## Abstract

The dynamics of a class of non-autonomous, convex (or concave) and monotone delay functional differential systems is studied. In particular, we provide an attractivity result when two completely strongly ordered minimal subsets *K*_{1}≪_{C}*K*_{2} exist. As an application of our results, sufficient conditions for the existence of global or partial attractors for non-autonomous delayed Hopfield-type neural networks are obtained.

## 1. Introduction

This paper is devoted to the study of the dynamical structure of a monotone skew-product semi-flow, induced by a family of non-autonomous differential equations. We obtain an ergodic representation theorem for the upper Lyapunov exponent of each minimal subset, which permits us to introduce tools of differential calculus in the infinite dimensional phase space in the line of Shen & Yi (1998) and Chicone & Latushkin (2002). Under convenient assumptions, including the existence of two strongly ordered minimal subsets, we deduce global or partial attractivity results when at least one of the equations of the family is eventually strongly convex (or concave).

The above dynamical situation has been extensively studied in different contexts (see Johnson *et al*. 2000, 2001; Alonso & Obaya 2003; Novo & Obaya 2004; Novo *et al*. 2005). In this paper we extend the applicability of the results of Novo *et al*. (2005) to a general family of non-autonomous and monotone functional differential equations with finite delay, which has significant implications in Mathematical Biology, Engineering and other Applied Sciences.

There has recently been increasing interest in the potential applications of artificial neural networks' dynamics in signal and image processing, as well as in biological modelling, cognitive simulation or numerical computation. We refer the reader to Wu (2001) and references therein for a complete introduction to the subject. This paper deals with the case referred to in the literature as Hopfield-type neural networks. The model was proposed by Hopfield (1982, 1984) and the time delay was incorporated by Marcus & Westervelt (1989). The autonomous case, with or without delay, has been intensively investigated with different techniques (see Belair 1993; Gopalsamy & He 1994; Van den Driessche & Zou 1998; Wu 1998 among many others). The non-autonomous periodic or almost-periodic case has recently been considered in a few papers (see Chen & Cao 2003; Liu & Liao 2004).

Most of the research has focused on neural networks with constant delays. However, in practice, delays appearing in neural networks are often of a variable nature. We study an interesting non-autonomous Hopfield-type case of finite time-varying delays to show the applicability of skew-product semi-flow and monotone dynamical system techniques to this kind of problem.

This paper is arranged as follows. The skew-product semi-flow considered, which is generated by non-autonomous delay functional differential equations and satisfies the assumptions of monotonicity and eventual strong convexity at one point, is detailed in §2. In §3, under the only assumption of monotonicity, we obtain an ergodic representation of the upper Lyapunov exponent of a minimal subset. In addition, when the flow is eventually strongly convex at one point and there are two completely strongly ordered minimal subsets *K*_{1}≪_{C}*K*_{2}, we show that *K*_{1} is an attractor subset, which is a copy of the base *Ω* and that every trajectory starting strongly below *K*_{2} tends asymptotically to *K*_{1} so that it is asymptotically almost-periodic when the base is almost-periodic. The main result holds when we change the minimal set *K*_{2} to a non-invariant compact set, defined in terms of a continuous super-equilibrium. Although the results are stated for the convex case, all of them apply to the concave one.

Finally, §4 shows the importance of the application of our results to non-autonomous Hopfield-type neural networks with time-varying delays, providing sufficient conditions for the existence of global or partial attractors. We describe the almost-periodic case, but similar conclusions apply for the almost-automorphic or in general recurrent case. First, we analyse the case without external inputs, in which we find one unique attractor in the positive cone and another in the negative one. In the presence of external inputs, and depending on the sign, we provide conditions which still guarantee the existence of partial or global attractors. The case of inhibitory interconnections inducing a monotone semi-flow in a different cone is also considered.

## 2. Monotone and eventually strongly convex skew-product semi-flows generated by delay differential equations

Let *r*>0 be a real number. We consider the Banach space with normal positive cone , where the partial ordering in is defined in the following way (*y*_{i} denotes the *i*th component of *y* (or, respectively, *z*_{i} denotes the *i*th component of *z*))Since is non-empty we obtain a strong ordering on *X* defined byWe endow with the maximum norm, which is monotone, and *X* with the supremum norm, , which is also monotone, that is, ‖*x*‖_{∞}≤‖*v*‖_{∞} whenever 0≤*x*≤*v*.

