Adn`ene ARBIFarouk CH′ERIF　Chaouki AOUITIAbderrahmen TOUATI.Department of Mathematics，F(xiàn)acult′e des Sciences de Bizerte，University of Carthage，Jarzouna 70，Bizerte，Tunisia.Department of Computer Science，ISSATS，Laboratory of Math Physics，University of Sousse;Specials Functions and Applications LRES35，Ecole Sup′erieure des Sciences et de Technologie，Sousse 400，TunisiaE-mail:adnen.arbi@enseignant.edunet.tn;faroukcheriff@yahoo.fr;chaouki.aouiti@fsb.rnu.tn;Abder.Touati@fsb.rnu.tn

DYNAMICS OF NEW CLASS OF HOPFIELD NEURAL NETWORKS WITH TIME-VARYING AND DISTRIBUTED DELAYS?

Adn`ene ARBI1?Farouk CH′ERIF2Chaouki AOUITI1Abderrahmen TOUATI1
1.Department of Mathematics，F(xiàn)acult′e des Sciences de Bizerte，University of Carthage，Jarzouna 7021，Bizerte，Tunisia
2.Department of Computer Science，ISSATS，Laboratory of Math Physics，University of Sousse;Specials Functions and Applications LR11ES35，Ecole Sup′erieure des Sciences et de Technologie，Sousse 4002，Tunisia
E-mail:adnen.arbi@enseignant.edunet.tn;faroukcheriff@yahoo.fr;
chaouki.aouiti@fsb.rnu.tn;Abder.Touati@fsb.rnu.tn

In this paper，we investigate the dynamics and the global exponential stability of a new class of Hopfield neural network with time-varying and distributed delays.In fact，the properties of norms and the contraction principle are adjusted to ensure the existence as well as the uniqueness of the pseudo almost periodic solution，which is also its derivative pseudo almost periodic.This results are without resorting to the theory of exponential dichotomy. Furthermore，by employing the suitable Lyapunov function，some delay-independent sufficient conditions are derived for exponential convergence.The main originality lies in the fact that spaces considered in this paper generalize the notion of periodicity and almost periodicity. Lastly，two examples are given to demonstrate the validity of the proposed theoretical results.

delayed functional differential equations;neural networks;pseudo-almost periodic solution;global exponential stability;time-varying and distributed delays; fixed point theorem

2010 MR Subject Classification34K05;92B20;34C27;34D20;34K40;47H10

1　Introduction

On the other hand，the study of the existence periodic solutions，as well as its numerous generalizations to almost periodic solutions［11，22-27］，pseudo almost periodic solutions，weighted pseudo almost periodic solutions，almost automorphic solutions［13］，and so forth［12］，is one of the most attracting topics in the qualitative theory of differential equations due both to its mathematical interest as well as to their applications in various areas of applied science. In［7］，Bai investigated the almost periodic solutions of the following HNNs:

By means of a suitable fixed point theorem and differential inequality techniques he obtained sufficient conditions for the existence and exponential stability of almost periodic solutions for（1.1）.As we all know，many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. Such a class which is formed by pseudo almost periodic functions is a natural generalization of the concept of almost periodicity.A natural problem is to ask whether the results expressed in［7］and［17］can be extended to pseudo almost periodic solutions.In particular，we are concerned with the study of the existence，uniqueness and the behavior of the solutions of the considered model in the space PAP（1）（R，Rn）.Roughly speaking，in this paper，and in order to give answer to the above question we shall investigate the continuously differentiable pseudo almost periodic solution for a more general model;that is，the following HNNs with mixed delays（continuously time-varying and distributed delays）:

To the best of our knowledge，no paper in the literature has investigated the globallyexponential stability，existence and uniqueness of PAP（1）（R，Rn）solution for system（1.2）.Our goal in this paper is to study the dynamics of Hopfield model（1.2）.Hence，by applying fixed point theorem and differential inequality techniques，we give some sufficient conditions ensuring the existence，uniqueness and globally exponential stability of a continuously differentiable pseudo almost periodic solution of system（1.2），which are new and complement the previously known results.

