In this paper, a kind of shunting inhibitory cellular neural network with a neutral delay was considered. By using the Banach fixed point theorem, we established a result about the existence and uniqueness of the almost periodic solution for the shunting inhibitory cellular neural network

Fang, Zheng

Yang, Yongqing

Electronic Journal of Qualitative Theory of Differential Equations

Zheng Fang

Yongqing Yang

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Electronic journal of qualitative theory of differential equations

Existence of almost periodic solution for SICNN with a neutral delay

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Electronic Journal of Qualitative Theory of Differential Equations2011, No. 30, 1-12; http://www.math.u-szeged.hu/ejqtde/Existence of almost periodic solution for SICNN with a neutral delayZheng Fang1 Yongqing Yang2School of Science, Jiangnan University, Wuxi, 214122, PR ChinaAbstractIn this paper, a kind of shunting inhibitory cellular neural network witha neutral delay was considered. By using the Banach fixed point theorem,we established a result about the existence and uniqueness of the almostperiodic solution for the shunting inhibitory cellular neural network.Keywords: shunting inhibitory cellular neural network, neutral delay,almost periodic solution, fixed point theorem.1. IntroductionShunting inhibitory cellular neural network (SICNN) is a kind of very importantmodel and has been investigated by many authors (see [1, 2, 3, 4] and the referencetherein) due to its wide applications in practical fields such as robotics, adaptive patternrecognition and image processing. In [1], Ding studied the following SICNNx′ij = −aijxij −∑Ckl∈Nr(i,j)Cklij f [xkl(t− τ(t))]xij(t) + Lij(t),Most of the existing SICNN models are concerns with the delays in the state.However, it is not enough for it can not describe the phenomenon precisely. It isnatural and useful to consider the model with a neutral delay, it means that the systemdescribing the model depends on not only the past information of the state but alsothe information of the derivative of the state, i.e., the decay rate of the cells. This kindof model is described by a differential equation with a neutral delay. The neutral typedifferential equations have many applications, for more details we refer to [6]. Someauthors have considered the Hopfield neural networks with neutral delays, see [7, 8].To the best of our knowledge, there is few consideration about the shunting inhibitorycellular neural network with a neutral delay. In this paper, we consider the following1e-mail: fangzhengjd@126.com, corresponding author2e-mail: yyq640613@gmail.comEJQTDE, 2011 No. 30, p. 1shunting inhibitory cellular neural network with a neutral