Let *Y* be a Banach space. A function is said to be *admissible* if the family is equicontinuous at every *x*_{0}∈*X* and if the set is a relatively compact subset of *Y* for each *x*_{0}∈*X*. A function is *C*^{2} *admissible* if *f* is *C*^{2} in the variable *x*∈*X* and if the maps , , , , and are admissible.

It is said that is uniformly almost-periodic if *f* is admissible and almost-periodic in (or, respectively, uniformly almost-automorphic if *f* is admissible and almost-automorphic).

We consider the system of delay functional differential equations(2.1)where is a *C*^{2} admissible function, and for each *t*≥0 the function *y*_{t} is defined as *y*_{t}(*s*)=*y*(*t*+*s*) for each *s*∈[−*r*,0]. We assume in addition that for each bounded set , *f* takes into a bounded set.

Let *Ω* be the hull of *f*, namely, the closure of the set of mappings with , in the topology of uniform convergence on compact sets. *Ω* is a compact metric space because *f* is admissible and *X* is separable (see Hino *et al*. 1991). The translation , , with , defines a continuous flow *σ* on *Ω*.

We will assume that is a minimal flow, which, among other cases, is satisfied when *f* is a uniformly almost-periodic or a uniformly almost-automorphic function. Each *ω*∈*Ω* is also a *C*^{2} admissible function and *f* has a unique extension to a continuous function , , which can also be differentiated twice with respect to *v*. Thus, we can consider the family of systems(2.2)which coincides with the initial system (2.1) when *ω*=*f*.

By the standard theory of delay differential equations (see Hino *et al*. 1991; Hale & Verduyn Lunel 1993) for each *ω*∈*Ω* and each *x*∈*X*, the system (2.2) locally admits a unique solution *y*(*t*, *ω*, *x*) with initial value *x*, that is, *y*(*t*, *ω*, *x*)=*x*(*t*) for each *t*∈[−*r*,0]. Therefore, the family (2.2) induces a local skew-product semi-flow(2.3)where and for *s*∈[−*r*,0].

Since for each bounded set , *F*(*Ω*×*B*) is a bounded set of , it is easy to check that if *y*(*t*, *ω*, *x*) is a bounded solution of equation (2.2) for *t* in its interval of existence, then *u*(*t*, *ω*, *x*) exists for all *t*>0, the forward orbit is relatively compact in *X* for any *δ*>0 and the omega limit set for the point (*ω*, *x*) makes sense.

Next, we recall some definitions concerning the stability of the trajectories of the semi-flow. A forward orbit of the skew-product semi-flow equation (2.3) is said to be *uniformly stable* if for every *ϵ*>0 there is a *δ*=*δ*(*ϵ*)>0, called the *modulus of uniform stability*, such that if *s*≥0 and thenA forward orbit of the skew-product semi-flow equation (2.3) is said to be *uniformly asymptotically stable* if it is uniformly stable and there is a *δ*_{0}>0 with the following property: for each *ϵ*>0 there is a *t*_{0}(*ϵ*)>0, such that if *s*≥0 and thenWe refer the reader to Fink (1974) for the definitions and properties of almost-periodic functions. Shen & Yi (1998) carried out a survey on almost-periodic and almost-automorphic dynamics in skew-product semi-flows.

The function *f* will satisfy the following *quasi-monotone condition* which generalizes the Kamke condition for ordinary differential equations (Smith 1995).

Whenever *x*≤*v* and *x*_{i}(0)=*v*_{i}(0) holds for some *i*∈{1, …, *n*}, then *f*_{i}(*t*, *x*)≤*f*_{i}(*t*, *v*) for each .

It is easily seen that if *f* satisfies assumption 2.3, the same happens for each function in the hull *Ω*. As usual, denotes the linear differential operator with respect to *x*, which is continuous in with *t*>0, and for any *v*∈*X*, uniformly in every compact subset of *Ω*×*X*. We note that *u*_{x} satisfies the following semi-cocycle property(2.4)

*If* *assumption 2.3* *is satisfied, then**for each* (*ω*, *x*)∈*Ω*×*X*.

