Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 132790, 19 pages doi:10.1155/2010/132790 Research Article Existence and Stability of Antiperiodic Solution for a Class of Generalized Neural Networks with Impulses and Arbitrary Delays on Time Scales Yongkun Li, Erliang Xu, and Tianwei Zhang Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China Correspondence should be addressed to Tianwei Zhang, 1200801347@stu.ynu.edu.cn Received 14 June 2010; Accepted 16 August 2010 Academic Editor: Kok Lay Teo Copyright q 2010 Yongkun Li et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited By using coincidence degree theory and Lyapunov functions, we study the existence and global exponential stability of antiperiodic solutions for a class of generalized neural networks with impulses and arbitrary delays on time scales Some completely new sufficient conditions are established Finally, an example is given to illustrate our results These results are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses Introduction In this paper, we consider the following generalized neural networks with impulses and arbitrary delays on time scales: xΔ t Δx tk A t, x t B t, x t x tk − x t− k F t, xt , I k x tk , t t ∈ T, t / tk , tk , k ∈ N, 1.1 where T is an ω/2-periodic time scale and if t ∈ T, θ ∈ E, then t θ ∈ T, E is a subset −∞, , A t, x t diag a1 t, x1 t , a2 t, x2 t , , an t, xn t , B t, x t of R− f1 t, xt , , fn t, xt T , fi t, xt b1 t, x1 t , b2 t, x2 t , , bn t, xn t T , F t, xt xi t θ , t ∈ T, θ ∈ E, i 1, 2, , n, and x tk , x t− fi t, x1t , x2t , , xnt , xit θ k represent the right and left limits of x tk in the sense of time scales, {tl } is a sequence of real numbers such that < t1 < t2 < · · · < tn → ∞ as l → ∞ There exists a positive integer q such tl ω/2, Ik q u −Ik −u , l ∈ Z, u ∈ R Without loss of generality, we also that tl q Journal of Inequalities and Applications assume that 0, ω/2 T ∩ {tl : l ∈ N} {t1 , t2 , , tq } For each interval I of R, we denote that T ∩ 0, ∞ IT I ∩ T, especially, we denote that T System 1.1 includes many neural continuous and discrete time networks 1–9 For examples, the high-order Hopfield neural networks with impulses and delays see : ⎡ xi t −ai xi t ⎣bi xi t n − aij t gj xj t − j − n n n bij t gj xj t − τj t j 1.2 ⎤ bijl t gj xj t − τj t gl xl t − τl t Ii t ⎦, t / tk , j l Δxi tk xi tk − xi t− k eik xi tk , i 1, 2, , n, k 1, 2, , 1.3 the Cohen-Grossberg neural networks with bounded and unbounded delays see : ⎡ xi t −ai xi t ⎣bi xi t n − cij t fj xj t j − dij t hj xi tk − xi t− k cij t gj xj t − τij t 1.4 ⎤ Kij u xj t − u du Ii t ⎦, t / tk , j Δxi tk n j ∞ n − lik xi tk , i 1, 2, , n, k 1, 2, , 1.5 and so on Arising from problems in applied sciences, it is well known that anti-periodic problems of nonlinear differential equations have been extensively studied by many authors during the past twenty years; see 10–21 and references cited therein For example, antiperiodic trigonometric polynomials are important in the study of interpolation problems 22, 23 , and anti-periodic wavelets are discussed in 24 Recently, several