delay:x′ij = −aij(t)xij −∑Ckl∈Nr(i,j)Cklij (t)f [xkl(t− τ(t))]xij(t)−∑Ckl∈Ns(i,j)Dklij (t)g[x′kl(t− σ(t))]xij(t) + Iij(t). (1.1)In this model, Cij represents the cell at the (i, j) position of the lattice, ther−neighborhood Nr(i, j) of Cij is defined as followsNr(i, j) ={Ckl : max{| k − i |, | l − j |} ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n},xij(t) describes the state of the cell Cij, the coefficient aij(t) > 0 is the passive decayrate of the cell activity, Cklij (t), Dklij (t) are connection weights or coupling strength ofpostsynaptic activity of the cell Ckl transmitted to the cell Cij , and f, g are continuousactivity functions, representing the output or firing rate of the cell Ckl , and τ(t), σ(t)correspond to the transmission delays.In the following, we give some basic knowledge about the almost periodic functionsand almost periodic solutions of differential equations, please refer to [9, 10] for moredetails.Definition 1.1 (See [10]) Let u : R → Rn be continuous in t. u is said to be almostperiodic on R if, for any ǫ > 0, the set T (u, ǫ) = {δ :| u(t+ δ)− u(t) |< ǫ, ∀ t ∈ R} isrelatively dense, i.e., for ∀ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, forany interval with length l(ǫ), there exists a number δ = δ(ǫ) in this interval such that| u(t+ δ)− u(t) |< ǫ, for all t ∈ R.Definition 1.2 ([9, 10]) If u : R → Rn is continuously differentiable in t, u(t) andu′(t) are almost periodic on R, then u(t) is said to be a continuously differentiablealmost periodic function.Let AP (R,Rm×n) and AP 1(R,Rm×n) be the set of continuous almost periodic func-tions, and continuously differentiable almost periodic functions from R to Rm×n, re-spectively. For each ϕ ∈ AP 1(R,Rm×n), define‖ϕ‖0 = supt∈Rmaxi,j{| ϕi,j |},‖ϕ‖ = max{‖ϕ‖0, ‖ϕ′‖0}.It is easy to check that (AP (R,Rm×n), ‖ · ‖0) and (AP1(R,Rm×n), ‖ · ‖) are all Banachspaces.Definition 1.3 ([9, 10]) Let x ∈ Rn and Q(t) be an n × n continuous matrixdefined on R. The linear systemx′(t) = Q(t)x(t) (1.2)EJQTDE, 2011 No. 30, p. 2is said to admit an exponential dichotomy on R if there exist positive constants k, α,projection P and the fundamental solution matrix X(t) of (1.2) satisfying‖ X(t)PX−1(s) ‖< ke−α(t−s), t ≥ s,‖ X(t)(I − P )X−1(s) ‖< ke−α(s−t), t ≤ s.Lemma 1.1 ([9, 10]) If the linear system (1.2) admits an exponential dichotomy,then almost periodic systemx′(t) = Q(t)x(t) + g(t) (1.3)has a unique almost periodic solution x(t), andx(t) =∫ t−∞X(t)PX−1(s)ds−∫ +∞tX(t)(I − P )X(s)ds. (1.4)Lemma 1.2 ([9, 10]) Let c(t) be an almost periodic function on R andM [ci] = limT→+∞1T∫ t+Ttci(s)ds > 0, i = 1, 2, · · ·n.Then the linear systemx′(t) = diag(−c1(t),−c2(t), · · · , cn(t))x(t)admits an exponential dichotomy on R.2. Main resultsFirstly, We give some assumptions.