We fix the point (*ω*, *x*)∈*Ω*×*X* and drop it from some of the notations. It is easy to check that for each *s*∈[−*r*,0], where *z*(*t*, *v*) is the solution of the variational equation(2.5)with the initial condition *z*(*s*, *v*)=*v*(*s*) for each *s*∈[−*r*,0]. Therefore, the result is an easy consequence of theorem 1.1 and lemma 1.3 in chapter 5 of Smith (1995) once we have checked that *L*_{i}(*t*)*v*≥0 if *v*≥0 and *v*_{i}(0)=0 for some *i*∈{1, …, *n*}, which is immediate from assumption 2.3. ▪

In particular, note that the above proposition implies the monotone character of the skew-product semi-flow, that is,and that the strong inequality also holds when *x*_{2}≫*x*_{1}. We refer the reader to Amann (1976), Krasnoselskii *et al*. (1989) and Smith (1995) for the basic definitions and results of the theory of *monotone dynamical systems*, that is, dynamical systems on an ordered metric space, which have the property that ordered initial states lead to ordered subsequent states.

Finally, we supply some conditions which guarantee the eventually strong convexity property for the skew-product semi-flow equation (2.3) at some point *ω*_{1}∈*Ω*.

There is an *ω*_{1}∈*Ω* such that the function *F*(*ω*_{1}.*t*, *v*) is

convex in

*v*, that iswhenever*v*_{1}≤*v*_{2},*λ*∈[0,1] and ;strongly convex in

*v*for*t*∈[0,*r*], that iswhenever*v*_{1}≪*v*_{2},*λ*∈(0,1) and*t*∈[0,*r*].

First of all, note that (i) implies that all the functions in the hull are convex. In particular, the above assumption is satisfied when for each *ω*∈*Ω* the function *F*(*ω*, *v*) is strongly convex in *v*. However, we are less restrictive because we are only assuming a strong convexity property for one of the functions in the hull and for *t*∈[0,*r*]. In applications, one often checks the above assumption for the function *f*.

*If* *assumptions 2.3 and 2.5* *are satisfied, then τ is an eventually strongly convex semi-flow at ω*_{1}*, that is,* *whenever x*_{1}≤*x*_{2} *then**for each t*≥0, *λ*∈[0,1], *ω*∈*Ω and**for each t*>*r*, *x*_{1}≪*x*_{2} and *λ*∈(0,1).

We fix *λ*∈(0,1) and we consider . We assume that all the solutions are defined for *t*≥0. Therefore, since *x*_{2}≥*x*_{1} implies *u*(*t*, *ω*, *x*_{2})≥*u*(*t*, *ω*, *x*_{1}) and *F* is convex, we deduce that for *t*≥0(2.6)Thus, comparison theorems for this kind of delay functional differential equation (see Smith 1995) lead to for each *t*≥0. Therefore, and the skew-product semi-flow is convex.

Next, we assume that *x*_{1}≪*x*_{2}, which implies *u*(*t*, *ω*_{1}, *x*_{1})≪*u*(*t*, *ω*_{1}, *x*_{2}) for each *t*∈[0,*r*], and as a consequence of assumption 2.5(ii)As before, comparison theorems provide for each *t*∈(0,*r*]. Moreover, from equation (2.6) and , comparison theorems again yield to for each *t*≥*r*, and in consequence for each *t*>*r*, as stated. ▪

## 3. Attractor minimal sets

First we consider the skew-product semi-flow equation (2.3) satisfying assumption 2.3 and obtain an ergodic representation formula for the upper Lyapunov exponent of a minimal subset , which permits the introduction of differential calculus tools in the infinite dimensional phase space in the line of Shen & Yi (1998) and Chicone & Latushkin (2002). We simplify the arguments given in Novo *et al*. (2005) for a single delay and extend its applicability to a general family of non-autonomous and monotone differential equations with finite delay.

Let be a compact, positively invariant set of the skew-product semi-flow equation (2.3). For (*ω*, *x*)∈*K*, we define the Lyapunov exponent *λ*(*ω*, *x*) asThe number is called the upper Lyapunov exponent on *K*. *K* is said to be linearly stable if *λ*_{K}≤0.

