of impulsive Hopfield-type neural network system with piecewise constant argument

Exponential periodic attractor

of impulsive Hopfield-type neural network system with piecewise constant argument

Manuel Pinto

, Daniel Sepúlveda

and Ricardo Torres

1Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile.

2Departamento de matemáticas, Universidad Tecnológica Metropolitana, Santiago, Chile.

3Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Campus Isla Teja, Valdivia, Chile.

Received 29 January 2018, appeared 30 May 2018 Communicated by Eduardo Liz

Abstract. In this paper we study a periodic impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Under general con- ditions, existence and uniqueness of solutions of such systems are established using ergodicity, Green functions and Gronwall integral inequality. Some sufficient condi- tions for the existence and stability of periodic solutions are shown and a new stability criterion based on linear approximation is proposed. Examples with constant and non- constant coefficients are simulated, illustrating the effectiveness of the results.

Keywords: piecewise constant arguments, Cauchy and Green matrices, hybrid equa- tions, stability of solutions, Gronwall’s inequality, periodic solutions, impulsive differ- ential equations, cellular neural networks.

2010 Mathematics Subject Classification: 34K13, 34K20, 34K34, 34K45, 92B20.

1 Introduction

1.1 Scope

In [45],A. D. Myshkisnoticed that there was no theory for differential equations with discon- tinuous argumenth(t),

x⁰(t) = f(t,x(t),x(h(t))).

These equations are also calledDifferential Equations with Piecewise Constant Arguments(in short DEPCA). The systematic study of problems related to piecewise constant argument began in the 80’s in [52]. Since then, these equations have been deeply studied by many researchers of diverse fields like biomedicine, chemistry, biology, physics, population dynamics and me- chanical engineering. See [17,32,35,43,46]. In [18],S. BusenbergandK. L. Cookewere the first to introduce a mathematical model that involved such types of deviated arguments in the study

BCorresponding author. Email: ricardo.torres@uach.cl; ricardotorresn@gmail.com

of models of vertically transmitted diseases, reducing their study to discrete equations. Very good sources of DEPCA theory are [30,55].

In [6],M. U. Akhmetconsiders the equation

x⁰(t) = f(t,x(t),x(γ(t))),

whereγ(t)is apiecewise constant argument of generalized type, that is, given(t_k)_k∈Zand(ζ_k)_k∈Z such that t_k < t_k+₁,∀k ∈ _Z with lim_k→±∞t_k = ±_∞ and t_k ≤ ζ_k ≤ t_k+₁, then if t ∈ I_k = [t_k,t_k+1), then γ(_t) = ζ_k. These equations are called Differential Equations with Piecewise Con- stant Argument of Generalized Type(in short DEPCAG). They have continuous solutions, even whenγ(t)is not, producing a recursive law on t_k i.e., a discrete equation. These equations combine discrete and continuous dynamics, this is the reason why they are called hybrids.

Stability, approximation of solutions, oscillation and periodicity have been studied in this con- text, see [6,13–15,28,31,33,34,36,37,40–42,44,49,50,54]. In the DEPCAG case, when continuity at the endpoints of intervals of the form I_k = [t_k,t_k+1) is not considered, i.e when a jump condition is defined at these points, give rise toImpulsive Differential Equations with Piecewise Constant Argument of Generalized Type(in shortIDEPCAG),

x⁰(t) = f(t,x(t),x(γ(t))), t6=t_k

∆x|_t=tk =Q_k(x(t⁻_k )), t =t_k. where ∆x|_t=tk = x(t_k)−x(t⁻_k) with x(t⁻_k ) = limt→t_k

t<tk

x(t). See [5,53,56]. In the last years the scientific community has been paying much attention to cellular neural networks (CNN’s).

