜑()Int𝐾 and define𝐶𝜑=𝜙𝐶thereexist𝛾0suchthat𝛾𝜑𝑀𝜙𝑀,𝛾𝜑𝜙𝜑=inf𝛾0𝛾𝜑𝑀𝜙𝑀.𝛾𝜑(2.17)

It is easy to check that (𝐶𝜑,𝜙𝜑) is a Banach space, 𝐾𝑀=𝐶𝜑𝐾𝑀 is a cone with nonempty interior Int𝐾𝑀 (see [20]), and 𝑖𝐶𝜑𝐶 is continuous. Using the smoothing property of the semiflow Φ on 𝐶+ and fundamental theory of abstract functional differential equations, we deduce that for all 𝑡>𝑟, Φ𝑡𝐶𝐶𝐶𝜑, Φ𝑡𝐶𝐶𝐶𝜑 is continuous, and Φ𝑡(𝜓𝜙)Int𝐾𝑀 for any 𝜓,𝜙𝐶 with 𝜓>𝑀𝜙. Thus, from Theorem 2.2, type-K strongly order-preserving property can be obtained.

Theorem 2.3. Assume that (WQM) holds. If 𝜓>𝑀𝜙, then 𝑥𝑡(𝜓)𝑀𝑥𝑡(𝜙) in 𝐾𝑀 for all 𝑡𝑟.

In order to obtain the main result of this paper, which says that the generic solution converges to equilibrium, the corresponding compactness assumption will be required.(A1)𝑓 maps bounded subset of 𝐷 to bounded subset of 𝑛. Moreover, for each compact subset 𝐴 of 𝐷, there exists a closed and bounded subset 𝐵=𝐵(𝐴) of 𝐷 such that 𝑥𝑡(𝜙)𝐵 for each 𝜙𝐴 and all large 𝑡.

Theorem 2.4. Assume that (WQM) and (A1) hold. Then the set of convergent points in 𝐷 contains an open and dense subset. If 𝐄 consists of a single point, it attracts all solutions of (2.7). If the initial value 𝑥0𝐾0(𝑥0𝐾0) and 𝐄 consists of two points or more, we conclude that all solutions converge to one of these.

Proof. By Theorem 2.3, the semiflow is eventually strongly monotone in 𝐾𝑀. Let ̂̂̂̂̂𝑒=(1,,1,1,,1)𝐾, where ̂1 denotes a constant mapping defined on 𝐶; that is, ̂1(𝑠)=1 for all 𝑠[𝑟,0]. Obviously, ̂𝑒𝑀̂0. For any 𝜓𝐷, either the sequence of points 𝜓+(1/𝑛)̂𝑒 or 𝜓(1/𝑛)̂𝑒 is eventually contained in 𝐷 and approaches 𝜓 as 𝑛, and, hence, each point of 𝐷 can be approximated either from above or from below in 𝐷 with respect to 𝑀. The assumption (A1) implies the compactness; that is, 𝑂(𝑥) has compact closure in 𝑋 for each 𝑥𝑋 (see [6]). Therefore, from [6, Theorem  1.4.3], we deduce that the set of quasiconvergent points contains an open and dense subset of 𝐷. From the proof of [6, Theorem  6.3.1], we know that the set 𝐄 is totally ordered by 𝑀. Reference [6, Remark  1.4.2] implies that the set of convergent points contains an open and dense subset of 𝐷. The last two assertions can be obtained from [6, Theorems  2.3.1 and 2.3.2].

Remark 2.5. The above theorem implies that there exists an equilibrium attracting all solutions with initial values in the cone 𝐾. If 𝐄 consists of a single element, the equilibrium attracts all solutions with initial values in 𝐷.

3. Delayed Hopfield-Type Neural Networks

In this section, we will apply our main result to the following system of delayed differential equations:𝑥𝑖(𝑡)=𝑎𝑖𝑥𝑖(𝑡)+𝑛𝑗=1𝑎𝑖𝑗𝑓𝑗𝑥𝑗𝑡𝑟𝑗+𝐼𝑖,𝑖=1,2,,𝑛,(3.1) where 𝑎𝑖>0 and 𝑟𝑗0 are constant, 𝑖,𝑗=1,,𝑛. The interconnection matrix (𝑎𝑖𝑗)𝑛×𝑛 is type-K monotone with the elements in the diagonal being nonnegative. In this situation, the interaction among neurons is not only excitatory but also inhibitory. The external input functions 𝐼𝑖 are constants or periodic. The activation functions 𝑓=(𝑓1,,𝑓𝑛)𝐷, where 𝐷 is an open subset of 𝑋=𝐶([𝑟,0],𝑛) with 𝑟=max{𝑟𝑗|𝑗𝑁}, satisfy (A1) and following property. (A2) There exist constants 𝐿𝑗 such that |𝑓𝑗(𝑥)𝑓𝑗(𝑦)|𝐿𝑗|𝑥𝑦| for 𝑗=1,,𝑛.

First, we consider the case that the external input functions 𝐼𝑖 are constants.

Theorem 3.1. Equation (3.1) has an equilibrium which attracts all its solutions coming from the initial value 𝜙𝐾0 with 𝜙(0) being bounded.

