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

Authors: Bin-Guo Wang
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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
(1.1)x′(t)=g(x(t),x(t-r)),
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 Section3, 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,Xi+),i∈N={1,2,…,n}, be ordered Banach spaces with IntXi+≠∅. For xi, yi∈Xi, we write xi≤Xiyi if yi-xi∈Xi+; xi<Xiyi if yi-xi∈Xi+∖{0}; xi≪Xiyi if yi-xi∈IntXi+. 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=1i=nXi which generates two cones X+=∏i=1i=nXi+ and K=∏i=1i=κXi+×∏i=κ+1i=n(-Xi+) with nonempty interiors IntX+=∏i=1i=nIntXi+ and IntK=∏i=1i=κIntXi+×∏i=κ+1i=n(-IntXi+). The ordering relation on X+ and K is defined in the following way:(2.1)x≤Xy⟺xi≤Xiyi,∀i∈N,x<Xy⟺x≤y,xi<Xiyi,forsomei∈N,thatis,x≤Xy,x≠y,x≪Xy⟺xi≪Xiyi,∀i∈N,x≤Ky⟺xi≤Xiyi,∀i∈I,xi≥Xiyi,∀i∈J,x<Ky⟺x≤Ky,xi<Xiyi,forsomei∈Iorxi>Xiyi,forsomei∈J,x≪Ky⟺xi≪Xiyi,∀i∈I,xi≫Xiyi,∀i∈J.A semiflow onX is a continuous mapping Φ: X×ℝ+→X, (x,t)→Φ(x,t), which satisfies (i) Φ0=id and (ii) Φt·Φs=Φt+s for t,s∈ℝ+. Here, Φt(x)≡Φ(x,t) for x∈X and t≥0. The orbit of x is denoted by O(x):(2.2)O(x)={Φt(x):t≥0}.
An equilibrium point is a point x for which Φt(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 semiflowΦ is said to be type-K monotone provided(2.3)Φt(x)⩽KΦt(y)wheneverx⩽Ky∀t≥0.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 that(2.4)Φt(U)⩽KΦt(V)∀t≥t0.
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.Ann×n matrix M is said to be type-K monotone if it has the following manner:(2.5)M=(A¯-B¯-C¯D¯),
where A¯=(aij)k×k satisfies (aij)⩾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.
IfM 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.Letr>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 by(2.6)K̃M={ϕ=(ϕ1,ϕ2,…,ϕn)∈C:ϕ(s)≥K0,s∈[-r,0]ϕ(t)≥KeM(t-s)ϕ(s),0≥t≥s≥-r}.
It is easy to see that K̃M 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 equation(2.7)x′(t)=f(xt),
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,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 ϕ∈D, (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 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, then(2.8)f(ψ)-f(ϕ)≥KM(ψ(0)-ϕ(0)).Theorem 2.2.
Suppose that(WQM) holds. If ψ≥Mϕ, then xt(ψ)≥Mxt(ϕ) for all t≥0.Proof.
Letη∈IntK. For any ɛ>0, define fɛ(ϕ)=f(ϕ)+ɛη for ϕ∈D, and let xtɛ(ψ) be a unique solution of the following equation:
(2.9)x′(t)=fɛ(xt),t≥0,x(s)=ψ(s),-r≤s≤0.
Letyɛ(t)=xϵ(t,ψ)-x(t,ϕ) and define
(2.10)S={t∈[0,∞):ytϵ≥M0}.
Since ψ≥Mϕ, S is closed and nonempty. We first prove the following two claims.
Claim 1.
Ift0∈S, there exists δ0>0 such that [t0,t0+δ0]⊂S.
According to the integral expression of (2.9) we have
(2.11)yϵ(t)=eM(t-s)yϵ(s)+∫steM(τ-s)[f(xτϵ(ψ))-f(xτ(ϕ))-M(xϵ(τ,ψ)-x(τ,ϕ))+ϵη]dτ.
Since t0∈S and (WQM) hold, we have
(2.12)f(xtϵ(ψ))-f(xt(ϕ))-M(xϵ(t,ψ)-x(t,ϕ))+ϵη|t=t0≥Kϵη≫K0.
By the characteristic of a cone, there is δ0>0 such that
(2.13)f(xtϵ(ψ))-f(xt(ϕ))-M(xϵ(t,ψ)-x(t,ϕ))+ϵη≥K0,∀t∈[t0,t0+δ0].
By Lemma 2.1, we have
(2.14)yϵ(t)≥KeM(t-s)yϵ(s),∀t0≤s≤t≤t0+δ0,
which, together with the definition of K̃M, implies that
(2.15)xtϵ(ψ)≥Mxt(ϕ),∀t∈[t0,t0+δ0].
Claim 2.
LetS1={t:[0,t]⊂S}. Then supS1=∞.
