9_314_Wu.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 3, 2009, (448-461) ISSN 1307-5543 – www.ejpam.com Exponential Stability of Almost Periodic Solution for Shunting Inhibitory Cellular Neural Networks with Time-Varying and Distributed Delays Ailong Wu∗ and Chaojin Fu College of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China Abstract. In this paper, shunting inhibitory cellular neural networks (SICNNs) with time- varying and distributed delays are considered. Without assuming the global Lipschitz condi- tions of activation functions, some new sufficient conditions for the existence and exponential stability of the almost periodic solutions are established. Finally, a numerical example is given to demonstrate the effectiveness of the obtained result. 2000 Mathematics Subject Classifications: 92B20; 93D05 Key Words and Phrases: Almost periodic, Shunting inhibitory cellular neural networks, Ex- ponential stability. 1. Introduction Recently, the dynamical behaviors of almost periodic solutions for shunting in- hibitory cellular neural networks (SICNNs) have been extensively studied (see [1− ∗Corresponding author. Email addresses: alwu83�gmail. om( A. Wu), haojinfu�126. om (C. Fu) http://www.ejpam.com 448 c© 2009 EJPAM All rights reserved. A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 449 11]), due to SICNNs have been extensively applied in psychophysics, speech, per- ception, robotics, adaptive pattern recognition, vision, and image processing. Many important results have been established and successfully applied to signal process- ing, pattern recognition, associative memories, and so on. However, in the existing literatures (see [1− 3, 5− 9]), almost all results on the stability of almost periodic solutions for SICNNs are obtained under global Lipschitz neuron activations. When neuron activation functions do not satisfy global Lipschitz conditions, people want to know whether the SICNNs is stable. In practical engineering applications, people also need to present new neural networks. Therefore, developing a new class of SICNNs without global Lipschitz neuron activation functions and giving the conditions of the stability of new SICNNs are very interesting and valuable. Consider the following SICNNs with time-varying and distributed delays: x ′ i j (t) =− ai j(t)x i j(t)− ∑ Ckl∈Nr (i, j) C kl i j (t) f (xkl(t −τ(t)))x i j(t) − ∑ Ckl∈Nq(i, j) Bkl i j (t) ∫ ∞ 0 Ki j(u)g(xkl(t − u))dux i j(t) + Li j(t), (1.1) where i = 1, · · · , m, j = 1, · · · , n, Ci j is the cell at the (i, j) position of the lattice, the r-neighborhood Nr(i, j) of Ci j is Nr(i, j) = ¦ Ckl : max(|k− i| , � �l − j � �)≤ r, 1≤ k ≤ m, 1≤ l ≤ n © , Nq(i, j) is similarly