The two main motivation issues are the own theoretical development and the wide applica- bility of the theory. In the former type of works the focus has been put in the mathematical foundations, the mathematical models formulation, and the qualitative and numerical analy- sis of those models, see for instance [16,26,27,38,58] and the references cited therein. Now, in the case of applications the topics are disperse, we refer for instance to signal processing, image processing, pattern recognition. See [9,26,27]. It is well known that we can find several mathematical models or approaches to describe the behavior in neural networks. The nature of existing models is diverse and the unification or construction of an hybrid model with all the distinct optics is a hard problem. However, there are some general distinctions. For in- stance we distinguish between discrete and continuous models, when the time is considered as discrete or a continuous variable, respectively. Another general classification is given by the dynamics of the cells by considering the deterministic or probabilistic behavior. A well known class of continuous deterministic CNN’s mathematical model is given by the following nonlinear ordinary differential system

dx_i(t)

dt = −a_i(t)x_i(t) +

∑

b_ijf_j(x_j(t)) +d_i(t), i=1, . . . ,m, (1.1) wherem corresponds to the number of units in the neural network, x_i = x_i(t)is the activity or the membrane potential of theith cell at time t, d_i = d_i(t) is the external input to theith cell, a_i = a_i(t) represents the passive decay rate of the ith cell activity, b_ijis the connection or coupling strength of postsynaptic activity of the ith cell transmitted to the jth cell, and the function f_j(x_j) is a continuous function representing the output or firing rate of the ith cell. The construction of (1.1) is given by using the electrochemical properties of the neural networks and assuming that the circuit is formed by resistors. The analysis of the neural

dynamic system (1.1) involves the study of several properties like stability, periodic and almost periodic oscillatory behavior, chaos and bifurcation. See [3,19,21–23,39,40,58–61].

Stimulated by two facts some new relevant generalized versions of the (1.1) are recently formulated. First, by considering that the circuit is constituted by memristors instead of re- sistors we get that the model equation includes a term with a piecewise argument. Second, if we consider that the representation of the state-variable trajectories in some experimental processes, we note that the model solutions are of the type of an impulsive differential equa- tion (IDE) solution. Then CNNs models of the mixed type IDE-DEPCA can be found in the mathematical literature of the last decades [5,8,13,56,57].

1.2 Cellular neural networks with piecewise constant argument

Cellular neural networks (1.1) in the DEPCAG and IDEPCAG cases have been deeply inves- tigated by many authors. Huang et al. [39] considered the following neural network with piecewise constant argument

y⁰_i(t) =−a_i([t])y_i(t) +

∑

b_ij([t])f_j(y_j([t])) +d_i([t]),

where [·] denotes to the greatest integer function and[t] = k ift ∈ I_k = [k,k+1), k ∈ _{N. In} this caset_k = γ_k =k, k∈N. Some sufficient conditions of existence and attractivity of almost periodic sequence solution were given for the discrete-time analogue

y_i(n+1) =y_i(n)e^−ai(n)+¹−a_i(n) a_i(n)

∑

b_ij(n)f_j(y_j(n)) +d_i(n)

! .

In [40], Huang et al. investigated the following neural network with piecewise constant argu- ment

y⁰_i(t) =−a_iy_i(t) +

∑

b_ijf_j

y_j

δ t

δ

+d_i(t), where δ_t

δ

= kδ ift ∈ I_k = [kδ,(k+1)δ), k ∈_Nandδ >0. In this caset_k = γ_k = kδ, k∈ _N.

The authors obtained several sufficient conditions for the existence and exponential attractivity of a uniqueδ-almost periodic sequence solution of the following discrete-time neural network

y_i((n+1)δ) =y_i(nδ)e^{−Rnδ(n+1)δai(u)du}+

∑

_{Z (n+1)δ}

nδ e^{−Rs(n+1)δai(u)du}f_j(y_j(nδ))

b_ij(s)ds +

Z _(n+1)δ

nδ

e^{−Rs(n+1)δai(u)du}d_i(s)_.

In [8], Akhmet et al. obtained some sufficient conditions for the globally asymptotically stable periodic solution of the following constant coefficients delayed IDEPCAG system:

y⁰_i(t) =−a_iy_i(t) +

∑

b_ijf_j(y_j(t)) +

∑

c_ijg_j(y_j(γ(t))) +d_i, t =t_k

∆y_i|_t=tk = I_i,k(y_i(t⁻_k )),

wheret,y_i ∈_R⁺_, a_i >_0, i=1, 2, . . . ,mandγ(t) =t_k ift_k ≤t< t_k+1,k ∈_N.

In [24], K.-S. Chiu et al. studied some new and simple sufficient conditions for the existence and uniqueness of periodic solutions of the following DEPCAG system:

y⁰_i(t) =−a_iy_i(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(γ(t))) +d_i(t)_,

whereγ(t) =γ_k if t_k ≤ γ_k < t_k+₁, k ∈ _N, θ⁺ = γ_k−t_k, θ⁻ = t_k+₁−γ_k, and a positive real numberθsuch thatt_k+1−t_k =θ⁺+θ⁻≤θ.