The remainder of this paper is organized as follows:In Section 2，we will introduce some necessary notations，definitions and fundamental properties of the space PAP（1）（R，Rn）which will be used in the paper.In Section 3，some sufficient conditions are derived ensuring the existence of the continuously differentiable pseudo almost periodic solution.Section 4 is devoted to the exponential stability of the continuously differentiable pseudo almost periodic solution of（1.2）.At last，two illustrative examples are given.It should be mentioned that the main results of this paper include Theorem 3.4，Theorem 4.2 and Theorem 4.4.

2　Preliminaries and Function Spaces

In this paper，we denote by Rn（R=R1）the set of all n-dimensional real vectors（real numbers）.Now，we introduce necessary notations，definitions and fundamental properties of the space PAP（1）（R，Rn）which will be used later.

2.1The Classical Almost Periodic Functions

Denote by BC（0）（R，Rn），the set of bounded continued functions from R to Rn.Note thatis a Banach space wheredenotes the sup norm.When，we setthe set of functions defined from R to Rnsuch that all the derivative in order i（0≤i≤k），are continuous.Besides，we define

Definition 2.1（see［2，13］）LetWe say that f is almost periodic or uniformly almost periodic（u.a.p），when the following property is satisfied

A set D of R is called relatively dense in R when

And so，introducing the sets

we can formulate the definition of the Bohr almost periodicity of f∈C（0）（R，Rn）in the following manner:for each∈＞0，the set E（f，∈）is relatively dense in R.An element of E（f，∈）is calledan∈-period of f.Consequently，a Bohr almost periodic function is a continuous function which possesses very much almost periods.We denote bythe set of the Bohr A.P. functions from

2.2The C（1）Almost Periodic Functions

Define

It is easy to check that endowed with the norm

Remark 2.2Directly from definition it follows that the spacecontains strictlyFor instance，the function（see［9］，Example 4.5）

For some preliminary results on almost periodic functions，we refer the reader to（see［8］and［12］）.

2.3The C（1）-Pseudo Almost Periodic Functions

The concept of pseudo almost periodicity（PAP）was introduced by Zhang（see for example［28］）in the early nineties.It is a natural generalization of the classical almost periodicity. Define the class of functionsas follows:

Definition 2.3（see［2，13］） A functionis called continuously differentiable pseudo almost periodic if it can be expressed as

Remark 2.4The functions h and φ in above definition are respectively called the almost periodic component and the ergodic perturbation of the continuously differentiable pseudo almost periodic function f.Besides the decomposition given in definition above is unique.

2.4The Model

Hence，we will consider the model

Denote:

Let us list some assumptions which will be used in this paper.

（H1）For all，the functionis almost periodic，the functionis continuously differentiable almost periodic and the functionsare continuously differentiable pseudo almost periodic.

（H2）The functionsare continuously differentiable and satisfy the Lipschitz condition，i.e.，there are constantssuch that for alland forone has

（H3）For allthe delay kernelsare continuous， integrable and satisfy

The initial conditions associated with system（1.2），are of the form:

3　Existence and Uniqueness of Continuously Differentiable Pseudo Almost Periodic Solution

In order to prove the first main result we shall state the following lemmas.

Lemma 3.1Suppose that assumption（H3）holds anddu belongs to

ProofWe see in fact，immediately，that the function φijsatisfies

which proves that the improper integral）ds is absolutely convergent and the function φijis bounded.Now，we have to prove the continuity of the function φij.Letbe a sequence of real numbers such that The continuity of the function θ implies that for all∈＞0，there exists N∈N such that

Thus，for all n≥N，one has

Let us end the proof，we show that the function φijbelongs to PAP（1）（R，R）.In fact，note that according to the well-known composition theorem of pseudo-almost periodic functions［3］and Lemma 3.1，we obtain immediately the following:for all1≤i，j≤n the functionbelongs toHence，for all 1≤j≤n，one has

Let us prove the almost periodicity of the functionwe consider，in view of the almost periodicity of uj，a number Lεsuch that in any intervalone finds a number δ，with property that:

Afterwards，we can write

On the other hand，by the Fubini’s theorem，one has

By applying the dominated convergence theorem to the right-hand side of the above inequality，we get immediately

Similarly，one has

Lemma 3.2Suppose that assumption（H1）holds andthendu belongs to

Now，we prove the generalized result for translation invariance of continuously differentiable pseudo almost periodic property.