specified. x i j is the activity of the cell Ci j, Li j(t) is the external input to Ci j, ai j(t) > 0 is the passive decay rate of the cell activity, C kl i j (t) ≥ 0 and Bkl i j (t) ≥ 0 are the connections or coupling strengths of postsynaptic activity of the cells in Nr(i, j) and Nq(i, j) transmitted to the cell Ci j, respectively. The activity func- tions f (xkl) and g(xkl) are continuous functions representing the output or firing rate of cell Ckl, and τ(t)≥ 0 is the transmission delay. Throughout this paper, we will assume that τ(t) : R → R is an almost periodic function, and 0≤ τ(t)≤ τ, where τ≥ 0 is a constant. A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 450 Set ¦ x i j(t) © = (x11(t), · · · , x1n(t), · · · , xm1(t), · · · , xmn(t)), for ∀x = ¦ x i j(t) © ∈ Rm×n, we define the norm ‖x‖=max (i, j) ¦� �x i j(t) � � © . Set B = ¦ ϕ | ϕ = ¦ ϕi j(t) © = (ϕ11(t), · · · ,ϕ1n(t), · · · ,ϕm1(t), · · · ,ϕmn(t)) © , where ϕ is is an almost periodic function on R. For ∀ϕ ∈ B, we define the norm ϕ B = sup t∈R ϕ(t) , then B is a Banach space. The initial conditions associated with system (1.1) are of the form x i j(s) = ϕi j(s), s ∈ (−∞, 0], i = 1, · · · , m, j = 1, · · · , n, (1.1) where ϕ = ¦ ϕi j(t) © ∈ C((−∞, 0], Rm×n). Definition 1.1. Let k ∈ Z+. A continuous function u : R→ Rk is called almost periodic if for each ǫ > 0 there exists a constant l(ǫ) > 0 such that every interval of length l(ǫ) contains a number δ with the property that ‖u(t +δ)− u(t)‖< ǫ for all t ∈ R. Definition 1.2. Let x ∈ Rn and Q(t) be a n× n continuous matrix defined on R. The linear system x ′(t) = Q(t)x(t) (1.3) is 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 (1.3) satisfying X (t)PX−1(s) ≤ ke−α(t−s) for t ≥ s, X (t)(I − P)X−1(s) ≤ ke−α(s−t) for t ≤ s. A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 451 Lemma 1.1. [12]. If the linear system (1.3) admits an exponential dichotomy, then almost periodic system x ′(t) = Q(t)x(t)+ g(t) (1.4) has a unique almost periodic solution x(t), and x(t) = ∫ t −∞ X (t)PX−1(s)g(s)ds − ∫ +∞ t X (t)(I − P)X−1(s)g(s)ds. Lemma 1.2. [12]. Let ci(t) be an almost periodic function on R and M[ci] = lim T→+∞ 1 T ∫ t+T t ci(s)ds > 0, i = 1, · · · , n. Then the linear system x ′(t) = diag(−c1(t), · · · ,−cn(t))x(t) admits an exponential dichotomy on R. 2. Existence of Almost Periodic Solutions Theorem 2.1. Assume that (H1) For i = 1, · · · , m, j = 1, · · · , n, the delay kernels Ki j : [0,∞)→ R are continuous and integrable, ai j, C kl i j , Bkl i j , Li j ∈ B; (H2) there exists a continuous function L : R+→ R+ such that for each r > 0, � � f (u)− f (v) � �≤ L(r) |u− v| , |u| , |v| ≤ r; � �g(u)− g(v) � �≤ L(r) |u− v| , |u| , |v| ≤ r. (H3) there