Later, in [25], the same author investigated some sufficient conditions for the existence, uniqueness and globally exponentially stability of solutions of the following IDEPCAG system with alternately retarded and advanced piecewise constant argument:

y⁰_i(t) =−a_iy_i(t) +

∑

b_ijf_j(y_j(t)) +

∑

c_ijg_j

y_j

m t+l

m

+d_i, t 6=t_k

∆yi|_t=t_k = J_i,k y_i(t⁻_k ).

In this caset_k =mk−landγ_k =mk, with 0≤ l<k, k∈_N.

Finally, in [1], S. Abbas and Y. Xia investigated existence, uniqueness and exponential attractivity of almost automorphic solution of the following IDEPCAG system with alternately retarded and advanced piecewise constant argument:

y⁰_i(t) =−a_iy_i(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j

y_j

2 t+1

2

+d_i(t), t6=t_k

∆yi|_t=tk = J_i,k y_i(t⁻_k ).

In this caset_k =2k−1 andγ_k =2k, k∈_N.

1.3 Aim of the paper

The main subjects under investigation in this paper are sufficient conditions for the existence, uniqueness, periodicity and stability of the following impulsive Hopfield-type neural network with piecewise constant arguments

y⁰_i(t) =−a_i(t)y_i(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(γ(t))) +d_i(t), t6=t_k, (1.2a)

∆yi|_t=t_k =−q_i,ky_i(t⁻_k) +I_i,k(y_i(t⁻_k)) +e_i,k, (1.2b) fori=1, 2, . . . ,m, wheremis the number of neurons in the network,

{t_k}_k∈N is a sequence of positive real numbers such that there is a positive number ¯θ such that 0<t_k+1−t_k ≤ θ^¯for all k∈_N,

(1.2c) γ:R⁺₀ →_R⁺₀ is the piecewise constant function, on every interval,

[t_k,t_k+1), argument precisely, it is a function such that γ([t_k,t_k+1[) ={t_k}, for allk∈ _N.







(1.2d) The length of every discontinuity ofy_i(t)on t=t_k is∆yi =y_i(t_k)−y_i(t⁻_k )

wherey_i(t⁻_k ) = _lim

t→t_k t<tk

y_i(t)_.







(1.2e)

The functions and parameters in (1.2a) and (1.2b) have the following meaning:

– The value of the functiony_i(t)corresponds to the state of theith unit at timet and the unknown functiony_i typically denotes the potential of theith cell of the network.

– The functionsa_i(t)>0, and 0<q_i,k <1 are the rates of reseting potential for the uniti.

– The functions f_j(y_i(t)) and g_j(y_i(γ(t)) represent the measure of the activations to the incoming potential of unit jon uniti.

– The functions b_ij(t) and c_ij(t) represent the activation connection weighs of unit j on unit i.

– The functionse_i andd_i(t)represent the input from outside on the uniti.

– The functions I_i,k(y_i(t⁻_k ))represent the activation connection weighs of the unition the unit ifor every impulse, such that I_i,k(y_i(t⁻_k )) =_limt→tk

t<t_k

I_i,k(y_i(t))_.

– The functionse_i,k represent the input from outside on the unitifor every impulse.

Here, N andR⁺₀ = [0,∞) denote the sets of natural and nonnegative real numbers, respec- tively. Note that (1.2) is a perturbed system of the impulsive differential linear nonhomoge- neous system

y⁰_i(t) =−a_i(t)y_i(t) +d_i(t)_, t6= t_k, (1.3a)

∆y_i|_t=tk =−q_i,ky_i(t⁻_k ) +e_i,k. (1.3b) Additional notation has been taken from the standard theory of impulsive and differential equations with piecewise continuous argument, see for instance [2,10,11,20,29,45].

