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

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 [7–18]).

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, 8–12, 14–17] 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(𝑡−𝑟)),(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],ℝ), 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 (𝑋_𝑖,𝑋^+𝑖),𝑖∈𝑁={1,2,…,𝑛}, be ordered Banach spaces with Int𝑋^+𝑖≠∅. For 𝑥_𝑖, 𝑦_𝑖∈𝑋_𝑖, we write 𝑥_𝑖≤_𝑋𝑖𝑦_𝑖 if 𝑦_𝑖−𝑥_𝑖∈𝑋^+𝑖; 𝑥_𝑖<_𝑋𝑖𝑦_𝑖 if 𝑦_𝑖−𝑥_𝑖∈𝑋^+𝑖⧵{0}; 𝑥_𝑖≪_𝑋𝑖𝑦_𝑖 if 𝑦_𝑖−𝑥_𝑖∈Int𝑋^+𝑖. For 𝑘∈𝑁, we denote 𝐼={1,2,…,𝜅} and 𝐽=𝑁⧵𝐼={𝜅+1,…,𝑛}. Thus, we can define the product space 𝑋=∏^{𝑖=𝑛𝑖=1}𝑋_𝑖 which generates two cones 𝑋⁺=∏^{𝑖=𝑛𝑖=1}𝑋^+𝑖 and 𝐾=∏^{𝑖=𝜅𝑖=1}𝑋^+𝑖×∏^{𝑖=𝑛𝑖=𝜅+1}(−𝑋^+𝑖) with nonempty interiors Int𝑋⁺=∏^{𝑖=𝑛𝑖=1}Int𝑋^+𝑖 and Int𝐾=∏^{𝑖=𝜅𝑖=1}Int𝑋^+𝑖×∏^{𝑖=𝑛𝑖=𝜅+1}(−Int𝑋^+𝑖). The ordering relation on 𝑋⁺ and 𝐾 is defined in the following way:𝑥<_𝑋𝑥≤_𝑋𝑦⟺𝑥_𝑖≤_𝑋𝑖𝑦_𝑖,∀𝑖∈𝑁,𝑦⟺𝑥≤𝑦,𝑥𝑥≪_𝑖<_𝑋𝑖𝑋𝑦_𝑖,forsome𝑖∈𝑁,thatis,𝑥≤_𝑋𝑦,𝑥≠𝑦,𝑥≤_𝐾𝑦⟺𝑥_𝑖≪_𝑋𝑖𝑦_𝑖,∀𝑖∈𝑁,𝑥<_𝐾𝑦⟺𝑥_𝑖≤_𝑋𝑖𝑦_𝑖,∀𝑖∈𝐼,x_𝑖≥_𝑋𝑖𝑦_𝑖,∀𝑖∈𝐽,𝑦⟺𝑥𝑥≤≪_𝐾𝑦_𝐾,𝑥_𝑖<_𝑋𝑖𝑦_𝑖,forsome𝑖∈𝐼or𝑥_𝑖>_𝑋𝑖𝑦_𝑖,forsome𝑖∈𝐽,𝑦⟺𝑥_𝑖≪_𝑋𝑖𝑦_𝑖,∀𝑖∈𝐼,𝑥_𝑖≫_𝑋𝑖𝑦_𝑖,∀𝑖∈𝐽.(2.1)