Let *C* be the compact subset of *X* defined bywhere the closure is taken in the topology of the norm ‖.‖_{∞}. For each we denote . Let *M* be a minimal set of *Ω*×*X*.

We consider the open subset and the map(3.1)It is easy to check that if *v*∈*C* then . Moreover, from proposition 2.4, we know that if *v*≫0 then *u*_{x}(*r*, *ω*, *x*)*v*≫0, which implies that *T* is well defined and maps into itself. The next lemma shows its continuity.

*The map* *is continuous*.

The first two components of the map are clearly continuous. Then, it is enough to show the continuity of the third component. It is also clear thatis a continuous map. Consequently, if we check that the map , is also continuous, the proof is finished. As in proposition 2.4, for each *s*∈[−*r*,0] where *z*(*t*,*v*) is the solution of the variational equation (2.5) and *z*(*s*,*v*)=*v*(*s*) for each *s*∈[−*r*,0]. Therefore, we deduce thatwhere . Thus, the continuity of the map , and the equicontinuity for *t*∈[0,*r*] of the familyfinish the proof. ▪

Next we consider the map(3.2)whose restriction to the above open set is also continuous. The discrete skew-product semi-flow induced by *T* on *M*×*C* will be denoted by (*M*×*C*, *T*). The next result shows that the Lyapunov exponent of a point of *M* can be obtained at discrete times for the 1-norm.

*For each* (*ω*, *x*)∈*M* and *e*≫0

We denote by . Then, as in proposition 4.3 of Novo *et al*. (2005), we can show thatfrom which we check that *λ*(*ω*, *x*)≤*λ*_{1}(*ω*, *x*). Moreover, as in lemma 3.2, sinceis uniformly bounded for each *t*≥0 and (*ω*, *x*)∈*M*, andthere is a positive constant *K*_{0} independent on *e* such that(3.3)for each (*ω*, *x*)∈*M*. Consequently, applying equation (2.4) and the latter bound(3.4)which yields to *λ*_{1}(*ω*, *x*)≤*λ*(*ω*, *x*), finishes the proof. ▪

From the above lemma, a similar proof to the one in proposition 4.3 of Novo *et al*. (2005), which is now omitted, shows the following result.

*Let* *be a minimal subset and T, h be the maps defined by expressions* *(3.1)* *and* *(3.2)**, respectively*. *Then for each* (*ω*, *x*, *e*)∈, *n*≥1,*and*, *consequently*

Finally, we obtain the following ergodic representation of the upper Lyapunov exponent on the minimal set *M*. We also omit the proof that is analogous to the one stated in theorem 4.4 of Novo *et al*. (2005). Use is made of inequality (3.3).

*Let* *be a minimal subset and T, h be the maps defined by expressions* *(3.1) and (3.2)*, *respectively*. *Then*, *there is a T-ergodic measure μ*, *that is*, *ergodic for the discrete semi-flow* (*M*×*C*, *T*), *such that h*∈*L*^{1}(*M*×*C*, *μ*) *and*

Let *K*_{1}≪_{C}*K*_{2} be two completely strongly ordered minimal subsets, that is, *x*_{1}≪*x*_{2} for all (*ω*, *x*_{1})∈*K*_{1} and (*ω*, *x*_{2})∈*K*_{2}. We show that, under *assumptions 2.3 and 2.5*, we are in the hyperbolic case, that is, , *K*_{1} is a copy of the base *Ω*, and an attractor subset of *Ω*×*X*. We denote the natural projection as .

*Let us assume that K*_{1}≪_{C}*K*_{2} *and* *assumptions 2.3 and 2.5* *hold*. *Then,* , *and*

*the minimal subset K*_{1}*is uniformly asymptotically stable*,*that is*,*for each*(*ω*,*x*_{1})∈*K*_{1}*the forward orbit**is uniformly asymptotically stable*;*K*_{1}*is a copy of the base Ω*,*that is,**for each ω*∈*Ω*,*and we can denote**;**for each*(*ω*,*x*)∈*Ω*×*X such that x*≪*x*_{2}*for some*(*ω*,*x*_{2})∈*K*_{2}*for each*(*ω*,*x*)∈*Ω*×*X such that x*≫*x*_{2}*for some*(*ω*,*x*_{2})∈*K*_{2}*, the forward orbit**is not bounded*.