1.4 General assumptions

In this paper in order to obtain the results for (1.2), we consider the following general assump- tions:

(H1) The functions a_i,b_ij,c_ij,d_i are real valued and ω-periodic with ω > 0. Moreover, there exists p∈ _Nsuch that the sequences{t_k}_k∈N,{q_i,k}_k∈N, {e_i,k}_k∈Nand{I_i,k}_k∈Nsatisfy

[0,ω]∩ {t_k}_k∈N=t₁, . . . ,t_p , t_k+p= t_k+ω, q_i,k+p =q_i,k,

e_i,k+p =e_i,k, I_i,k+p = I_i,k, ∀k∈_N, ∀i∈ {1, . . . ,m}. (H2) (Non-critical case) The functiona_i and the sequence{q_i,k}_k∈Nare such that

∏

1−q_i,k exp

−

Z _ω

0 a_i(u)du

6=1, ∀i∈ {1, . . . ,m}.

(H3) The functions f_j andg_j are Lipschitz, i.e. there existsL_j, ¯L_j >0 such that

|f_j(u)− f_j(v)| ≤ L_j|u−v|_, |g_j(u)−g_j(v)| ≤ L^¯_j|u−v|_, ∀u,v∈_R^m_, ∀j∈ {_{1, . . . ,}m}_.

(H4) The functionsI_i,k are Lipschitz, i.e. there existsl_i,k >0 such that

|I_i,k(u)−I_i,k(v)| ≤l_i,k|u−v|, ∀u,v∈_R^m, ∀k∈ _N, ∀i∈ {1, . . . ,m}. (H5) The functions f_j,g_j and I_i,k satisfy f_j(0) =g_j(0) =I_i,k(0) =0, (H3) and (H4) for

|u|,|v| ≤R.

(H6) There existsσ>0 such that Z _t

s a_i(u)du+

∑

s≤tk<t

ln(1+q_i,k)≥ σ(t−s), ∀k∈ _N,∀i∈ {1, . . . ,m}. This condition follows from

¯

a+_{ln 1}+q⁺

≥_σ,

where ¯a=min_{i∈{1,...,m}}inf_t∈R⁺ a_i(t)andq⁺=max_{i∈{1,...,m}}sup_k∈N q_i,k.

Furthermore, in various results of this paper, the following assumptions will be needed:

(H7) We assume that

ρ=sup

n∈_N Z _tn+1

tn

eb(s) +_ec(s)ds<1, whereeb(s)andec(s)are defined as follows

eb(s) =

∑

∑

|b_ij(s)|L_i and ec(s) =

∑

∑

|c_ij(s)|L_i. (1.4) Here,L_i, L_i is the notation defined on (H3).

(H8) We consider that

K

ωM(eb+_ec) +_pMel

<1, (1.5)

where K is the norm of the Green function of the system (1.2) defined in (3.2), el_k is defined as

el_k =

∑

l_i,k, (1.6)

and

M(eb) = ¹ ω

Z _ω

0

eb(u)du, M(el) = ¹ p

∑

el_k denote the means ofebandelrespectively.

Condition (1.5) follows from

K

eb⁺+_ec⁺+el⁺

<_1.

(H9) There existsσ>0 such that ωM

eb+e^θσ¯ ec

+_pMln(1+el)<σ,

withωand pas is given on (H1),eb,ecandelthe notation in (1.4) and (1.6), respectively.

This condition follows from

eb⁺+e^θσ¯ ec⁺+_ln(₁+el⁺)<_σ.

Remark 1.1. We stand out the following facts:

(a) The hypothesis (H6) follows fromωM(a_i) +_pM(ln(1+q_i))>σ.

(b) In (H8), whena_i(t) =a_i andq_i,k =q_i are constants, we can take:

K = ¹

1−(1−α)^pexp(−ωa)^{, a}= min

1≤i≤ma_i and α= min

1≤i≤mq_i.

2 Existence and uniqueness of solutions for (1.2)

2.1 A useful Gronwall type result

The following lemma will be adopted throughout this paper and its proof is almost identical to the verification of Lemma 2.2 in [47] with slight changes which are caused by the impulsive effect.