A semiflow on 𝑋 is a continuous mapping Φ: 𝑋×ℝ₊→𝑋, (𝑥,𝑡)→Φ(𝑥,𝑡), which satisfies (i) Φ₀=𝑖𝑑 and (ii) Φ_𝑡⋅Φ_𝑠=Φ_𝑡+𝑠 for 𝑡,𝑠∈ℝ₊. Here, Φ_𝑡(𝑥)≡Φ(𝑥,𝑡) for 𝑥∈𝑋 and 𝑡≥0. The orbit of 𝑥 is denoted by 𝑂(𝑥):𝑂(𝑥)=Φ_𝑡(𝑥)∶𝑡≥0.(2.2) An equilibrium point is a point 𝑥 for which Φ_𝑡(𝑥)=𝑥 for all 𝑡≥0. 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 𝑥∈𝑋 is called a quasiconvergent point if 𝜔(𝑥)⊂𝐄. The set of all such points is denoted by Q. A point 𝑥∈𝑋 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 providedΦ_𝑡(𝑥)⩽_𝐾Φ_𝑡(𝑦)whenever𝑥⩽_𝐾𝑦∀𝑡≥0.(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 𝑥∈𝑈, 𝑦∈𝑉 and 𝑡₀>0, such thatΦ_𝑡(𝑈)⩽_𝐾Φ_𝑡(𝑉)∀𝑡≥𝑡₀.(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 Φ_𝑡(𝑥)≪_𝐾Φ_𝑡(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 Φ_𝑡0(𝑥)≪_𝐾Φ_𝑡0(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=⎛⎜⎜⎝−𝐴−𝐵𝐶𝐷⎞⎟⎟⎠,(2.5) where 𝐴=(a_𝑖𝑗)_𝑘×𝑘 satisfies (𝑎_𝑖𝑗)⩾0 if 𝑖≠𝑗, similarly for the (𝑛−𝑘)×(𝑛−𝑘) matrix 𝐷 and 𝐵⩾0, 𝐶⩾0.

In this paper, the following lemma is necessary.

Lemma 2.1. If 𝑀 is a type-K monotone matrix, then 𝑒^𝑀𝑡 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 by𝐾_𝑀=𝜙=𝜙₁,𝜙₂,…,𝜙_𝑛∈𝐶∶𝜙(𝑠)≥_𝐾0,𝑠∈[−𝑟,0]𝜙(𝑡)≥_𝐾𝑒^{𝑀(𝑡−𝑠)}𝜙(𝑠),0≥𝑡≥𝑠≥−𝑟.(2.6) It is easy to see that 𝐾_𝑀 is a closed cone in 𝐶 and generates a partial ordering on 𝐶 which is written by ≥_𝑀. Assume that 𝜙∈𝐶 is differentiable on (−𝑟,0), a similar argument to [18, lemma 2.1] implies that 𝜙≥_𝑀0 if and only if 𝜙(−𝑟)≥_𝐾0 and 𝑑𝜙(𝑠)/𝑑𝑠−𝑀𝜙(𝑠)≥_𝐾0 for all 𝑠∈(−𝑟,0).

Consider the abstract functional differential equation𝑥^(𝑡)=𝑓𝑥_𝑡,(2.7) where 𝑓∶𝐷→𝑋 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,𝜎_𝜙). Moreover, if 𝜎_𝜙>𝑟, then 𝑥(𝑡,𝜙) is a classical solution of (2.7) for 𝑡∈(𝑟,𝜎_𝜙). In this section, for simplicity, we assume that, for each 𝜙∈𝐷, (2.7) admits a solution 𝑥(𝑡,𝜙) defined on [0,∞). Therefore, (2.7) generates a semiflow on 𝐶 by Φ_𝑡(𝜙)≡𝑥_𝑡(𝜙), where 𝑥_𝑡(𝜙)(𝑠)=𝑥(𝑡+𝑠,𝜙) for 𝑡≥0 and −𝑟≤𝑠≤0.

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

(WQM) Whenever 𝜙,𝜓∈𝐷,𝜓≥_𝑀𝜙, then𝑓(𝜓)−𝑓(𝜙)≥_𝐾𝑀(𝜓(0)−𝜙(0)).(2.8)

Theorem 2.2. Suppose that (WQM) holds. If 𝜓≥_𝑀𝜙, then 𝑥_𝑡(𝜓)≥_𝑀𝑥_𝑡(𝜙) for all 𝑡≥0.