Once we have obtained the ergodic representation of the upper Lyapunov exponent, the proof is completely similar to that of theorem 6.3 in Novo *et al*. (2005) and we omit it. We only want to remark that again use is made of inequalities (3.3) and (3.4). ▪

As explained before, all the conclusions apply to the concave case by changing the positive cone to . Consequently, in that case, if we have two completely strongly ordered minimal sets *K*_{1}≪_{C}*K*_{2} (with the usual order) the attractor turns out to be *K*_{2}.

A similar result holds when we change the minimal set *K*_{2} for a compact set, defined in terms of a continuous super-equilibrium in the following way. First, we recall that a continuous map , defines a continuous super-equilibrium for the monotone skew-product semi-flow equation (2.3) ifNote that this is equivalent to the positive invariance of the setThus, if we consider the compact set and we assume that *K*_{1}≪_{C}*K*_{2}, that is, *x*_{1}≪*x*_{2}(*ω*) for each (*ω*, *x*_{1})∈*K*_{1}, with slight changes we can show that assertions (i), (ii) and (iii) of theorem 3.6 hold, whereas we can no longer predict the behaviour of the trajectories above *K*_{2}. Analogously, in the concave case, we can change *K*_{1} for a compact set defined in terms of a continuous subequilibrium.

## 4. Delayed Hopfield-type neural networks

The non-autonomous system of delay functional differential equations(4.1)describes the dynamics of a network of *n* neurons (or amplifiers) with delayed outputs, which are coupled with a non-constant and almost-periodic interconnection matrix [*w*_{ij}(*t*)], whose entries are non-negative (i.e. *w*_{ij}(*t*)≥0 for each ), which means that the interaction among neurons is excitatory. The external input functions *I*_{i}(*t*) are almost-periodic. We will also assume that *a*_{i}(*t*), *τ*_{ij}(*t*) are positive almost-periodic functions for each *i*, *j*=1,…,*n*, and that the smooth real signal or activation functions , *j*=1,…,*n* satisfy the following conditions of monotonicity, convexity and concavity:

*f*_{j}(0)=0 and for each ;, for

*s*≠0, that is,*f*_{j}is strongly convex for*s*<0 and strongly concave for*s*>0;existing limits of .

This model corresponds to the so-called delayed Hopfield-type neural network (Hopfield 1982, 1984; Marcus & Westervelt 1989). The autonomous case, with or without delay, has been intensively investigated (Belair 1993; Gopalsamy & He 1994; Van den Driessche & Zou 1998; Van den Driessche *et al*. 2001).

For simplicity we start with the analysis of the non-autonomous almost-periodic case without external input functions, that is,(4.2)and we provide a global attractivity result in *X*_{+} and *X*_{−}. Similar conclusions can be obtained changing almost-periodic for almost-automorphic or recurrent. Note that when all the coefficients and delays are *T*-periodic, the attracting solutions are also *T*-periodic. We will denoteWe will also consider the function(4.3)which is continuous, admits right and left derivatives at each point, is strongly convex for *s*<0 and strongly concave for *s*>0 and are existing.

*Let us assume that there exist positive constants* 0<*α*≤*β*, 0<*β*^{*}≤*α*^{*} such that*and* , *j*=1,…,*n*. *Then*, *there is a unique almost-periodic solution y*^{*}(*t*)≫0 *and a unique almost-periodic solution y*_{*}(*t*)≪0 *of equation* *(4.2)* *such that*

Since both cases are completely analogous, we will only discuss the existence of *y*_{*}(*t*)≪0 which corresponds, as we will now show, to the convex case.

We take with and let *Ω* be the hull of the function defined by system (4.2)which is *C*^{2} admissible. From the almost-periodicity of the coefficients and delays, *Ω* is minimal and almost-periodic and we consider the corresponding family including system (4.2).