Lemma 2.1. Let I an interval and u,η₁,η2 be three functions from I ⊂ _R to R⁺₀ such that u is continuous;η₁,η₂are locally integrable andη:{t_k} →_R⁺₀.Letγ(t)be a piecewise constant argument of generalized type,i.e.a step function such that γ(t) =ζ_k for all t∈ I_k = [t_k,t_k+1),with t_k ≤ζ_k ≤ t_k+1,∀k∈ _Nsatisfying

υ⁺_k =

Z _ζk

t_k

(η₁(s) +η₂(s))ds≤ν=sup

k∈_N

υ⁺_k <1, u(t)≤ u(τ) +

Z _t

τ

(η₁(s)u(s) +η₂(s)u(γ(s)))ds+

∑

τ≤t_k<t

η(t_k)u(t⁻_k ). Then, the inequalities

u(t)≤

∏

τ≤tk<t

(1+η(t_k))

! exp

Z _t

τ

η₁(s) + ^η2(s) 1−ν

ds

u(τ) u(ζ_k)≤ (1−ν)⁻¹u(t_k)

are valid for all t≥τ.

Corollary 2.2. Let I an interval and u,η₁,η₂ be three functions from I ⊂ _R to R⁺₀ satisfying the hypothesis described in Lemma2.1and consider the step function defined asγ(t) =t_k for all t ∈ I_k = [t_k,t_k+1),∀k∈ _N.If

u(t)≤ u(τ) +

Z _t

τ

η₁(s)u(s) +η₂(s)u(γ(s))ds+

∑

τ≤t_k<t

η(t_k)u(t⁻_k ) holds, then the inequality

u(t)≤

∏

τ≤tk<t

(1+η(t_k))

! exp

_{Z t}

τ

(η₁(s) +η₂(s))ds

u(τ). is valid for all t≥τ.

2.2 Existence and uniqueness of solutions of (1.2a) for t ∈[tr,tr+1) withr ∈ _N In this section we consider the analysis of (1.2a) with initial conditiony(ξ) =y₀and restricted to the case thatξ,t ∈ [t_r,t_r+1)with t_randt_r+1 two arbitrary consecutive impulsive times. In- deed, for convenience of the presentation of the results and proofs, we consider the following system

y⁰_i(t) =−a_i(t)y_i(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(tr)) +d_i(t), y_i(ξ) =y⁰_i with arbitrary initial momentξ ∈ [t_r,t_r+1), t∈[ξ,t_r+1)andr ∈_N.







(2.1)

Note that, in the third term of (2.1), we have used the fact that γ(t) = t_r for t ∈ [t_r,t_r+1). Moreover we note that (2.1) is equivalent to the following integral equation

z_i(t) =_Hi(z(t),ξ,y₀), z(t) = (z₁(t), . . . ,z_m(t)), t ∈[ξ,t_r+1], (2.2) where

Hi(z(t),ξ,y₀)

= exp

−

Z _t

ξ

a_i(u)du y⁰_i +

Z _t

ξ

exp

−

Z _t

s a_i(u)du ^m

j

∑

b_ij(s)f_j(z_j(s)) +

∑

c_ij(s)g_j(z_j(t_r)) +d_i(s)

! ds.

(2.3)

The following lemmata provide the conditions for the uniqueness and existence of solutions for (2.1).

Lemma 2.3. Consider that there are solutions of (2.1)for y₀= (y⁰₁, . . . ,y⁰_m)^T ∈_R^mandξ ∈[t_r,t_r+1]. If (H3) and (H6) are satisfied, then the solution y(t) = y(t,ξ,y₀) = (y₁(t), . . . ,y_m(t))^T of (2.1) is unique for each y₀ andξ.

Proof. The proof is developed by contradiction. Indeed, we assume that z²_i and z¹_i are two distinct solutions of (2.2). Then, by application of the hypotheses (H3) and (H6), we have the estimate

|z²_i(t)−z¹_i(t)| ≤

Z _t

ξ

exp

−σ(t−s)

×

∑

b_ij(s)L_j

z²_j(s)−z¹_j(s) +

∑

c_ij(s)L¯_j

z²_j(tr)−z¹_j(tr)

! ds.

Then, using the notations (1.4) andk · k₁for the sum norm inR^m, we obtain that

z²(t)−z¹(t)

1≤

Z _t

ξ

exp −σ(t−s)eb(s)z²(s)−z¹(s)

1+_ec(s)z²(t_r)−z¹(t_r)

1

ds, which is rewritten as it follows

u(t)≤

Z _t

ξ

eb(s)u(s) +_ec(s)u(γ(s))ds with u(t) =exp(σt)z²(t)−z¹(t)

1.

From Lemma2.1 we deduce thatu(t)≡0, sinceu(ξ) =0. Now, we have that z² = z¹, which contradicts our initial assumption. Hence, we have the uniqueness of solutions for (2.2) or equivalently the uniqueness of solutions for (2.1).