exists a constant r0 > 0 such that D[F(0)r0+ L(r0)r 2 0 ] + L ≤ r0, DF(0) + 2DL(r0)r0 < 1, A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 452 where D =max (i, j) { ∑ Ckl∈Nr (i, j) C kl i j + ∑ Ckl∈Nq(i, j) B kl i j ∫∞ 0 � �Ki j(u) � �du a i j }> 0, F(0) = max ¦� � f (0) � � , � �g(0) � � © , L = max (i, j) L i j a i j , L i j = sup t∈R � �Li j(t) � �, C kl i j = sup t∈R C kl i j (t), B kl i j = sup t∈R Bkl i j (t), a i j = inf t∈R ai j(t)> 0. Then SICNNs (1.1) has a unique almost periodic solution in the region E := ¦ ϕ ∈ B : ϕ B ≤ r0 © . Proof. For any given ϕ ∈ B, we consider the following almost periodic differential equation: x ′ i j (t) =− ai j(t)x i j(t)− ∑ Ckl∈Nr (i, j) C kl i j (t) f (ϕkl(t −τ(t)))ϕi j(t) − ∑ Ckl∈Nq(i, j) Bkl i j (t) ∫ ∞ 0 Ki j(u)g(ϕkl(t − u))duϕi j(t) + Li j(t). (2.1) Then, notice that M[ai j] > 0, from Lemma 1.2, the linear system x ′ i j (t) = −ai j(t)x i j(t), i = 1, · · · , m, j = 1, · · · , n, (2.2) admits an exponential dichotomy on R. Thus, by Lemma 1.1, we obtain that the system (2.1) has exactly one almost periodic solution: xϕ(t) = ∫ t −∞ e− ∫ t s ai j(u)du � − ∑ Ckl∈Nr (i, j) C kl i j (s) f (ϕkl(s−τ(s)))ϕi j(s) − ∑ Ckl∈Nq(i, j) Bkl i j (s) ∫ ∞ 0 Ki j(u)g(ϕkl(s− u))duϕi j(s) + Li j(s) � ds. Now, we define a nonlinear operator on B by T (ϕ)(t) = xϕ(t),∀ϕ ∈ B. Next, we will prove T (E)⊂ E. For any given ϕ ∈ E, it suffices to prove that T (ϕ) B ≤ r0. By (H2) and (H3), we have T (ϕ) B = sup t∈R max (i, j) {| ∫ t −∞ e− ∫ t s ai j(u)du � − ∑ Ckl∈Nr (i, j) C kl i j (s) f (ϕkl(s− τ(s)))ϕi j(s) A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 453 − ∑ Ckl∈Nq(i, j) Bkl i j (s) ∫ ∞ 0 Ki j(u)g(ϕkl(s− u))duϕi j(s) + Li j(s) � ds |} ≤ sup t∈R max (i, j) {| ∫ t −∞ e−ai j(t−s) � ∑ Ckl∈Nr (i, j) C kl i j f (ϕkl(s−τ(s)))ϕi j(s)ds + ∑ Ckl∈Nq(i, j) B kl i j ∫ ∞ 0 Ki j(u)g(ϕkl(s− u))duϕi j(s)ds � |}+max (i, j) L i j a i j ≤ sup t∈R max (i, j) {| ∫ t −∞ e−ai j(t−s) � ∑ Ckl∈Nr (i, j) C kl i j ( � � f (0) � �+ L(r0) | ϕkl(s −τ(s)) |) � �ϕi j(s) � �ds+ ∑ Ckl∈Nq(i, j) B kl i j ( � �g(0) � �+ L(r0) � �ϕkl(s−τ(s)) � �)· ∫ ∞ 0 � �Ki j(u) � �du � �ϕi j(s) � �ds � |}+ L ≤ sup t∈R max (i, j) {| ∫ t −∞ e−ai j(t−s) � ∑ Ckl∈Nr (i, j) C kl i j ( � � f (0) � �+ L(r0)r0)r0ds + ∑ Ckl∈Nq(i, j) B kl i j ( � �g(0) � �+ L(r0)r0)r0 ∫ ∞ 0 � �Ki j(u) � �duds � |}+ L ≤ D[F(0)r0+ L(r0)r 2 0 ] + L ≤ r0. Therefore, T (E)⊂ E. Taking ϕ,ψ ∈ E, combining (H2) and (H3), we deduce that T (ϕ)− T (ψ) B = sup t∈R T (ϕ)(t)− T (ψ)(t) = sup t∈R max (i, j) {| ∫ t −∞ e− ∫ t s ai j(u)du ∑ Ckl∈Nr (i, j) −C kl i j (s) � f (ϕkl(s−τ(s)))ϕi j(s) − f (ψkl(s−τ(s)))ψi j(s) � ds |+ | ∫ t −∞ e− ∫ t s ai j(u)du ∑ Ckl∈Nq(i, j) −Bkl i j (s)· ∫ ∞ 0 Ki j(u) � g(ϕkl(s− u))duϕi j(s)− g(ψkl(s− u))duψi