Ift*=supS1<∞, then there is a sequence {tn}⊂S1⊂S such that tn→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,∞).
Sincefϵ→f uniformly on bounded subset of D as ϵ→0+, then
(2.16)lim⁡ϵ→0+xtϵ(ψ)=xt(ψ),∀t≥0.
Letting ϵ→0+ in ytϵ=xtϵ(ψ)-xt(ϕ)≥M0, we have xt(ψ)-xt(ϕ)≥M0, which implies that xt(ψ)≥Mxt(ϕ).By the definition of the semiflowΦt, it is easy to see from (WQM) that Φt is monotone with respect to ≥M in the sense that Φt(ψ)≥MΦt(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 coneK̃M 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 define(2.17)Cφ={ϕ∈C:thereexistγ≥0suchthat-γφ≤Mϕ≤Mγφ},‖ϕ‖φ=inf⁡{γ≥0:-γφ≤Mϕ≤Mγφ}.It is easy to check that(Cφ,∥ϕ∥φ) is a Banach space, KM=Cφ∩K̃M is a cone with nonempty interior IntKM (see [20]), and i:Cφ→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, ΦtC⊂C∩Cφ, Φt:C→C∩Cφ is continuous, and Φt(ψ-ϕ)∈IntKM 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 xt(ψ)≫Mxt(ϕ) in KM 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 ℝn. 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 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 Theorem2.3, the semiflow is eventually strongly monotone in KM. 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, ê≥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 coneK. 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:(3.1)xi′(t)=-aixi(t)+∑j=1naijfj(xj(t-rj))+Ii,i=1,2,…,n,
where ai>0 and rj≥0 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→ℝ, where D is an open subset of X=C([-r,0],ℝn) with r=max{rj|j∈N}, satisfy (A1) and following property. (A2)
There exist constantsLj such that |fj(x)-fj(y)|≤Lj|x-y| for j=1,…,n.First, we consider the case that the external input functionsIi 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.
Forϕ∈X, we define
(3.2)Fi(ϕ)=-aiϕi(0)+∑j=1naijfj(ϕj(-rj))+Ii.Choosing M=diag{-μ,…,-μ} with μ>0, and denoting L=max1≤j≤nLj, α=max1≤i,j≤n|aij| and β=max1≤j≤naj. Since ϕ(0) is bounded, for ψ, ϕ∈D with ψ≥Mϕ, there exist m¯≥0 and m̲≥0 with m¯≥m̲ such that
(3.3)m̲≤ψj(0)-ϕj(0)≤m¯,∀i∈I,-m¯≤ψj(0)-ϕj(0)≤-m̲,∀i∈J.From (A2) and the definition of K̃M, if ψ≥Mϕ, then
(3.4)Fi(ψ)-Fi(ϕ)+μ(ψi(0)-ϕi(0))=(μ-ai)(ψi(0)-ϕi(0))+∑j=1naij(fj(ψj(-rj))-fj(ϕj(-rj)))≥(μ-ai)(ψi(0)-ϕi(0))-∑j=1kaijLj(ψj(-rj)-ϕj(-rj))-∑j=k+1naijLj(ψj(-rj)-ϕj(-rj))≥(μ-ai)(ψi(0)-ϕi(0))-∑j=1kaijLjeμrj(ψj(0)-ϕj(0))-∑j=k+1naijLjeμrj(ψj(0)-ϕj(0))≥(μ-βm¯m̲-nαLeμrm¯m̲)m̲,for all i∈I. By a similar argument we have
(3.5)Fi(ψ)-Fi(ϕ)+μ(ψi(0)-ϕi(0))≤(μ-βm¯m̲-nαLeμrm¯m̲)(-m̲)
for all i∈J. Let H=βm¯/m̲ and let G=nαLm¯/m̲, and define g(μ)=μ-H-Geμr. If r=0, we have g(μ)≥0 for μ≥H+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 functionsIi being periodic functions, we have following result.Theorem 3.2.
For any periodic external input functionI(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 ϕ≥K0 with ϕ(0) being bounded converge to it as t→∞.Proof.
Letx(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
(3.6)x(t+ω)=x(t+ω,ϕ)=x(t,x(ω,ϕ)).
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,
(3.7)lim⁡n→∞x(nω,ϕ)=ϕ*.We have, therefore,
(3.8)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*.
From (3.6) and (3.8) we deduce that
(3.9)x(t+ω,ϕ*)=x(t,x(ω,ϕ*))=x(t,ϕ*).Therefore, x(t,ϕ*)=:x*(t) is a unique periodic solution of (3.1). Using the conclusion of Theorem 2.4 again, we have
(3.10)lim⁡t→∞x(t,ϕ)=lim⁡t→∞x(t,x(t,ϕ))=lim⁡t→∞x(t,ϕ*).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.



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