Lemma 2.4. Let (H3), (H6) and (H7) be satisfied. Then for each y₀ = (y⁰₁, . . . ,y⁰_m)^T ∈ _R^m and ξ ∈[tr,tr+₁), there exists a solution y(t) =y(t,ξ,y0) = (y₁(t), . . . ,ym(t))^Tof (2.1)on[ξ,tr+₁]such that y(ξ) =y₀.

Proof. In order to prove the lemma, it is enough to show that the equation (2.2) has a unique solution z(t) = (z₁(t), . . . ,z_m(t))^T on [ξ,t_r+1]. Indeed, let us define the norm kzk₀ = max_t∈[tr,tr+1]kz(t)k₁ and construct the following sequence{zⁿ_i(t)}_{n∈Nsuch that}

z⁰_i(t) =_Hi(0,ξ,y₀) and zⁿ_i⁺¹(t) =_Hi(z_iⁿ(t),ξ,y₀) forn∈_N,

whereHi is defined in (2.3). By application of (H3), (H6) and using the notation (1.4), we can see that

kzⁿ⁺¹(t)−zⁿ(t)k₁ ≤

∑

Z _t

ξ

exp

−

Z _t

s a_i(u)du

×

∑

|b_ij(s)|L_j|zⁿ_j(s)−zⁿ_j⁻¹(s)|

∑

|c_ij(s)|L_j|zⁿ_j(tr)−zⁿ_j⁻¹(tr)|

! ds

≤ kzⁿ−zⁿ⁻¹k₀

Z _t

ξ

e^−σ(t−s)

eb(s) +_ec(s)ds

≤ρkzⁿ−zⁿ⁻¹k₀,

whereρ is the notation defined on (H7). Now, using mathematical induction, we get that kzⁿ⁺¹−zⁿk₀ ≤ρⁿ⁺¹kz⁰k₀.

Hence, by (H7), the sequence {zⁿ(t)}_n∈N is convergent and its limit satisfies the integral equation (2.2) on[ξ,t_r+1]. The existence is proved.

Remark 2.5. The previous results extend the corresponding constant coefficient case given by Akhmet et al. in [8].

2.3 Existence and uniqueness of solutions for (1.2) on [t0,t]⊂_R⁺₀

Using the impulsive condition, the solutions of (2.1) can be extended inductively onk∈ _Nto construct a solution of (1.2a) on the interval[t0,t]. Indeed, we will give a theorem that allows us to construct a unique solution of equation (1.2) on[t₀,t]⊂_R⁺.

Theorem 2.6. Assume that conditions (H3)–(H4), (H6) and (H7) are fulfilled. Then, for (t₀,y₀) ∈ R⁺₀ ×_R^m, there exists y(t) =y(t,t₀,y⁰) = (y₁(t),y₂(t), . . . ,y_m(t))^T for t≥t₀, a unique solution of (1.2), such that y(t₀) =y₀.

Proof. We proceed inductively, using the sequence of impulsive times. Indeed, in the following we describe the first two steps. First, fix t0 ∈ _R⁺₀. Then, there exists r ∈ _N such that t₀ ∈[t_r−1,t_r)and by Lemmas2.3and2.4withξ =t₀we obtain the unique solution y(t,t₀,y⁰) on [ξ,t_r]. Now, we apply the impulse condition (1.2b) to evaluate uniquely the solution at t=tr:

y_i(t_r,t₀,y⁰) =y_i(t⁻_r ,t₀,y⁰)−q_i,ry_i(t_r⁻,t₀,y⁰) +I_i,r(y(t⁻_r )) +e_i,r

= (1−q_i,r)y_i(t⁻_r,t₀,y⁰) +I_i,r(y_i(t⁻_r ,t₀,y⁰)) +e_i,r.

Next, on the interval[t_r,t_r+1]the solution satisfies the ordinary differential equation (2.1) with ξ = t_k andy⁰_i = y_i(tr,t₀,y⁰). Then, by a new application of Lemmas2.3and2.4 we have that the new system has a unique solution y(t,t_r,y(t_r,t₀,y⁰)). Thus, by construction, we have the unique solution of (1.2) on[t_r,t_r+1]. The mathematical induction completes the proof.