authors 25–30 have investigated the anti-periodic problems of neural networks without impulse by similar analytic skills However, to the best of our knowledge, there are few papers published on the existence of anti-periodic solutions to neural networks with impulse The main purpose of this paper is to study the existence and global exponential stability of anti-periodic solutions of system 1.1 by using the method of coincidence degree theory and Lyapunov functions The initial conditions associated with system 1.1 are of the form x0 φ, that is, xi θ φi θ , θ ∈ E, i 1, 2, , n 1.6 Throughout this paper, we assume that H1 t, u ∈ C T × R, R , t ω/2, −u t, u , and there exist positive constants am , aM such that < am < t, u < aM for all t ∈ T, u ∈ R, i 1, 2, , n; i i i i Journal of Inequalities and Applications H2 bi t, u ∈ C T × R, R , bi t and Lb such that i ∂bi t, u ≥ μi , ∂u for all t ∈ T, u, v ∈ R, i ω/2, −u −bi t, u There exist positive constants μi |bi t, u − bi t, v | ≤ Lb |u − v|, i 0, bi t, 1.7 1, 2, , n; H3 fi ∈ C T × Rn , R , fi t constants ci such that ω/2, −u −fi t, u , for i ≤ ci fi t, x1t , , xnt − fi t, y1t , , ynt 1, 2, , n There exist positive n xjt − yjt , 1.8 j for all t, x1t , , xnt , t, y1t , , ynt ∈ T × Rn and fi t, 0, , 0, i 1, 2, , n; H4 Iik ∈ C R, R and there exist positive constants LI such that ik |Iik u − Iik v | ≤ LI |u − v|, ik 1.9 for all u, v ∈ R, k ∈ N, i 1, 2, , n For convenience, we introduce the following notation: hM max |h t |, t∈ 0,ω hm T |h t |, t∈ 0,ω T ω h 1/2 |h t |2 Δt , 1.10 where h is an ω-periodic function The organization of this paper is as follows In Section 2, we introduce some definitions and lemmas In Section 3, by using the method of coincidence degree theory, we obtain the existence of the anti-periodic solutions of system 1.1 In Section 4, we give the criteria of global exponential stability of the anti-periodic solutions of system 1.1 In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections and The conclusions are drawn in Section Preliminaries In this section, we will first recall some basic definitions and lemmas which can be found in books 31, 32 Definition 2.1 see 31 A time scale T is an arbitrary nonempty closed subset of real numbers R The forward and backward jump operators σ, ρ : T → T and the graininess μ : T → R are defined, respectively, by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, μ t σ t − t 2.1 Journal of Inequalities and Applications Definition 2.2 see 31 A function f : T → R is called right-dense continuous provided it is continuous at right-dense point of T and left-side limit exists finite at left-dense point of Crd T T The set of all right-dense continuous functions on T will be denoted by Crd Crd T, R If f is continuous at each right-dense and left-dense point, then f is said to be a continuous function on T, the set of continuous function will be denoted by C T Definition 2.3 see 31 For x : T → R, one defines the delta derivative of x t , xΔ t to be the number if it exists with the property that for a given ε > 0, there exists a neighborhood U of t such that x σ t −x t − xΔ t σ t − s ≤ ε|σ t − s|, 2.2 for all s ∈ U Definition 2.4 see 31 If F Δ t f t , then one defines the delta integral by t f s Δs F t −F a 2.3 a Definition 2.5 see 33 For each t ∈ T, let N be a neighborhood of t Then, one defines the generalized derivative or dini derivative , D uΔ t to mean that, given ε > 0, there exists a right neighborhood N ε ⊂ N of t such that u σ t −u s < D uΔ t μ t, s ε, 2.4 for each s ∈ N ε , s > t, where μ t, s σ t − s In case t is right-scattered and u t is continuous at t, this reduces to D uΔ t u σ t −u t σ t −t 2.5 Similar to 34 , we will give the definition of anti-periodic function on a time scale as following Definition 2.6 Let T / R be a periodic time scale with period p One says that the function f : T → R is ω/2-anti-periodic if there exists a natural number n such that ω/2 np, f t ω/2 −f t for all t ∈ T and ω is the smallest number such that f t ω/2 −f t If T R, one says that f is ω/2-anti-periodic if ω/2 is the smallest positive number such that f t ω/2 −f t for all t ∈ T Definition 2.7 see 31 A function p : T → R is called regressive if μ t p t / for all σ t − t is the graininess function If p is regressive and right-dense t ∈ Tk , where μ t continuous function, then the generalized exponential function ep is defined by t ep t, s exp s ξμ τ p τ Δτ , 2.6 Journal of Inequalities and Applications for s, t ∈ T, with the cylinder transformation ⎧ ⎪ Log hz ⎨ , h ⎪ ⎩z, ξh z if h / 0, if h 2.7 Let p, q : T → R be two regressive functions, we define p⊕q : p q p: − μpq, p , μp p p⊕ q q 2.8 Then the generalized exponential function has the following properties Lemma 2.8 see 31, 32 Assume that p, q : T → R are two regressive functions, then i e0 t, s ≡ and ep t, t ≡ 1; ii ep σ t , s iii ep t, σ s e vii p t, s ; 1/ep s, t vi ep t, s ep s, r Δ ep ·, s μ s p s ; ep t, s / iv 1/ep t, s v ep t, s μ t p t ep t, s ; e s, t ; p ep t, r ; pep ·, s Lemma 2.9 see 31 Assume that f, g : T → R are delta differentiable at t ∈ Tk Then fg Δ t fΔ t g t f σ t gΔ t f t gΔ t fΔ t g σ t 2.9 The following lemmas can be found in 35, 36 , respectively Lemma 2.10 Let t1 , t2 ∈ 0, w T If x : T → R is ω-periodic, then x t ≤ x t1 ω xΔ s Δs, x t ≥ x t2 − ω xΔ s Δs 2.10 Lemma 2.11 Let a, b ∈ T For rd-continuous functions f, g : a, b → R, one has b a f t g t Δt ≤ 1/2 b f t a Δt 1/2 b g t Δt 2.11 a ∗ ∗ ∗ x1 t , x2 t , , xn t T of system 1.1 is Definition 2.12 The anti-periodic solution x∗ t said to be globally exponentially stable if there exist positive constants and M M ≥ 1, Journal of Inequalities and Applications for any solution x t φ1 t , φ2 t , , φn t T x1 t , x2 t , , xn t T of system 1.1 with the initial value φ t ∈ C ET , Rn , such that n xi t − xi∗ t ≤M e φ − x∗ , t, α 2.12 α ∈ ET 2.13 i where n φ − x∗ sup φi s − xi∗ s , i s∈ET The following continuation theorem of coincidence degree theory is crucial in the arguments of our main results Lemma 2.13 see 37 Let X, X be two Banach spaces, Ω ⊂ X be open bounded and symmetric with ∈ Ω Suppose that L : D L ⊂ X → Y is a linear Fredholm operator of index zero with D L ∩ Ω / ∅ and N : Ω → Y is