(H1) aij(t), Cklij (t), Dklij (t), Iij(t), τ(t), σ(t) are all almost periodic functions,i = 1, 2, · · · , n, j = 1, 2, · · · , m;(H2) the activity functions f and g are Lipschtiz functions, i.e., there exist L > 0,l > 0 such that|f(x)− f(y)| ≤ L|x− y|, ∀x, y ∈ R,|g(x)− g(y)| ≤ l|x− y|, ∀x, y ∈ R;(H3) supt∈R|Cklij (t)| = Cklij < +∞, supt∈R|Dklij (t)| = Dklij < +∞, supt∈R|aij(t)| = a+ij <+∞, aij(u) ≥ aij∗ > 0.EJQTDE, 2011 No. 30, p. 3Letϕ0 ={ t∫−∞e−R tsaij(u)duIij(s)ds}, ‖ϕ0‖ = R0θ1 = maxi,j{ ∑Ckl∈Nr(i,j)Cklij(2LR + |f(0)|)+∑Ckl∈Ns(i,j)Dklij(2lR+ |g(0)|)};θ2 = maxi,j{ ∑Ckl∈Nr(i,j)Cklij(4LR + |f(0)|)+∑Ckl∈Ns(i,j)Dklij(4lR+ |g(0)|)}.where R is a constant with R ≥ R0.Theorem 2.1 If (H1), (H2), (H3) and the following conditions are satisfied(1) R0 ≤ R < +∞;(2) θ1 ·maxi,j{1aij∗, 1 +a+ijaij∗}≤ 12;(3) θ = θ2 ·maxi,j{1aij∗, 1 +a+ijaij∗}< 1,then Eq.(1.1) has a unique almost periodic solution.Proof. For any given ϕ = {ϕij} ∈ AP1(R,Rm×n), we consider the almost periodicsolution of the following differential equationx′ij = −aij(t)xij −∑Ckl∈Nr(i,j)Cklij (t)f(ϕkl(t− τ(t)))ϕij(t)−∑Ckl∈Ns(i,j)Dklij (t)g(ϕ′kl(t− σ(t)))ϕij(t) + Iij(t). (2.1)Since ϕij(t), aij(t), Cklij (t), Dklij (t), τ(t), σ(t) and Iij(t) are all almost periodic func-tions, and M [aij ] > 0, according to Lemma 1 and lemma 2, we know that Eq.(2.1) hasa unique almost periodic solution xϕ = {xϕij}, which can be expressed as followsxϕij =t∫−∞e−R tsaij(u)du[−∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))ϕij(s)−∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))ϕij(s) + Iij(s)]ds.We define a nonlinear operator on AP 1(R,Rm×n) as followsT (ϕ)(t) = xϕ(t), ∀ϕ ∈ AP 1(R,Rm×n).EJQTDE, 2011 No. 30, p. 4It is obvious that the fixed point of T is a solution of Eq.(1.1). In the following we willshow that T is a contract mapping, thus the Banach fixed point theorem assert that Thas a fixed point.Let E be defined as followsE = {ϕ ∈ AP 1(R,Rm×n)| ‖ϕ− ϕ0‖ ≤ R}.Firstly, we show that T (E) ⊆ E. For each ϕ ∈ E, we have ‖ϕ − ϕ0‖ ≤ R,‖ϕ‖ ≤ ‖ϕ− ϕ0‖+ ‖ϕ0‖ ≤ R +R ≤ 2R. Thus‖Tϕ− ϕ0‖0= supt∈Rmaxi,j{∣∣∣t∫−∞e−R tsaij (u)du[−∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))ϕij(s)−∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))ϕij(s)]ds∣∣∣}≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du∣∣∣− ∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))ϕij(s)−∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))ϕij(s)∣∣∣ds}≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du[ ∑Ckl∈Nr(i,j)|Cklij (s)||f(ϕkl(s− τ(s)))||ϕij(s)|+∑Ckl∈Ns(i,j)|Dklij (s)||g(ϕ′kl(s− σ(s)))||ϕij(s)|]ds}≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du[ ∑Ckl∈Nr(i,j)Cklij(|f(ϕkl(s− τ(s)))− f(0)|+ |f(0)|)+∑Ckl∈Ns(i,j)Dklij(|g(ϕ′kl(s− σ(s))− g(0)|+ |g(0)|))]ds}2R≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du[ ∑Ckl∈Nr(i,j)Cklij (L|ϕkl(s− τ(s))|+ |f(0)|)+∑Ckl∈Ns(i,j)Dklij (l|ϕ′kl(s− σ(s))|+ |g(0)|)]ds}2R≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du[ ∑Ckl∈Nr(i,j)Cklij (2LR + |f(0)|)EJQTDE, 2011 No. 30, p. 5+∑Ckl∈Ns(i,j)Dklij (2lR+ |g(0)|)]ds}2R≤ supt∈Rmaxi,j{ t∫−∞e−aij∗(t−s)ds}2Rθ1≤ 2Rθ1 maxi,j{ 1aij∗}≤ R (2.2)and‖(Tϕ− ϕ0)′‖0= supt∈Rmaxi,j{∣∣∣∫ t−∞−aij(t)e−R tsaij(u)du ·[−∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))−∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))]ϕij(s)ds+[−∑Ckl∈Nr(i,j)Cklij (t)f(ϕkl(t− τ(t)))−∑Ckl∈Ns(i,j)Dklij (t)g(ϕ′kl(t− σ(t)))]ϕij(t)∣∣∣}= supt∈Rmaxi,j{∣∣∣∫ t−∞aij(t)e−R tsaij(u)du ·[ ∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))+∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))]ϕij(s)ds−[ ∑Ckl∈Nr(i,j)Cklij (t)f(ϕkl(t− τ(t)))+∑Ckl∈Ns(i,j)Dklij (t)g(ϕ′kl(t− σ(t)))]ϕij(t)∣∣∣}≤ supt∈Rmaxi,j{∫ t−∞|aij(t)|e−R tsaij(u)du ·[ ∑Ckl∈Nr(i,j)|Cklij (s)| |f(ϕkl(s− τ(s)))|+∑Ckl∈Ns(i,j)|Dklij (s)| |g(ϕ′kl(s− σ(s)))|]|ϕij(s)|dsEJQTDE, 2011 No. 30, p. 6+[ ∑Ckl∈Nr(i,j)|Cklij (t)| |f(ϕkl(t− τ(t)))|+∑Ckl∈Ns(i,j)|Dklij (t)| |g(ϕ′kl(t− σ(t)))|]|ϕij(t)|}≤ supt∈Rmaxi,j{∫ t−∞a+ije−aij∗(t−s)ds ·[ ∑Ckl∈Nr(i,j)Cklij(2LR + |f(0)|)+∑Ckl∈Ns(i,j)Dklij(2lR+ |g(0)|)]+[ ∑Ckl∈Nr(i,j)Cklij(2LR + |f(0)|)+∑Ckl∈Ns(i,j)Dklij(2lR+ |g(0)|)]}· 2R≤ 2Rθ1 maxi,j{1 +a+ijaij∗}≤ R. (2.3)From (2.2) and (2.3), we have ‖Tϕ− ϕ0‖ ≤ R, thus T (E) ⊆ E.Let ϕ, ψ ∈ E, denote byI1(s) =∑Ckl∈Nr(i,j)Cklij (s)[f(ϕkl(s− τ(s)))ϕij(s)− f(ψkl(s− σ(s)))ψij(s)],I2(s) =∑Ckl∈Ns(i,j)Dklij (s)[g(ϕ′kl(s− σ(s)))ϕij(s)− g(ψ′kl(s− σ(s)))ψij(s)].We have|I1(s)|=∣∣∣ ∑Ckl∈Nr(i,j)Cklij (s)[f(ϕkl(s− τ(s)))ϕij(s)− f(ψkl(s− σ(s)))ψij(s)]∣∣∣≤∑Ckl∈Nr(i,j)Cklij[∣∣∣f(ϕkl(s− τ(s)))ϕij(s)− f(ϕkl(s− τ(s)))ψij(s)∣∣∣+∣∣∣f(ϕkl(s− τ(s)))ψij(s)− f(ψkl(s− σ(s)))ψij(s)∣∣∣]=∑Ckl∈Nr(i,j)Cklij[∣∣∣f(ϕkl(s− τ(s)))∣∣∣ · ∣∣∣ϕij(s)− ψij(s)∣∣∣EJQTDE, 2011 No. 30, p. 7+∣∣∣f(ϕkl(s− τ(s)))− f(ψkl(s− σ(s)))∣∣∣ · ∣∣∣ψij(s)∣∣∣]≤∑Ckl∈Nr(i,j)Cklij[∣∣∣f(ϕkl(s− τ(s)))∣∣∣ · ∣∣∣ϕij(s)− ψij(s)∣∣∣+L ·∣∣∣ϕkl(s− τ(s))− ψkl(s− σ(s))∣∣∣ · ∣∣∣ψij(s)∣∣∣]≤∑Ckl∈Nr(i,j)Cklij[(∣∣∣f(ϕkl(s− τ(s)))− f(0)∣∣∣+ ∣∣∣f(0)∣∣∣) ·∣∣∣ϕij(s)− ψij(s)∣∣∣+L ·∣∣∣ϕkl(s− τ(s))− ψkl(s− σ(s))∣∣∣ · ∣∣∣ψij(s)∣∣∣]≤∑Ckl∈Nr(i,j)Cklij[(2LR + |f(0)|)·∣∣∣ϕij(s)− ψij(s)∣∣∣+2LR ·∣∣∣ϕkl(s− τ(s))− ψkl(s− σ(s))∣∣∣]≤∑Ckl∈Nr(i,j)Cklij(4LR + |f(0)|)· ‖ϕ− ψ‖Similarly,|I2(s)|=∣∣∣ ∑Ckl∈Ns(i,j)Dklij (s)[g(ϕ′kl(s− τ(s)))ϕij(s)− g(ψ′kl(s− σ(s)))ψij(s)]∣∣∣≤∑Ckl∈Ns(i,j)Dklij(4lR+ |g(0)|)· ‖ϕ− ψ‖‖Tϕ− Tψ‖0= supt∈Rmaxi,j{∣∣∣t∫−∞e−R tsaij(u)du ·[−∑Ckl∈Nr(i,j)Cklij (s)([f(ϕkl(s− τ(s)))ϕij(s)− f(ψkl(s− σ(s)))ψij(s))−∑Ckl∈Ns(i,j)Dklij (s)(g(ϕ′kl(s− σ(s)))ϕij(s)− g(ψ′kl(s− σ(s)))ψij(s))]ds∣∣∣}≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)du(|I1(s)|+ |I2(s)|)ds}≤ supt∈Rmaxi,j{ t∫−∞e−R tsaij(u)duds[ ∑Ckl∈Nr(i,j)Cklij(4LR + |f(0)|)EJQTDE, 2011 No. 30, p. 8+∑Ckl∈Ns(i,j)Dklij(4lR+ |g(0)|)]}· ‖ϕ− ψ‖≤ supt∈Rmaxi,j{ t∫−∞e−aij∗(t−s)ds}· θ2 · ‖ϕ− ψ‖≤ θ2 ·maxi,j{1aij∗} · ‖ϕ− ψ‖ (2.4)And‖(Tϕ− Tψ)′‖0= supt∈Rmaxi,j{∣∣∣∣∫ t−∞−aij(t)e−R tsaij (u)du ·{[−∑Ckl∈Nr(i,j)Cklij (s)f(ϕkl(s− τ(s)))−∑Ckl∈Ns(i,j)Dklij (s)g(ϕ′kl(s− σ(s)))]ϕij(s)+[ ∑Ckl∈Nr(i,j)Cklij (s)f(ψkl(s− τ(s)))+∑Ckl∈Ns(i,j)Dklij (s)g(ψ′kl(s− σ(s)))]ψij(s)}ds+{[−∑Ckl∈Nr(i,j)Cklij (t)f(ϕkl(t− τ(t)))−∑Ckl∈Ns(i,j)Dklij (t)g(ϕ′kl(t− σ(t)))]ϕij(t)+[ ∑Ckl∈Nr(i,j)Cklij (t)f(ψkl(t− τ(t)))+∑Ckl∈Ns(i,j)Dklij (t)g(ψ′kl(t− σ(t)))]ψij(t)}∣∣∣∣}≤ supt∈Rmaxi,j{∫ t−∞|aij(t)|e−R tsaij(u)du(|I1(s)|+ |I2(s)|)ds+(|I1(t)|+ |I2(t)|)}≤ supt∈Rmaxi,j{∫ t−∞|aij(t)|e−R tsaij(u)duds+ 1}· θ2 · ‖ϕ− ψ‖≤ supt∈Rmaxi,j{∫ t−∞a+ije−aij∗(t−s)ds+ 1}· θ2 · ‖ϕ− ψ‖EJQTDE, 2011 No. 30, p. 9≤ θ2 ·maxi,j{1 +a+ijaij∗}· ‖ϕ− ψ‖ (2.5)From (2.4) and (2.5), we have ‖Tϕ− Tψ‖ ≤ θ2 ·maxi,j{1aij∗, 1 +a+ijaij∗}· ‖ϕ− ψ‖ =θ · ‖ϕ − ψ‖. According to the condition of this theorem we know θ < 1, therefore Thas a unique fixed point.Example We consider the following SICCN with neutral a delay:x′ij = −aij(t)xij −∑Ckl∈Nr(ij)Cklij (t)f [xkl(t− τ(t))]xij(t)+∑Ckl∈Ns(i,j)Dklij (t)g[x′kl(t− σ(t))]xij(t) + Iij(t), (2.6)where i = 1, 2, 3, j = 1, 2, 3, τ(t) = cos2 t, σ(t) = sin 2t, f(x) = 45sin x, g(x) = 34|x|,a11(t) a12(t) a13(t)a21(t) a22(t) a23(t)a31(t) a32(t) a33(t) =5 + | sin t| 5 + | sin 2t| 9 + | sin t|6 + | cos t| 6 + | sin t| 7 + | cos t|8 + | cos t| 8 + | sin t| 5 + | sin 2t|c11(t) c12(t) c13(t)c21(t) c22(t) c23(t)c31(t) c32(t) c33(t) =0.004| sin 3t| 0.002| sin 3t| 0.001| sin 3t|0.002| sin 3t| 0.001| sin 3t| 0.001| sin 3t|0.001| sin 3t| 0.002| sin 3t| 0.001| sin 3t|d11(t) d12(t) d13(t)d21(t) d22(t) d23(t)d31(t) d32(t) d33(t) =0.001| cos 2t| 0.001| cos 2t| 0.002| cos 2t|0.001| cos 2t| 0.002| cos 2t| 0.003| cos 2t|0.002| cos 2t| 0.002| cos 2t| 0.001| cos 2t|I11(t) I12(t) I13(t)I21(t) I22(t) I23(t)I31(t) I32(t) I33(t) =sin t sin t cos tsin t cos t cos tcos t cos t cos tIn the following, we will check that all assumptions of the theorem are satisfied. Bycomputing,we havea+11 a+12 a+13a+21 a+22 a+23a+31 a+32 a+33 =6 6 107 7 89 9 6 ,a11∗ a12∗ a13∗a21∗ a22∗ a23∗a31∗ a32∗ a33∗ =5 5 96 6 78 8 5EJQTDE, 2011 No. 30, p. 10c11 c12 c13c21 c22 c23c31 c32 c33 =0.004 0.002 0.0010.002 0.001 0.0010.001 0.002 0.001 ,d11 d12 d13d21 d22 d23d31 d32 d33 =0.001 0.001 0.0020.001 0.002 0.0030.002 0.002 0.001 .Note that f and g are Lipschtz functions with f(0) = g(0) = 0, the Lipschtzconstants of f and g, L, l, are less than 1, we take L = l = 1.