2.4 Integral equations associated to (1.2)

Let us establish the integral equation associated to (1.2) in the following two lemmas. We will prove only the first one, the proof for the second one is similar and omitted.

Lemma 2.7. A function y(_t) = _y(_t,t0,y0) = (_y1(_t)_{, . . . ,ym}(_t))^T_{, where t0}is a fixed real number, is a solution of (1.2)onR⁺₀ if and only if it is a solution, onR⁺₀, of the following integral equation:

y_i(t) =y⁰_i +

Z _t

t0

−a_i(s)y(s) +

∑

b_ij(s)f_j(y_j(s)) +

∑

c_ij(s)g_j(y_j(γ(s))) +d_i(s)

! ds

+

∑

t0≤tk<t

(₁−q_i,k)y_i(t⁻_k ) +h^˜_i,k y_i(t⁻_k )_, whereh˜_i,k y_i(t⁻_k ) = I_i,k(y_i(t⁻_k )) +e_i,k, for i=_{1, . . . ,}m,t≥t₀.

Proof. Sufficient part of this lemma can be easily proved. Therefore, we only prove the neces- sity part of this lemma. Fixi= 1, . . . ,m. Assume thaty(t) = (y₁(t), . . . ,y_m(t))^T is a solution of (1.2) onR⁺₀. Denote byϕ_i the following function

ϕ_i(t) =y⁰_i +

Z _t

t0

−a_i(s)y(s) +

∑

b_ij(s)f_j(y_j(s)) +

∑

c_ij(s)g_j(y_j(γ(s))) +d_i(s)

! ds

+

∑

t0≤t_k<t

(1−q_i,k)y_i(t⁻_k ) +h^˜_i,k y_i(t⁻_k ). (2.4) It is clear that the expression in the right side exists for allt. Assume that t ∈ (t_r−1,t_r), then differentiatingϕ_i we get

ϕ⁰_i(t) =−a_i(t)y(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(γ(t))) +d_i(t). Also, we have that

y⁰_i(t) =−a_i(t)y(t) +

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(γ(t))) +d_i(t). Hence, fort 6=t_k, k∈N, we obtain

(ϕ_i(t)−y_i(t))⁰ =0.

Moreover, it follows from (2.4) that

∆ϕi(t_r) = ϕ_i(t_r)−ϕ_i(t⁻_r) =−q_i,rϕ_i(t⁻_r ) +h^˜_i,r ϕ_i(t⁻_r ), which implies that

ϕ_i(tr) = (1−q_i,r)ϕ_i(t⁻_r ) +h^˜_i,r ϕ_i(t⁻_r ). (2.5) One can see that ϕ_i(t₀) = y⁰_i. Then, by (2.5), we have that ϕ_i(t) = y_i(t) on [t₀,tr), which implies ϕ_i(t⁻_r ) =y_i(t⁻_r ). Next, using (2.5) and the last equation, we obtain

ϕ_i(t_r) = (1−q_i,r)ϕ_i(t⁻_r ) +h^˜_i,k ϕ_i(t_r⁻) = (1−q_i,r)y_i(t⁻_r ) +h^˜_i,r y_i(t⁻_r ) =y_i(t_r). Therefore, one can conclude that ϕ_i(t) = y_i(t) _for t ∈ [t_r,t_r+1). Similarly, as shown in the discussion above, one can also obtain with variation of constant formula that ϕ_i(t) = y_i(t) on [t_r,t_r+1]. We can complete the proof by using mathematical induction and a variation of constant formula.

Lemma 2.8. A function y(t) =y(t,t₀,y₀) = (y₁(t), . . . ,y_m(t))^T, where t₀ is a fixed real number, is a solution of (1.2)onR⁺₀ if and only if it is a solution, onR⁺₀, of the following integral equation:

y_i(t) =

k(t) l=

∏

(1−q_i,l)exp

−

Z _t

t₀ a_i(u)du

y⁰_i

+

Z _t

t0

k(t) l=

∏

(1−q_i,l)exp

−

Z _t

s a_i(u)du

×

∑

b_ij(s)f_j(y_j(s)) +

∑

c_ij(s)g_j(y_j(γ(s))) +d_i(s)

! ds

+

∑

t0≤t_k<t k(t) l=

∏

(1−q_i,l)exp

−

Z _t

t_k a_i(u)du

h˜_i,k(y(t⁻_k )),

for i =1, . . . ,m,t ≥t₀,where k=k(t)is the unique k∈_Nsuch that t ∈[t_k,t_k+1).