It is not hard to show, by checking the initial system (4.2) for assumptions 2.3 and 2.5, that the semi-flow is monotone and eventually strongly convex at one point in the negative cone . In the positive cone, the induced semi-flow is monotone and eventually strongly concave. Moreover, from *f*_{j}(0)=0 for each *j*=1,…,*n* we deduce that is a minimal subset. Next, we take the function , which belongs to and satisfies the same assumptions as all the signal functions *f*_{j}. Moreover, *f*(*s*)≤*f*_{j}(*s*)≤*h*(*s*) for each *s*≥0 and *h*(*s*)≤*f*_{j}(*s*)≤*f*(*s*) for each *s*≤0, where *f* is the continuous function defined in terms of *f*_{j} by equation (4.3). Therefore, by denotingwe deduce thatfor each *v*∈*X*_{−} and *i*=1,…,*n*. Therefore, we can compare the non-autonomous system (4.2) with the simpler ones(4.4)(4.5)with constant interconnection matrices. In particular, if *y*_{α}(*t*, *v*_{0}) denotes the solution of system (4.4) (respectively, *y*_{β}(*t*, *v*_{0}) denotes the solution of system (4.5)) with initial condition *v*_{0}≪0, from the comparison theorem for delay functional differential equations with the quasi-monotone condition given in assumption 2.3, we deduce that(4.6)for each *t*≥0. Moreover, since *h*′(0), , *h*′(0)>*α*/*α*^{*}, and , system (4.4) has a synchronized negative equilibrium where *α*^{*}*h*(*a*)=*αa* (respectively, system (4.5) has where *β*^{*}*f*(*b*)=*βb*) with . In addition, applying our results we deduce that is an attractor for equation (4.4) in Int *X*_{−} and consequentlyThus, from equation (4.6) and for each *t*≥0, we conclude that is bounded for *t*≥0.

Hence, the omega limit set of , with *ω*_{0}=*g*, contains a second minimal subset *K*_{1}, satisfying *K*_{1}≪_{C}*K*_{2}. Finally, theorem 3.6 asserts that *K*_{1} is an attractor which is a copy of the base *Ω*, from which the conclusions of the theorem follow. ▪

Theorem 4.2 provides a global attractivity result in Int *X*_{−} (respectively, partial attractivity result in Int *X*_{+}) when external almost-periodic inputs exist. Again, similar conclusions can be obtained changing almost-periodic for almost-automorphic or recurrent. We maintain the notation of theorem 4.1 and, in addition, we denoteWhen , *j*=1,…,*n*, we denote by *s*_{0}^{+}>0 the unique positive point such that , where *f* is the continuous function defined in terms of *f*_{j} by equation (4.3).

*Let us assume that there exist positive constants* 0<*α*≤*β*, 0<*β*^{*}≤*α*^{*} *such that*

Again, with and let *Ω* be the hull of the function defined by system (4.1)From the almost-periodicity of the coefficients and delays, *Ω* is minimal and almost-periodic and we consider the corresponding family of systems including equation (4.1)(4.7)

(i) By checking assumptions 2.3 and 2.5 for the initial system (4.1), we deduce that, in the negative cone *X*_{−}, the semi-flow is monotone and eventually strongly convex at one point. Now, we consider the compact, though not necessarily minimal, subset given in terms of the null super-equilibrium (because the external inputs are non-positive).

As in theorem 4.1, in *X*_{−} we can compare the non-autonomous system (4.1) with the simpler ones(4.8)(4.9)with constant interconnection matrices. In particular, if denotes the solution of system (4.8) (respectively, denotes the solution of system (4.9)) with initial condition *v*_{0}≪0, from the comparison theorem we deduce that(4.10)for each *t*≥0. If *I*^{+}<0 and consequently *I*^{−}<0, it is easy to check from , and *β*/*β*^{*}>*α*/*α*^{*}, that where *α*^{*}*h*(*a*)=*αa*−*I*^{−} is a synchronized negative equilibrium of (4.8) (respectively, where *β*^{*}*f*(*b*)=*β**b*−*I*^{+} is a synchronized negative equilibrium of (4.9)) and . Moreover, from remark 3.8 we deduce that attracts the solutions of system (4.8) in the interior of the negative cone *X*_{−}. Therefore, we haveand, from equation (4.10), we conclude that is bounded for *t*≥0.