j(s) � ds |} ≤ sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nr (i, j) C kl i j | f (ϕkl(s− τ(s))) || ϕi j(s)−ψi j(s) | ds} A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 454 + sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nr(i, j) C kl i j | f (ϕkl(s−τ(s)))− f (ψkl(s −τ(s))) | · |ψi j(s) | ds}+ sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nq(i, j) B kl i j · ∫ ∞ 0 | Ki j(u)g(ϕkl(s− u))du | · | ϕi j(s)−ψi j(s) | ds} + sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nq(i, j) B kl i j ∫ ∞ 0 | Ki j(u) � g(ϕkl(s− u)) − g(ψkl(s− u)) � du | · |ψi j(s) | ds} ≤ sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nr (i, j) C kl i j ( � � f (0) � �+ L(r0)r0)ds} · ϕ−ψ B + sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nr(i, j) C kl i j L(r0)r0ds} · ϕ−ψ B + sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nq(i, j) B kl i j ( � �g(0) � �+ L(r0)r0) ∫ ∞ 0 � �Ki j(u) � �duds} · ϕ−ψ B + sup t∈R max (i, j) { ∫ t −∞ e−ai j(t−s) ∑ Ckl∈Nq(i, j) B kl i j L(r0)r0 ∫ ∞ 0 � �Ki j(u) � �duds} · ϕ−ψ B ≤ D(F(0) + L(r0)r0) · ϕ−ψ B + DL(r0)r0 · ϕ−ψ B ≤ [DF(0) + 2DL(r0)r0] · ϕ−ψ B < ϕ−ψ B So T is a contraction from E to E. Since E is a closed subset of B, T has a unique fixed point in E, which means system (1.1) has a unique almost periodic solution in E. 3. Exponential Stability of the Almost Periodic Solution Theorem 3.1. Suppose (H1) − (H3) hold, let x∗(t) = n x∗ i j (t) o be the unique almost periodic solution of SICNNs (1.1) in the region ϕ B ≤ r0. Further we assume that A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 455 (H4) there exists a constant r1 ≥ r0 such that F(0) + L(r0)r0+ L(r1)r1 < 1 D , where F(0) =max ¦� � f (0) � � , � �g(0) � � © ; (H5) For i = 1, · · · , m, j = 1, · · · , n, there exists a constant λ0 > 0 such that ∫ ∞ 0 � �Ki j(s) � � eλ0sds <+∞. Then there exists a constant λ > 0 such that for any solution x(t) = ¦ x i j(t) © of SICNNs (1.1) with initial value sup t∈(−∞,0] ϕ(t) ≤ r1, ‖x(t)− x∗(t)‖ ≤ Me−λt , ∀t > 0, where M = sup t∈(−∞,0] ϕ(t)− x∗(t) . Proof. Set Γi j(α) =α− a i j + ∑ Ckl∈Nr (i, j) C kl i j ( � � f (0) � �+ L(r0)r0+ L(r1)r1eατ) + ∑ Ckl∈Nq(i, j) B kl i j · � ( � �g(0) � �+ L(r0)r0) ∫ ∞ 0 � �Ki j(s) � �ds+ L(r1)r1 ∫ ∞ 0 � �Ki j(s) � � eαsds � where i = 1, · · · , m, j = 1, · · · , n. It is easy to prove that Γi j are continuous functions on [0,λ0]. Moreover, by (H4) and (H5), we have Γi j(0) =−a i j + ∑ Ckl∈Nr (i, j) C kl i j � � � f (0) � �+ L(r0)r0+ L(r1)r1 � + ∑ Ckl∈Nq(i, j) B kl i j � � �g(0) � �+ L(r0)r0+ L(r1)r1 � ∫ ∞ 0 � �Ki j(s) � �ds ≤−a i j + ∑ Ckl∈Nr (i, j) C kl i j � F(0) + L(r0)r0+ L(r1)r1 � + ∑ Ckl∈Nq(i, j) B kl i j � F(0) + L(r0)r0 + L(r1)r1 � ∫ ∞ 0 � �Ki j(s) � �ds < 