∑ckl∈N1(1,1)ckl +∑ckl∈N1(1,1)dkl = 0.014,∑ckl∈N1(1,2)ckl +∑ckl∈N1(1,2)dkl = 0.021,∑ckl∈N1(1,3)ckl +∑ckl∈N1(1,3)dkl = 0.013,∑ckl∈N1(2,1)ckl +∑ckl∈N1(2,1)dkl = 0.021,∑ckl∈N1(2,2)ckl +∑ckl∈N1(2,2)dkl = 0.030,∑ckl∈N1(2,3)ckl +∑ckl∈N1(2,3)dkl = 0.019,∑ckl∈N1(3,1)ckl +∑ckl∈N1(3,1)dkl = 0.013,∑ckl∈N1(3,2)ckl +∑ckl∈N1(3,2)dkl = 0.019,∑ckl∈N1(3,3)ckl +∑ckl∈N1(3,3)dkl = 0.013,From computing we know ‖ϕ0‖ ≤115, we choose R = 3. Obviously maxi,j{1aij∗, 1 +a+ijaij∗}= 115. θ1 = 0.18, θ2 = 0.36, θ1 · maxi,j{ 1aij∗, 1 +a+ijaij∗} = 0.396 ≤ 12, θ = θ2 ·maxi,j{1aij∗, 1 +a+ijaij∗}= 0.792 < 1. So all the conditions of theorem 2.1 are satisfied,hence by the theorem 2.1, Eq.(1.1) has a unique almost periodic solution.AcknowledgementsThe authors would like to thank the referees for their valuable suggestions andcomments.The first author was supported by Fund for Young Teachers of Jiangnan University(2008LQN010) and Program for Innovative Research Team of Jiangnan University.The second author was supported by the National Natural Science Foundation ofChina under Grant 60875036, the Key Research Foundation of Science and Technologyof the Ministry of Education of China under Grant 108067, and Program for InnovativeResearch Team of Jiangnan University.EJQTDE, 2011 No. 30, p. 11References[1] H. S. Ding, J. Liang, T. J. Xiao, Existence of almost periodic solutions for SICNNswith time-varying delays, Physics Letters A, 372 (2008) 5411-5416.[2] Y. Xia , J. Cao, Zhenkun Huang, Existence and exponential stability of almostperiodic solution for shunting inhibitory cellular neural networks with impulses,Chaos, Solitons and Fractals, 34 (2007) 1599-1607.[3] L. Wang, Existence and global attractivity of almost periodic solutions for delayedhigh-ordered neural networks, Neurocomputing, 73 (2010) 802-808.[4] L. Chen, H. Zhao, Global stability of almost periodic solution of shunting in-hibitory cellular neural networks with variable coefficients, Chaos, Solitons andFractals, 35 (2008) 351-357.[5] C. Ou, Almost periodic solutions for shunting inhibitory cellular neural networks,Nonlinear Analysis: Real World Applications, 10 (2009) 2652-2658.[6] J. Hale, Theory of functional differential equations, Springer-Verlag, New York,1977.[7] B. Xiao, Existence and uniqueness of almost periodic solutions for a class of Hop-field neural networks with neutral delays, Applied Mathematics Letters, 22 (2009)528-533.[8] C. Bai, Existence and stability of almost periodic solutions of Hopfield neuralnetworks with continuously distributed delays, Nonlinear Analysis, 71 (2009) 5850-5859.[9] C. Y. He, Almost periodic differential equation, Higher Education PublishingHouse, Beijing, 1992 (in Chinese).[10] A. M. Fink, Almost periodic differential equation, in: Lecture Notes of Mathe-matics, Vol. 377, Springer, Berlin, 1974.(Received July 1, 2010)EJQTDE, 2011 No. 30, p. 12

Existence of almost periodic solution for SICNN with a neutral delay

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