Hence, the omega limit set of , with *ω*_{0}=*g*, contains a second minimal subset *K*_{1} satisfying *K*_{1}≪_{C}*K*_{2}. Finally, remark 3.8 asserts that *K*_{1} is an attractor which is a copy of the base *Ω* from which the conclusions of part (i) of the theorem follow. We omit the case *I*^{+}=0 and for each *j*=1,…,*n*, which is completely analogous.

(ii) In the positive cone *X*_{+} the semi-flow is monotone and eventually strongly concave at one point, and we can compare the non-autonomous system (4.1) with(4.11)(4.12)with constant interconnection matrices. In particular, if denotes the solution of system (4.11) (respectively, denotes the solution of system (4.12)) with the initial condition *v*_{0}≫0, from the comparison theorem, we deduce that for each *t*≥0(4.13)From and , we deduce that the equation *β*^{*}*f*(*x*)=*β**x*−*I*^{−} has two positive solutions: 0≤*c*<*d*. Then, is a synchronized positive equilibrium of equation (4.11), and consequently a subequilibrium of equation (4.7). Therefore, we consider the compact, though not necessarily minimal, subset .

Moreover, from *h*′(0)>*α*/*α*^{*} and there is , where *α*^{*}*h*(*a*)=*αa*−*I*^{+} and *a*>*c*, which is a synchronized positive equilibrium of (4.12) and an attractor in . Then, if we consider the other synchronized positive equilibrium of (4.11), , we deduce thatand from (4.13) we conclude that is bounded for *t*≥0.

Hence, the omega limit set of , with *ω*_{0}=*g*, contains a second minimal subset *K*_{2}, satisfying *K*_{1}≪_{C}*K*_{2}. Finally, applying remark 3.8 to the concave case, the conclusions of part (ii) of the theorem follow. ▪

We omit the proof of the following result which is similar to the previous one. Note that the case of negative inputs is studied in theorem 4.2, the case of positive inputs in theorem 4.3 and the case *I*^{+}*I*^{−}<0 is included in part (ii) of both results.

When , *j*=1,…,*n*, denotes the unique negative point so that where *f* is the continuous function defined in terms of *f*_{j} by (4.3).

*Let us assume that there exist positive constants* 0<*α*≤*β*, 0<*β*^{*}≤*α*^{*} *so that*

We can also study the case in which there are inhibitory interconnections, that is, some of the entries of the matrix [*w*_{ij}(*t*)] are negative but it is a *K*-matrix uniformly in (i.e. there are *m*_{i}∈{0,1}, *i*=1,…,*n*, such that for each *i*, *j*=1,…,*n* and ).

In this case, the change of variable , *i*=1,…,*n* takes equation (4.1) towhere and for each *i*, *j*=1,…,*n*, to which the above results can be applied to obtain sufficient conditions for the existence of global and partial attractors.

This is equivalent to consider the new conewhich generates a partial ordering ≤_{K} in in the usual way, that is, *x*≤_{K}*y* if, and only if, *y*−*x*∈*K*_{+} and the corresponding induced order in . With this new order, the induced skew-product semi-flow is monotone and eventually strongly convex at one point in the negative cone , and monotone and eventually strongly concave in the positive cone. Then we can state sufficient conditions for the existence of global and partial attractors as in theorems 4.2 and 4.3.

A more general delayed Hopfield-type neural network model to which our theory applies would be the one of finite distributed delayswhere the smooth and sigmoid-type real signal or activation functions satisfy the same conditions stated for system (4.1) at the beginning of the section, and

*a*_{i}(*t*)≥0,*w*_{ij}(*t*)≥0,*I*_{i}(*t*) are bounded, uniformly continuous functions;*μ*_{ij}(*t*, .) are positive Borel measures uniformly bounded and continuous in*t*with respect to the weak topology;the hull

*Ω*of the function , defined byfor each*i*=1,…,*n*, is minimal.

Sufficient conditions for the existence of global and partial attractors can also be stated for this situation.

## Acknowledgments

The authors were partly supported by Junta de Castilla y León under project VA024/03 and CICYT under project BFM2002-03815.

## Footnotes

- Received May 16, 2004.
- Accepted April 8, 2005.

- © 2005 The Royal Society