Proof. Let 𝜂∈Int𝐾. For any 𝜀>0, define 𝑓_𝜀(𝜙)=𝑓(𝜙)+𝜀𝜂 for 𝜙∈𝐷, and let 𝑥^𝜀𝑡(𝜓) be a unique solution of the following equation: 𝑥(𝑠)𝑥=^(𝜓𝑡)(=𝑠)𝑓,_𝜀𝑥−_𝑡𝑟,≤𝑠𝑡≤≥00,.(2.9)
Let 𝑦^𝜀(𝑡)=𝑥^𝜖(𝑡,𝜓)−𝑥(𝑡,𝜙) and define 𝑆=𝑡∈[0,∞)∶𝑦^𝜖𝑡≥_𝑀0.(2.10) Since 𝜓≥_𝑀𝜙, 𝑆 is closed and nonempty. We first prove the following two claims.
Claim 1. If 𝑡₀∈𝑆, there exists 𝛿₀>0 such that [𝑡₀,𝑡₀+𝛿₀]⊂𝑆.
According to the integral expression of (2.9) we have 𝑦^𝜖(𝑡)=𝑒^{𝑀(𝑡−𝑠)}𝑦^𝜖(𝑠)+^𝑡𝑠𝑒^{𝑀(𝜏−𝑠)}𝑓𝑥^𝜖𝜏(𝜓)−𝑓𝑥_𝜏(𝜙)−𝑀(𝑥^𝜖(𝜏,𝜓)−𝑥(𝜏,𝜙))+𝜖𝜂𝑑𝜏.(2.11) Since 𝑡₀∈𝑆 and (WQM) hold, we have 𝑓𝑥^𝜖𝑡(𝜓)−𝑓𝑥_𝑡(𝜙)−𝑀(𝑥^𝜖(𝑡,𝜓)−𝑥(𝑡,𝜙))+𝜖𝜂|_𝑡=𝑡0≥_𝐾𝜖𝜂≫_𝐾0.(2.12) By the characteristic of a cone, there is 𝛿₀>0 such that 𝑓𝑥^𝜖𝑡(𝜓)−𝑓𝑥_𝑡(𝜙)−𝑀(𝑥^𝜖(𝑡,𝜓)−𝑥(𝑡,𝜙))+𝜖𝜂≥_𝐾0,∀𝑡∈𝑡₀,𝑡₀+𝛿₀.(2.13) By Lemma 2.1, we have 𝑦^𝜖(𝑡)≥_𝐾𝑒^{𝑀(𝑡−𝑠)}𝑦^𝜖(𝑠),∀𝑡₀≤𝑠≤𝑡≤𝑡₀+𝛿₀,(2.14) which, together with the definition of 𝐾_𝑀, implies that 𝑥^𝜖𝑡(𝜓)≥_𝑀𝑥_𝑡(𝜙),∀𝑡∈𝑡₀,𝑡₀+𝛿₀.(2.15)
Claim 2. Let 𝑆₁={𝑡∶[0,𝑡]⊂𝑆}. Then sup𝑆₁=∞.
If 𝑡^∗=sup𝑆₁<∞, then there is a sequence {𝑡_𝑛}⊂𝑆₁⊂𝑆 such that 𝑡_𝑛→𝑡^∗ as 𝑛→∞. From the closeness of 𝑆 we have 𝑡^∗∈𝑆. By Claim 1, [𝑡^∗,𝑡^∗+𝛿^∗]⊂𝑆 for some 𝛿^∗>0, which contradicts the definition of 𝑡^∗. Therefore, sup𝑆₁=∞, which implies 𝑆=[0,∞).
Since 𝑓_𝜖→𝑓 uniformly on bounded subset of 𝐷 as 𝜖→0⁺, then lim_𝜖→0+𝑥^𝜖𝑡(𝜓)=𝑥_𝑡(𝜓),∀𝑡≥0.(2.16) Letting 𝜖→0⁺ in 𝑦^𝜖𝑡=𝑥^𝜖𝑡(𝜓)−𝑥_𝑡(𝜙)≥_𝑀0, we have 𝑥_𝑡(𝜓)−𝑥_𝑡(𝜙)≥_𝑀0, which implies that 𝑥_𝑡(𝜓)≥_𝑀𝑥_𝑡(𝜙).

By the definition of the semiflow Φ_𝑡, it is easy to see from (WQM) that Φ_𝑡 is monotone with respect to ≥_𝑀 in the sense that Φ_𝑡(𝜓)≥_𝑀Φ_𝑡(𝜙) whenever 𝜓≥_𝑀𝜙 for all 𝑡≥0.