Proof. From [21, Theorem  1], we deduce that (3.1) admits at least an equilibrium; that is, the equilibrium points set 𝐄 is nonempty.
For 𝜙𝑋, we define 𝐹𝑖(𝜙)=𝑎𝑖𝜙𝑖(0)+𝑛𝑗=1𝑎𝑖𝑗𝑓𝑗𝜙𝑗𝑟𝑗+𝐼𝑖.(3.2)Choosing 𝑀=diag{𝜇,,𝜇} with 𝜇>0, and denoting 𝐿=max1𝑗𝑛𝐿𝑗, 𝛼=max1𝑖,𝑗𝑛|𝑎𝑖𝑗| and 𝛽=max1𝑗𝑛𝑎𝑗. Since 𝜙(0) is bounded, for 𝜓, 𝜙𝐷 with 𝜓𝑀𝜙, there exist 𝑚0 and 𝑚0 with 𝑚𝑚 such that 𝑚𝜓𝑗(0)𝜙j(0)𝑚,𝑖𝐼,𝑚𝜓𝑗(0)𝜙𝑗(0)𝑚,𝑖𝐽.(3.3)From (A2) and the definition of 𝐾𝑀, if 𝜓𝑀𝜙, then 𝐹𝑖(𝜓)𝐹𝑖𝜓(𝜙)+𝜇𝑖(0)𝜙𝑖=(0)𝜇𝑎𝑖𝜓𝑖(0)𝜙𝑖+(0)𝑛𝑗=1𝑎𝑖𝑗𝑓𝑗𝜓𝑗𝑟𝑗𝑓𝑗𝜙𝑗𝑟𝑗𝜇𝑎𝑖𝜓𝑖(0)𝜙𝑖(0)𝑘𝑗=1𝑎𝑖𝑗𝐿𝑗𝜓𝑗𝑟𝑗𝜙𝑗𝑟𝑗𝑛𝑗=𝑘+1𝑎𝑖𝑗𝐿𝑗𝜓𝑗𝑟𝑗𝜙𝑗𝑟𝑗𝜇𝑎𝑖𝜓𝑖(0)𝜙𝑖(0)𝑘𝑗=1𝑎𝑖𝑗𝐿𝑗𝑒𝜇𝑟𝑗𝜓𝑗(0)𝜙𝑗(0)𝑛𝑗=𝑘+1𝑎𝑖𝑗𝐿𝑗𝑒𝜇𝑟𝑗𝜓𝑗(0)𝜙𝑗(0)𝜇𝛽𝑚𝑚𝑛𝛼𝐿𝑒𝜇𝑟𝑚𝑚𝑚,(3.4)for all 𝑖𝐼. By a similar argument we have 𝐹𝑖(𝜓)𝐹𝑖𝜓(𝜙)+𝜇𝑖(0)𝜙𝑖(0)𝜇𝛽𝑚𝑚𝑛𝛼𝐿𝑒𝜇𝑟𝑚𝑚𝑚(3.5) for all 𝑖𝐽. Let 𝐻=𝛽𝑚/𝑚 and let 𝐺=𝑛𝛼𝐿𝑚/𝑚, and define 𝑔(𝜇)=𝜇𝐻𝐺𝑒𝜇𝑟. If 𝑟=0, we have 𝑔(𝜇)0 for 𝜇𝐻+𝐺. If 𝑟>0 and 𝐺𝑒𝐻𝑟𝑟<1/𝑒, we deduce that 𝑔(𝜇) reaches its positive maximum value at 𝜇=𝐻+(1/𝑟)ln(1/𝐺𝑒𝐻𝑟𝑟)>0. Thus, there exists a positive constant 𝜇 such that (WQM) holds; the conclusion can be obtained by Remark 2.5.

For the case of the external input functions 𝐼𝑖 being periodic functions, we have following result.

Theorem 3.2. For any periodic external input function 𝐼(𝑡)=(𝐼1(𝑡),,𝐼𝑛(𝑡)), 𝐼𝑖(𝑡+𝜔)=𝐼𝑖(𝑡), 𝑖=1,,𝑛, (3.1) admits a unique periodic solution 𝑥(𝑡) and all other solutions which come from the initial value 𝜙𝐾0 with 𝜙(0) being bounded converge to it as 𝑡.

Proof. Let 𝑥(𝑡)=𝑥(𝑡,𝜙) be the solution of (3.1) for 𝑡0 with 𝑥(𝑠)=𝜙(𝑠) for 𝑠[𝑟,0]. From the properties of the solution semiflow we have 𝑥(𝑡+𝜔)=𝑥(𝑡+𝜔,𝜙)=𝑥(𝑡,𝑥(𝜔,𝜙)).(3.6) From the proof of Theorem 3.1, we know that there exists a type-K monotone matrix such that (WQM) holds; Theorem 2.4 tells us that every orbit of (3.1) is convergent to a same equilibrium, denoted by 𝜙, and then, lim𝑛𝑥(𝑛𝜔,𝜙)=𝜙.(3.7)We have, therefore, 𝑥𝜔,𝜙=𝑥𝜔,lim𝑛𝑥(𝑛𝜔,𝜙)=lim𝑛𝑥(𝜔,𝑥(𝑛𝜔,𝜙))=lim𝑛𝑥((𝑛+1)𝜔,𝜙)=𝜙.(3.8)
From (3.6) and (3.8) we deduce that 𝑥𝑡+𝜔,𝜙=𝑥𝑡,𝑥𝜔,𝜙=𝑥𝑡,𝜙.(3.9)Therefore, 𝑥(𝑡,𝜙)=𝑥(𝑡) is a unique periodic solution of (3.1). Using the conclusion of Theorem 2.4 again, we have lim𝑡𝑥(𝑡,𝜙)=lim𝑡𝑥(𝑡,𝑥(𝑡,𝜙))=lim𝑡𝑥𝑡,𝜙.(3.10)Since 𝑥(𝑡) is a periodic solution, the proof is complete.

Remark 3.3. Neural networks have important applications, such as to content-addressable memory [22], shortest path problem [23], and sorting problem [24]. Generally, the monotonicity is always assumed. Here, we relax the monotone condition, and hence neural networks have more extensive applications.

Acknowledgments

This paper is supported by NSF of China under Grant 10926091 and the Fundamental Research Funds for the Central Universities.

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