0. A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 456 Thus, there exists a sufficiently small constant λ > 0 such that Γi j(λ)< 0, i = 1, · · · , m, j = 1, · · · , n. (3.1) Take ǫ > 0. Set Zi j(t) = � � �x i j(t)− x∗ i j (t) � � � eλt , i = 1, · · · , m, j = 1, · · · , n. It follows that: Zi j(t) ≤ M < M + ǫ,∀ t ∈ (−∞, 0], i = 1, · · · , m, j = 1, · · · , n. In the following, we will prove that Zi j(t)≤ M + ǫ,∀ t > 0, i = 1, · · · , m, j = 1, · · · , n. (3.2) If this is not true, then there exist i0 ∈ {1, · · · , m} and j0 ∈ {1, · · · , n} such that ¦ t > 0 | Zi0 j0 (t)> M + ǫ © 6= ;. (3.3) Let t i j =    inf ¦ t > 0 | Zi j(t)> M + ǫ © , ¦ t > 0 | Zi j(t)> M + ǫ © 6= ;, +∞, ¦ t > 0 | Zi j(t)> M + ǫ © = ;. Then t i j > 0 and Zi j(t)≤ M + ǫ,∀ t ∈ (−∞, t i j], i = 1, · · · , m, j = 1, · · · , n. (3.4) We denote tph = min (i, j) t i j, where p ∈ {1, · · · , m} and h ∈ {1, · · · , n}. From (3.3), we have 0 < tph < +∞. It follows from (3.4), we have Zi j(t)≤ M + ǫ,∀ t ∈ (−∞, tph], i = 1, · · · , m, j = 1, · · · , n. (3.5) In addition, noticing that tph = inf ¦ t > 0 | Zph(t)> M + ǫ © , we obtain Zph(tph) = M + ǫ, and D+Zph(tph)≥ 0. (3.6) Since x(t) and x∗(t) are solutions of Eq.(1.1), combining with (3.5)-(3.6), (H2) and (H3), we have 0≤ D+Zph(tph) = D+[ � � �xph(t)− x∗ ph (t) � � � eλt] |t=tph A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 457 ≤ � � �xph(tph)− x∗ ph (tph) � � �λeλtph − a ph � � �xph(tph)− x∗ ph (tph) � � � eλtph + ∑ Ckl∈Nr (p,h) C kl ph | f (xkl(tph−τ(tph)))xph(tph)− f (x∗ kl (tph−τ(tph)))x ∗ ph (tph) | · eλtph + ∑ Ckl∈Nq(p,h) B kl ph | ∫ ∞ 0 Ki j(u)g(xkl(tph− u))duxph(tph) − ∫ ∞ 0 Ki j(u)g(x ∗ kl (tph− u))dux∗ ph (tph) | eλtph ≤ (λ− a ph )Zph(tph) + ∑ Ckl∈Nr (p,h) C kl ph | f (x∗ kl (tph−τ(tph))) | · | xph(tph) − x∗ ph (tph) | eλtph + ∑ Ckl∈Nr (p,h) C kl ph | f (xkl(tph−τ(tph))) − f (x∗ kl (tph−τ(tph))) | · | xph(tph) | eλtph + ∑ Ckl∈Nq(p,h) B kl ph ∫ ∞ 0 � �Ki j(u) � � · � �g(x∗ kl (tph− u)) � �du· | xph(tph)− x∗ ph (tph) | eλtph + ∑ Ckl∈Nq(p,h) B kl ph · ∫ ∞ 0 � �Ki j(u) � � · | g(xkl(tph− u))− g(x∗ kl (tph− u)) | du· | xph(tph) | eλtph ≤ (λ− a ph )(M + ǫ) + ∑ Ckl∈Nr(p,h) C kl ph ( � � f (0) � �+ L(r0)r0) · Zph(tph) + ∑ Ckl∈Nr (p,h) C kl ph L(r1) � �xkl(tph−τ(tph))− x∗ kl (tph−τ(tph)) � � · eλ(tph−τ(tph))eλτ(tph) · r1+ ∑ Ckl∈Nq(p,h) B kl ph ( � �g(0) � �+ L(r0)r0)· ∫ ∞ 0 � �Ki j(u) � �du · Zph(tph) + ∑ Ckl∈Nq(p,h) B kl ph L(r1) ∫ ∞ 0 � �Ki j(u) � � · � �xkl(tph− u)− x∗ kl (tph− u) � � eλ(tph−u)eλudu · r1 ≤ (λ− a ph )(M + ǫ) + ∑ Ckl∈Nr(p,h) C kl ph ( � � f (0) � �+ L(r0)r0) · (M + ǫ) + ∑ Ckl∈Nr (p,h) C kl ph L(r1)r1eλτ · (M + ǫ) + ∑ Ckl∈Nq(p,h) B kl ph � � �g(0) � � A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 458 + L(r0)r0 � · ∫ ∞ 0 � �Ki j(u) � �du · (M + ǫ) + ∑ Ckl∈Nq(p,h) B kl ph L(r1)r1 ∫ ∞ 0 � �Ki j(u) � � eλudu · (M + ǫ) ≤ (λ− a ph )(M + ǫ) + ∑ Ckl∈Nr(p,h) C kl ph (F(0) + L(r0)r0) · (M + ǫ) + ∑ Ckl∈Nr (p,h) C kl ph L(r1)r1eλτ · (M + ǫ) + ∑ Ckl∈Nq(p,h) B kl ph � F(0) + L(r0)r0 � · ∫ ∞ 0 � �Ki j(u) � �du · (M + ǫ) + ∑ Ckl∈Nq(p,h) B kl ph L(r1)r1 ∫ ∞ 0 � �Ki j(u) � � eλudu · (M + ǫ) It follows that: λ− a ph + ∑ Ckl∈Nr(p,h) C kl ph (F(0) + L(r0)r0+ L(r1)r1eλτ) + ∑ Ckl∈Nq(p,h) B kl ph · � (F(0) + L(r0)r0) ∫ ∞ 0 � �Ki j(u) � �du+ L(r1)r1 ∫ ∞ 0 � �Ki j(u) � � eλudu � ≥ 0, that is Γph(λ)≥ 0. This contradicts with (3.1). Hence, (3.2) holds, i.e., � � �x i j(t)− x∗ i j (t) � � � eλt = Zi j(t)≤ M + ǫ,∀ t > 0, i = 1, · · · , m, j = 1, · · · , n. Therefore, ‖x(t)− x∗(t)‖=max (i, j) � � �x i j(t)− x∗ i j (t) � � �≤ (M + ǫ)e−λt ,∀ t > 0. Let ǫ→ 0, we get ‖x(t)− x∗(t)‖ ≤ Me−λt ,∀ t > 0. A. Wu and C. Fu / Eur. J. Pure Appl. Math, 2 (2009), (448-461) 459 4. Illustrative Example Consider SICNNs (1.1) described by i, j = 1, 2, 3, τ(t) = cos2 t , f (x) = g(x) = x4+1 6 , Ki j(u) = e−u sin u, ai j(t) =       5+ |sin t | 5+ � �sin p 2t � � 9+ |sin t | 6+ |sin t | 6+ |sin t | 7+ |sin t | 8+ |sin t | 8+ |sin t | 5+ � �sin p 3t � �       , Ci j(t) = Bi j(t) = � �sin p 3t � �       1 10 3 10 1 2 1 5 1 10 1 5 1 10 1 5 1 10       , Li j(t) =       sin t sin t cos t sin t+sin p 2t 2 cos t cos t cos t cos t+cos p 3t 2 sin t       . Obviously , let L(r) = 2 3 r3 and r0 = 1, then we get D ≤ 0.6, L = 0.2, so D[F(0)r0+ L(r0)r 2 0 ] + L ≤ 0.7 < 1 = r0, DF(0) + 2DL(r0)r0 ≤ 0.81 < 1. From Theorem 2.1, the system in example has a unique almost periodic solution in the region ϕ B ≤ 1. Take r1 = 4 Æ 51 50 , then D[F(0) + L(r0)r0+ L(r1)r1] < 1. From Theorem 3.1, all the solutions with initial value sup t∈[−1,0] ϕ(t) ≤ r1 converge exponentially to the unique almost periodic solution in the region ϕ B ≤ 1 as t → +∞. 5. Conclusion In this paper, some new sufficient conditions are established to ensure the exis- tence and exponential stability of almost periodic solutions for SICNNs with time- varying and distributed delays. Since we do not need the neuron activations to satisfy global Lipschitz conditions, the result in this paper is new, and it is also valuable in the design of neural networks which is used to solve efficiently problems arising in practical engineering applications. REFERENCES 460 ACKNOWLEDGEMENTS The work is supported by Key Science Foundation of Ed- ucational Department of Hubei Province under Grant D20082201, Innovation Teams of Hubei Normal University and Innovation Scientific Research Foundation of Hubei Normal University. References [1] B. Liu and L. 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