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 𝐾_𝑀 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 define𝐶_𝜑=𝜙∈‖𝐶𝜙‖∶_𝜑thereexist𝛾≥0suchthat−𝛾𝜑≤_𝑀𝜙≤_𝑀𝛾𝜑,=inf𝛾≥0∶−𝛾𝜑≤_𝑀𝜙≤_𝑀𝛾𝜑.(2.17)

It is easy to check that (𝐶_𝜑,‖𝜙‖_𝜑) is a Banach space, 𝐾_𝑀=𝐶_𝜑∩𝐾_𝑀 is a cone with nonempty interior Int𝐾_𝑀 (see [20]), and 𝑖∶𝐶_𝜑→𝐶 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, Φ_𝑡𝐶⊂𝐶∩𝐶_𝜑, Φ_𝑡∶𝐶→𝐶∩𝐶_𝜑 is continuous, and Φ_𝑡(𝜓−𝜙)∈Int𝐾_𝑀 for any 𝜓,𝜙∈𝐶 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 𝑥_𝑡(𝜓)≫_𝑀𝑥_𝑡(𝜙) in 𝐾_𝑀 for all 𝑡≥𝑟.

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 ℝ^𝑛. Moreover, for each compact subset 𝐴 of 𝐷, there exists a closed and bounded subset 𝐵=𝐵(𝐴) of 𝐷 such that 𝑥_𝑡(𝜙)∈𝐵 for each 𝜙∈𝐴 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 𝑥₀≥_𝐾0(𝑥₀≤_𝐾0) 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 𝐾_𝑀. Let 𝑒̂=(1̂,…,1̂,−1̂,…,−1̂)∈𝐾, where 1̂ denotes a constant mapping defined on 𝐶; that is, 1̂(𝑠)=1 for all 𝑠∈[−𝑟,0]. Obviously, 𝑒≥̂_𝑀0̂. For any 𝜓∈𝐷, either the sequence of points 𝜓+(1/𝑛)𝑒̂ or 𝜓−(1/𝑛)𝑒̂ 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 𝑥∈𝑋 (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:𝑥^𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝑟_𝑗+𝐼_𝑖,𝑖=1,2,…,𝑛,(3.1) where 𝑎_𝑖>0 and 𝑟_𝑗≥0 are constant, 𝑖,𝑗=1,…,𝑛. The interconnection matrix (𝑎_𝑖𝑗)_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 𝐼_𝑖 are constants or periodic. The activation functions 𝑓=(𝑓₁,…,𝑓_𝑛)∶𝐷→ℝ, where 𝐷 is an open subset of 𝑋=𝐶([−𝑟,0],ℝ^𝑛) with 𝑟=max{𝑟_𝑗|𝑗∈𝑁}, satisfy (A1) and following property. (A2) There exist constants 𝐿_𝑗 such that |𝑓_𝑗(𝑥)−𝑓_𝑗(𝑦)|≤𝐿_𝑗|𝑥−𝑦| for 𝑗=1,…,𝑛.