3 Green function and periodic solutions for (1.2), global and local conditions

In this section, we will prove the existence and uniqueness of a periodic solution of the CNN model (1.2). First, we obtain a Green function which reduces the problem to an integral equation. Then, we prove the existence and uniqueness of a periodic solution in two situa- tions: under global Lipschitz conditions (H3)–(H4) and under local Lipschitz conditions (H5) satisfied in the ball B[0,R].

3.1 Green function

Here, we will give the following version of the Poincaré criterion for system (1.2). One can easily prove the following lemma (see for instance [7]).

Lemma 3.1. Suppose that conditions (H1)–(H4) and (H7) hold. Then, a solution y(t) =y(t, 0,y₀) = (_{y1,y2, . . . ,ym})^T _{of (1.2)with y}(₀) =_y0isω-periodic if and only if y(ω) =_y0.

Lemma 3.2. Suppose that conditions (H1) and (H2) hold and y is aω-periodic solution of (1.2). Then y satisfies the integral equation

y_i(t) =

Z _ω

0 Ki(t,s)F_i(s,y(s))ds+

∑

Ki(t,t_k)h^˜_i,k(y(t⁻_k )), (3.1)

where

F_i(t,y(t)) =

∑

b_ij(t)f_j(y_j(t)) +

∑

c_ij(t)g_j(y_j(γ(t))) +d_i(t), h˜_i,k(y(t⁻_k )) =I_i,k(y_i(t⁻_k )) +e_i,k

Ki(t,s) = 1−

∏

1−q_i,l exp

−

Z _ω

0 a_i(u)du

!−1

(3.2)

×















k(t) l=

∏

(₁−q_i,l)_exp−Rt

s a_i(u)du

, 0≤s≤t ≤ω

k(t)+ω l=

∏

(1−q_i,l)exp

−Rt+ω

s a_i(u)du

, 0≤t<s ≤ω.

The functionKiis the Green function of the system(1.2).

Proof. LetPC_ω=ϕ∈PC(_R⁺_0,R^m)| ϕ(t+ω) =ϕ(t)_, t≥₀ be the linear space ofω-periodic functions. Using Lemma 2.8, one can show that if y ∈ PCω is a ω-periodic solution of the following system:

y⁰_i(t) =−a_i(t)y_i(t) +F_i(t,ϕ(t)), t6= t_k, (3.3a)

∆yi|_t=tk = −q_i,ky_i(t⁻_k ) +h^˜_i,k(ϕ(t⁻_k )), (3.3b) withi=1, 2, . . . ,m, k =1, 2, . . .p, theny_i(t, 0,y⁰_i)is given by

y_i(t) =

k(t)

∏

(1−q_i,l)exp

−

Z _t

0 a_i(u)du

y⁰_i +

t

Z

0 k(t) l=

∏

1−q_i,l exp

−

Z _t

s a_i(u)du

F_i(s,ϕ(s))ds

+

∑

0≤t_k<t k(t) l=

∏

1−q_i,l exp

−

Z _t

t_k a_i(u)du

h˜_i,k(ϕ(t⁻_k )). (3.4)

Then, evaluating att=ω we obtain y_i(ω) =

∏

1−q_i,l exp

−

Z _ω

0 a_i(u)du

y⁰_i +

Z _ω

0

∏

1−q_i,l exp

−

Z _ω

s a_i(u)du

F_i(s,ϕ(s))ds +

∑

∏

1−q_i,l exp

−

Z _ω

t_k a_i(u)du

h˜_i,k(ϕ(t⁻_k))_.

Now, in order to prove thatyis a periodic solution we need to verify thaty_i(ω) =y_i(0) =y⁰_i. Indeed, from (3.4) we have that

y_i(ω) =

∏

1−q_i,l exp

−

Z _ω

0 a_i(u)du

y⁰_i

+

ω

Z

0

∏

1−q_i,l exp

−

Z _ω

s a_i(u)du

F_i(s,ϕ(s))ds +

∑

∏

1−q_i,l exp

−

Z _ω

tk

a_i(u)du

eh_i,k(ϕ(t⁻_k ))

=y⁰_i