ProofFrom the uniform continuity of，one can choose，for any，a constant

Besides，from the theory of almost periodic functions it follows that for δ＞0，it is possible to find a real numberfor any interval with length l，there exists a numberin this interval such that for all

Combining（3.1）and（3.2），we obtain

Now，we shall prove that

By the following change of variablewe can obtain

Theorem 3.4Assume that assumptions（H1）-（H4）are fulfilled.Then the generalized Hopfield neural network（1.2）has a unique continuously differentiable pseudo almost periodic solution in the convex set

Therefore the space B defined in Theorem 3.4 is a closed convex subset ofand for any，we can obtain this estimation

For φ in B，define the nonlinear operator Γ by:for each

Now，we have proven that this operator（Γφ）belongs precisely to

Now，let us prove the almost periodicity ofwe consider，in view of the almost periodicity of，a number lεsuch that in any interval）one finds a number δ，with property that

Afterwards，we can write

On the other hand

So we can write

Consequently，Γφ∈B.At this stage we are nearer to the end of the proof.For φ，ψ∈B，one hasAccording to the well-known contraction principle there exists a unique fixed pointsuch thatis a continuously differentiable pseudo almost periodic solution of model（1.2）.This completes the proof.

4　The Stability of the Continuously Differentiable Pseudo Almost Periodic Solution

In this section，we establish some results for the stability of the pseudo almost periodic solution of（1.2）.In fact，the first result extends a very interesting result of（see［7］）concerning the global exponential stability of solutions of（1.2）.In the next step we shall explain and prove a result concerning asymptotic behavior of solutions of（1.2）.

Theorem 4.2If conditions（H1）-（H4）hold，then the unique continuously differentiable pseudo almost periodic solution of system（1.2）is globally exponentially stable.

Remark 4.3The proof is similar to Theorem 3.4 in［7］and Theorem 3 in［4］.Notice that the continuously differentiable pseudo almost periodicity is without importance in the proof of the above theorem.

Theorem 4.4Suppose that assumptions（H1）-（H4）hold.Letbe a continuously differentiable pseudo almost periodic solution of system（1.2）.If

then all solutions φ of（1.2）satisfying

converge to its unique continuously differentiable pseudo almost periodic solution x?.

ProofLet x?（·）be a solution of（1.2）and φ be a continuously differentiable pseudo almost periodic solution of（1.2）.First，one verifies without difficulty that

Now，consider the following（ad-hoc）Lyapunov Function

Let us calculate the upper right Dini derivative D+V（t）of V along the solution of the equation above.Then we get

Reasoning in a similar way to the above we obtain the following estimate

By using the inequality

By integrating the above inequality from t0to t，we get

Now，we remark that V（t）＞0.It follows that

The proof of this theorem is now completed.

Remark 4.5If we let aij=0 system（1.2）changes into the model of reference［7］and if cij=0 system（1.2）changes into the model of reference［4］.If we let cij=aij=0 system（1.2）changes into the model of reference［17］.Besides，our model is a natural continuation of references［4］and［18］.Due to the difference in the methods discussed，the results in this paper and those in the above references are different.Therefore，our results are novel and have some significance in theories as well as in applications of almost periodic oscillatory neural networks. On the other hand，these results generalize the theorems in［16］:a different approach is used to obtain several sufficient conditions for the existence and attractively of almost periodic solution for a new class of recurrent neural networks similar to（1.1）.

Remark 4.6One can partially extend this line of reasoning from PAP（1）（R，Rn）toand prove similar results.

5　Numerical Examples

In order to illustrate some features of our main results，in this section，we will apply our main results to some special two-dimensional system and three-dimensional system.These examples demonstrate the efficiencies of our criteria.In the numerical simulations，the fourthorder Runge-Kutta numerical scheme is used to solve the systems.