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

Theorem 3.1. Equation (3.1) has an equilibrium which attracts all its solutions coming from the initial value 𝜙≥_𝐾0 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 𝜙∈𝑋, we define 𝐹_𝑖(𝜙)=−𝑎_𝑖𝜙_𝑖(0)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝜙_𝑗−𝑟_𝑗+𝐼_𝑖.(3.2)Choosing 𝑀=diag{−𝜇,…,−𝜇} with 𝜇>0, and denoting 𝐿=max_{1≤𝑗≤𝑛}𝐿_𝑗, 𝛼=max_{1≤𝑖,𝑗≤𝑛}|𝑎_𝑖𝑗| and 𝛽=max_{1≤𝑗≤𝑛}𝑎_𝑗. Since 𝜙(0) is bounded, for 𝜓, 𝜙∈𝐷 with 𝜓≥_𝑀𝜙, there exist 𝑚≥0 and 𝑚≥0 with 𝑚≥𝑚 such that −𝑚≤𝜓_𝑗(0)−𝜙_j(0)≤𝑚,∀𝑖∈𝐼,𝑚≤𝜓_𝑗(0)−𝜙_𝑗(0)≤−𝑚,∀𝑖∈𝐽.(3.3)From (A2) and the definition of 𝐾_𝑀, if 𝜓≥_𝑀𝜙, then 𝐹_𝑖(𝜓=)−𝜇𝐹−_𝑖(𝜙𝑎_𝑖)+𝜇𝜓_𝑖(0)−𝜙_𝑖(0)≥𝜇−𝑎_𝑖𝜓_𝑖(0)−𝜙_𝑖(0)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝜓_𝑗−𝑟_𝑗−𝑓_𝑗𝜙_𝑗−𝑟_𝑗−𝜓_𝑖(0)−𝜙_𝑖(0)−^𝑘_𝑗=1𝑎_𝑖𝑗𝐿_𝑗𝜓_𝑗−𝑟_𝑗−𝜙_𝑗−𝑟_𝑗_𝑛𝑗=_𝑘+1≥𝜇−𝑎𝑎_𝑖𝑗𝑖𝐿_𝑗𝜓_𝑗−𝑟_𝑗−𝜙_𝑗−𝑟_𝑗−𝜓_𝑖(0)−𝜙_𝑖(0)−^𝑘_𝑗=1𝑎_𝑖𝑗𝐿_𝑗𝑒^𝜇𝑟𝑗𝜓_𝑗(0)−𝜙_𝑗(0)_𝑛_𝑗=_𝑘+1≥𝜇−𝑎𝛽_𝑖𝑗𝐿_𝑗𝑒^𝜇𝑟𝑗𝜓_𝑗(0)−𝜙_𝑗(0)𝑚𝑚−𝑛𝛼𝐿𝑒^𝜇𝑟𝑚𝑚𝑚,(3.4)for all 𝑖∈𝐼. By a similar argument we have 𝐹_𝑖(𝜓)−𝐹_𝑖(𝜙)+𝜇𝜓_𝑖(0)−𝜙_𝑖(0)≤𝜇−𝛽𝑚𝑚−𝑛𝛼𝐿𝑒^𝜇𝑟𝑚𝑚−𝑚(3.5) for all 𝑖∈𝐽. Let 𝐻=𝛽𝑚/𝑚 and let 𝐺=𝑛𝛼𝐿𝑚/𝑚, and define 𝑔(𝜇)=𝜇−𝐻−𝐺𝑒^𝜇𝑟. If 𝑟=0, we have 𝑔(𝜇)≥0 for 𝜇≥𝐻+𝐺. If 𝑟>0 and 𝐺𝑒^𝐻𝑟𝑟<1/𝑒, we deduce that 𝑔(𝜇) reaches its positive maximum value at 𝜇=𝐻+(1/𝑟)ln(1/𝐺𝑒^𝐻𝑟𝑟)>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 𝐼_𝑖 being periodic functions, we have following result.

Theorem 3.2. For any periodic external input function 𝐼(𝑡)=(𝐼₁(𝑡),…,𝐼_𝑛(𝑡)), 𝐼_𝑖(𝑡+𝜔)=𝐼_𝑖(𝑡), 𝑖=1,…,𝑛, (3.1) admits a unique periodic solution 𝑥^∗(𝑡) and all other solutions which come from the initial value 𝜙≥_𝐾0 with 𝜙(0) being bounded converge to it as 𝑡→∞.

Proof. Let 𝑥(𝑡)=𝑥(𝑡,𝜙) be the solution of (3.1) for 𝑡≥0 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, lim_𝑛→∞𝑥(𝑛𝜔,𝜙)=𝜙^∗.(3.7)We have, therefore, 𝑥𝜔,𝜙^∗=𝑥𝜔,lim_𝑛→∞𝑥(𝑛𝜔,𝜙)=lim_𝑛→∞𝑥(𝜔,𝑥(𝑛𝜔,𝜙))=lim_𝑛→∞𝑥((𝑛+1)𝜔,𝜙)=𝜙^∗.(3.8)
From (3.6) and (3.8) we deduce that 𝑥𝑡+𝜔,𝜙^∗=𝑥𝑡,𝑥𝜔,𝜙^∗=𝑥𝑡,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 lim_𝑡→∞𝑥(𝑡,𝜙)=lim_𝑡→∞𝑥(𝑡,𝑥(𝑡,𝜙))=lim_𝑡→∞𝑥𝑡,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.