L-compact Further, one also assumes that H Lx − Nx / λ −Lx − N −x for all x ∈ D L ∩ ∂Ω, λ ∈ 0, Nx has at least one solution on D L ∩ Ω Then the equation Lx Existence of Antiperiodic Solutions In this section, by using Lemma 2.13, we will study the existence of at least one anti-periodic solution of 1.1 Theorem 3.1 Assume that H1 – H4 hold Suppose further that H5 E eij n×n is a nonsingular M matrix, where, for i, j ⎧ ⎪ m ⎪ ⎪ ⎨ωa − ω2 am aM Lb − ωam eij i ⎪ ⎪ ⎪− ⎩ μi i i i 2q 2q LI − ik i k ωam i LI − μi k ik μi 1, 2, , n, ωam aM ωci , i i aM ωci , i i j, 3.1 i / j Then system 1.1 has at least one ω/2-anti-periodic solution Proof Let Ck 0, ω; t1 , , tq , tq , , t2q T {x : 0, ω T → Rn m |xk t is a piecewise continuous map with first-class discontinuity points in 0, ω T ∩{tk }, and at each discontinuity point it is continuous on the left} Take X x ∈ C 0, ω; t1 , , tq , tq , , t2q Y T :x t X×R n×q ω −x t , ∀t ∈ 0, ω T , 3.2 Journal of Inequalities and Applications are two Banach spaces with the norms n x X z Y |xi |0 , x X y , 3.3 i respectively, where |xi |0 Set maxt∈ 0,ω T |xi t |, i · is any norm of Rn×q 1, , n, x −→ xΔ , Δx t1 , , Δx tq L : Dom L ∩ X −→ Y, , 3.4 where x ∈ C1 0, ω; t1 , , t2q Dom L T ω :x t −x t , ∀t ∈ 0, N : X −→ Y, ⎞ ⎛ I11 x1 t1 I1q x1 tq A1 t ⎟ ⎜ ⎟ ⎜ ⎜⎜ ⎟ ⎜ ⎜⎜ ⎟ ⎜ ⎟, , ⎜ ⎜⎜ ⎟, ⎜ ⎠ ⎝ ⎝⎝ ⎠ ⎝ ⎞ ⎛ ⎛⎛ Nx An t In1 xn t1 ω T , ⎞⎞ 3.5 ⎟⎟ ⎟⎟ ⎟⎟, ⎠⎠ Inq xn tq where Ai t t, xi t bi t, xi t fi t, xt , i 1, 2, , n 3.6 It is easy to see that Ker L {0}, Im L z f, C1 , , Cq ∈ Y : ω f s Δs Y 3.7 Thus, dim KerL codim ImL, and L is a linear Fredholm mapping of index zero Define the projectors P : X → Ker L and Q : Y → Y by ω Px x s Δs 0, 3.8 Qz Q f, C1 , , Cq ω ω f s Δs, 0, , , 3.9 Journal of Inequalities and Applications respectively It is not difficult to show that P and Q are continuous projectors such that Im P Further, let L−1 P Ker L, Im L Ker Q Im I − Q L−1 is given by P L|Dom L∩Ker P and the generalized inverse KP t KP z f s Δs Ck − t>tk ω/2 3.10 q f s Δs − Ck , 2k 3.11 in which Cq i −Ci for all ≤ i ≤ q Similar to the proof of Theorem 3.1 in 38 , it is not difficult to show that QN Ω , KP I − Q N Ω are relatively compact for any open bounded set Ω ⊂ X Therefore, N is L-compact on Ω for any open bounded set Ω ⊂ X Corresponding to the operator equation Lx − Nx λ −Lx − N −x , λ ∈ 0, , we have xΔ t λ Δx tk G t, x − λ G t, −x , λ Gi t, x − λ − I k x tk 1 λ t ∈ T , t / tk , 3.12 λ Ik −x tk , t tk , k ∈ N, or xiΔ t λ λ λ Gi t, −x , t ∈ T , t / tk , 3.13 Δxi tk 1 λ Iik xi tk − λ λ Iik −xi tk , t tk , i 1, 2, , n, k ∈ N, where Gi t, x Gi t, −x t, −xi t t, xi t bi t, xi t bi t, −xi t fi t, xt fi t, −xt , , 3.14 i 1, 2, , n Journal of Inequalities and Applications t0 Set t0 ω 0, t2q ω, in view of 3.13 , H1 – H4 and Lemma 2.11, we obtain that 2q ≤ tk k xiΔ t Δt tk−1 ω k Gi t, x − 1 λ 1 λ ≤ ω λ λ − Gi t, −x Δt λ λ Iik −xi tk max{|Gi t, x |, |Gi t, −x |}Δt 2q λ λ λ Iik xi tk λ |Δxi tk | k λ 2q 2q xiΔ t Δt 1 ≤ λ max{|Iik xi tk |, |Iik −xi tk |} k ω max t, xi t bi t, xi t , fi t, xt t, −xi t bi t, −xi t fi t, −xt Δt 2q 3.15 max{|Iik xi tk |, |Iik −xi tk |} k ω ≤ aM