Example 5.1

Let us consider the following matrices

Therefore，all conditions of our results（Theorem 3.4，Theorem 4.2 and Theorem 4.4）are satisfied，then the delayed two-dimensional Hopfield neural network（5.1）has a unique continuously differentiable pseudo almost periodic solution in the region

Fig.1　Transient response of state variables x1and x2for system（5.1）for t∈［0，50］

Fig.2　The orbits of x1-x2for system（5.1）for t∈［0，500］

Fig.3　The phase trajectory of x1-x2according to t for system（5.1）for t∈［0，500］

For numerical simulation of system（5.1），the initial states are given by the random function. Fig.1 depicts the time responses of state variables of x1（t）and x2（t）with step h=0.02 of system（5.1）.Figs.2 and 3 depict the phase responses state variables of x1and x2.It confirms that the proposed conditions in our theoretical results are effective for model（5.1）.

Example 5.2

Let us consider the following matrices

Therefore，all conditions of our results（Theorem 3.4，Theorem 4.2 and Theorem 4.4）are satisfied，then the delayed three-dimensional Hopfield neural network（5.2）has a unique continuously differentiable pseudo almost periodic solution in the region

For numerical simulation of system（5.2），the initial states are given by the random function. Figure 4 depicts the time responses of state variables of x1（t），x2（t）and x3（t）with step h=0.02 of system（5.2）.Figs.5 and 6 depict the phase responses state variables of x1-x2，x1-x3，x2-x3and x1-x2-x3，respectively.It confirms that the proposed conditions in our theoretical results are effective for model（5.2）.

Fig.4　Transient response of state variables x1，x2and x3in system（5.2）at the interval［0，20］

Fig.5　In the left figure，the orbits of x1-x3for system（5.2）for t∈［0，100］.In the right figure，the orbits of x1-x2for system（5.2）for t∈［0，100］

Fig.6　In the left figure，the orbits of x2-x3for system（5.2）for t∈［0，100］.In the right figure，the orbits of x1-x2-x3for system（5.2）for t∈［0，100］

6　Conclusion and Future Works

The concept of pseudo almost periodicity is very well studied by many physical and natural systems.In recent years，many mathematicians and scientists argued that a more general class of functions is more suitable to explain many complicated processes which show behavior which is”almost”periodic but not purely periodic.In fact，the pseudo almost periodic functions generalize the almost periodic functions and the periodic functions.The main purpose of this paper is to study the existence and the global exponential stability of a continuously differentiable pseudo almost periodic solution of a new class of HNNs.Several novel sufficient conditions are obtained ensuring the existence and uniqueness of the continuously differentiable pseudo almost periodic solution for this model based on a special functional and analysis technique.Further，without resorting to the theory of dichotomy exponential，we will discuss the exponential stability of the continuously differentiable pseudo almost periodic solution by constructing suitable Lyapunov functions.The only restriction for the activation function is the lipschitz property. Finally，two illustrative examples are given to demonstrate the effectiveness of the obtained results.Notice that the method of this paper may be extended to study some other systems［1，16，20，21，27］.Besides，this work can be extended for a class of fractional order ordinary and delay differential equations.Also，our approach can be used to study and investigate the PAP（k）（R，Rn）-solutions of（1.2）（for k∈N｛0，1｝）.

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In the past two decades，neural networks considerable attention，and there were extensive research results presented about the stability analysis of neural network and its applications（see，e.g.，［4，7，12，17，19］）.In particular，the dynamical behaviors of delayed Hopfield neural networks（HNNs）point is globally asymptotically stable so as to avoid the risk of having spurious equilibria and local minima.In the case of global stability，there is no need to be specific about the initial conditions for the system since all trajectories starting from anywhere settle down at the same unique equilibrium.For these reasons，the stability research related to HNNs was extensively studied and developed in recent years［5，6，10］.

much attention due to their applications in many areas，we refer the reader to［5，15］and the references cited therein.In order to solve problems in the fields of associated memory，parallel computing，signal processing，pattern recognition，static image processing，and especially for solving some difficult optimization problems，neural networks have to be designed such that there is only one equilibrium point and this equilibrium

?January 25，2015;revised July 17，2015.

?Adn`ene ARBI.