i − bi t, |, |bi t, −xi t max{|bi t, xi t − bi t, |}Δt ω max fi t, x1t , , xnt − fi t, 0, , , Δt fi t, −x1t , , −xnt − fi t, 0, , 2q 2q − Iik |, |Iik −xi tk max{|Iik xi tk k ⎡ ≤ aM ⎣Lb i i ω ω |xi t |Δt ci √ ≤ aM Lb ω xi i i aM ci i ⎤ xjt Δt⎦ j xj √ 2q 2q |Iik | LI |xi |0 ik k 2q 2q LI |xi |0 ik ω |Iik | k k n j where i n − Iik |} k |Iik |, k 1, 2, , n Integrating 3.13 from to ω, we have from H1 – H4 that ω t, xi t bi t, xi t λ − λai t, −xi t bi t, −xi t λ Δt 10 Journal of Inequalities and Applications ω t, xi t bi t, xi t λ ω λai t, xi t bi t, xi t λ Δt t, xi t bi t, xi t Δt ω 1 λ 2q λk ω ≤ aM i 1 − Iik xi tk λ t, −xi t fi t, −xt Δt 2q λ ω λ t, xi t fi t, xt Δt − λk Iik −xi tk max fi t, x1t , , xnt − fi t, 0, , , Δt fi t, −x1t , , −xnt − fi t, 0, , 2q 2q max{|Iik xi tk − Iik |, |Iik −xi tk − Iik |} k |Iik | k √ n ≤ aM ci i xj j 2q 2q LI |xi |0 ik ω k |Iik |, i 1, 2, , n, k 3.16 by H2 , we obtain that ω t, xi t xi t Δt ≤ where i ω M n xj a ci μi i j √ 2q ω LI |xi | μi k ik 1, 2, , n From Lemma 2.10, for any ti , ti ∈ 0, ω T , i t, xi t xi t Δt ≤ ω t, xi t xi ti Δt 2q |Iik |, μi k ω 1, 2, , n, we have ω xiΔ t Δt Δt, t, xi t 3.17 3.18 ω t, xi t xi t Δt ≥ ω t, xi t xi ti Δt − ω ω xiΔ t Δt Δt t, xi t 0 3.19 Dividing by xi ti ≥ xi ti ≤ ω t, xi t Δt on the both sides of 3.18 and 3.19 , respectively, we obtain that ω t, xi t Δt ω t, xi t Δt ω t, xi t xi t Δt − ω 0 ω ω t, xi t xi t Δt xiΔ t Δt, i 1, 2, , n, 3.20 xiΔ t Δt, i 1, 2, , n Journal of Inequalities and Applications Let ti , ti ∈ 0, ω T , such that xi ti 11 max xi t , xi ti t∈ 0,ω mint∈ 0,ω T xi t , by the arbitrariness T of ti , ti in view of 3.15 , 3.17 , 3.20 , we have xi ti ≥ ω ω ≥− t, xi t Δt ω − ⎣aM Li i ≤ xi ti ≤ ω xj ω ω t, xi t Δt where i √ ω xj 2q LI |xi |0 ik ω k ω xj 2 √ n aM ci i xj j t Δt xiΔ t Δt ⎤ 2q |Iik |⎦ μi k 2q LI |xi | μi k ik ω |Iik |⎦, k xiΔ ω t, xi t xi t Δt ⎤ 3.21 ω t, xi t xi t Δt √ 2q 2q 2q LI |xi |0 ik ω k ⎤ |Iik |⎦ k 1, 2, , n Thus, we have from 3.21 that |xi |0 max |xi t | t∈ 0,ω T ⎡ ⎣1 M n xj ≤ a ci ωam μi i j i ⎡ ⎣aM Lb i i √ ω xi √ ω aM ci i ⎤ 2q |Iik |⎦ μi k 2q LI |xi | μi k ik √ n xj j where i √ n j t, xi t Δt ⎡ ⎣1 M n ≤ xj a ci ωam μi i j i ⎣aM Lb i i ⎤ 2q |Iik |⎦ μi k 2q LI |xi | μi k ik 0 ω ⎡ √ ω xiΔ t Δt aM ci i ω t, xi t xi t Δt − t, xi t Δt ⎡ ⎣1 M n ≥− m a ci xj ωai μi i j √ b xiΔ t Δt ω ⎡ ω t, xi t xi t Δt − 2q 2q LI |xi |0 ik ω k 3.22 ⎤ |Iik |⎦, k 1, 2, , n In addition, we have that ω xi 1/2 |xi s |Δs ≤ √ ω max |xi t | t∈ 0,ω T √ ω|xi |0 , i 1, 2, , n 3.23 12 Journal of Inequalities and Applications By 3.22 , we obtain that, ⎡ ωam |xi |0 i √ n xj ≤ ⎣ aM ci i μi j ⎤ 2q |Iik |⎦ μi k 2q LI |xi | μi k ik ω ⎡ √ ωam ⎣aM Lb ω xi i i i aM ci i xj j ⎡ LI |xi | μi k ik 0 ⎡ ωam ⎣aM Lb ω|xi |0 i i i aM ωci i 2q LI |xi |0 ik ω k xj ⎤ |Iik | ⎦ k 3.24 2q n j where i 2q ⎤ 2q |Iik |⎦ μi k 2q n xj ≤ ⎣ aM ωci μi i j √ n 2q LI |xi |0 ik k ⎤ |Iik |⎦, k 1, 2, , n That is, 2q 2q ωam − ω2 am aM Lb − ωam i i i i i LI − ik k 1 |Iik | μi k 1 μi ωam aM ωci xj i i |Iik | ωam i Di , i 3.25 2q 2q ≤ LI |xi |0 − μi k ik 1, 2, , n k Denote that, |x|0 |x1 |0 , |x2 |0 , , |xn |0 T , D D1 , D2 , , Dn T 3.26 Then 3.25 can be rewritten in the matrix form E|x|0 ≤ D 3.27 From the conditions of Theorem 3.1, E is a nonsingular M matrix, therefore, |x|0 ≤ E−1 D M1 , M2 , , Mn T 3.28 Let n Mi M Clearly, M is independent of λ 3.29 Ω {x ∈ X : x 3.30 i Take X < M} Journal of Inequalities and Applications 13 It is clear that Ω satisfies all the requirements in Lemma 2.13 and condition H is satisfied In view of all the discussions above, we conclude from Lemma 2.13 that system 1.1 has at least one ω/2-anti-periodic solution This completes the proof Global Exponential Stability of Antiperiodic Solution ∗ ∗ ∗ Suppose that x∗ t x1 t , x2 t , , xn t T is an ω/2-anti-periodic solution of system 1.1 In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of this anti-periodic solution Theorem 4.1 Assume that H1 – H5 hold Suppose further that H6 there exist positive constants La such that i |ai t, u − t, v | ≤ La |u − v|, i H7 for all u, v ∈ R, i ∀u, v ∈ R, i 1, 2, , n; 4.1 1, 2, , n, there exist positive constants Lab such that i t, u bi t, u − t, v bi t, v u − v ≤ 0, i 4.2 |ai t, u bi t, u − t, v bi t, v | ≥ Lab |u − v|, i H8 there are ω-periodic functions ri t such that ri t 1, 2, , n, i 1, 2, , n; supu∈R |fi t, u |, i 1, 2, , n; H9 there exists a positive constant such that Ψi , t −Lab i La riM i μt μ t − θ e t − θ, t aM ci > 0, i n 4.3 i 1, 2, , n; j H10 impulsive operator Iik xi tk satisfy Iik xi tk −γik xi tk , < γik < 2, i 1, , n, k ∈ N 4.4 Then the ω/2-anti-periodic solution of system 1.1 is globally exponentially stable Proof According to Theorem 3.1, we know that system 1.1 has an ω/2-anti-periodic ∗ ∗ solution x∗ t x1 t , x2 t , , x∗ t T with initial value x∗ s , s ∈ ET , suppose that n 14 Journal of Inequalities and Applications x t x1 t , x2 t , , xn t T is an arbitrary solution of system 1.1 with initial value φ s , s ∈ ET Then it follows from system 1.1 that xi t − xi∗ t Δ − t, xi∗ t bi t, xi∗ t t, xi t bi t, xi t ∗ − t, xi∗ t fi t, xt , t ∈ T , t / tk , −γik xi tk − xi∗ tk , Δ xi tk − xi∗ tk t In view of system 4.5 , for t ∈ T , t / tk , k ∈ N, i xi t − xi∗ t Δ t, xi t fi t, xt t, xi t bi t, xi t 4.5 tk , k ∈ N, i 1, 2, , n 1, 2, , n, we have − t, xi∗ t bi t, xi∗ t ∗ t, xi t fi t, xt − t, xi∗ t fi t, xt t, xi t 4.6 − t, xi∗ t bi t, xi∗ t t, xi t bi t, xi t − t, xi∗ t t, xi∗ t fi t, xt ∗ fi t, xt − fi t, xt Hence, we can obtain from H6 – H9 that D xi t − xi∗ t Δ ≤ −Lab xi t − xi∗ t i n La riM xi t − xi∗ t i ci xj t aM i ∗ θ − xj t La riM i xi t − xi∗ t θ 4.7 j −Lab i aM ci i n xj t ∗ θ − xj t θ , j for i 1, 2, , n, and we have from H10 that xi tk − xi∗ tk − γik xi tk − xi∗ tk , i 1, 2, , n, k ∈ N 4.8 For any α ∈ E, we construct the Lyapunov functional V t V1 t n V1 t V2 t , e t, α xi t − xi∗ t , i n n t V2 t i 1j t θ ∗ μ s − θ e s − θ, α aM ci xj s − xj s Δs i 4.9 Journal of Inequalities and Applications 15 For t ∈ T , t / tk , k ∈ N, calculating the delta derivative D V t system 4.5 , we can get D V1 t Δ n ≤ i n ≤ of V t along solutions of e σ t , α D xi t − xi∗ t Δ i ⎧ ⎨ e t, α xi t − xi∗ t ⎩ i n e t, α xi t − xi∗ t Δ e σ t ,α ⎡ × ⎣ −Lab i xi t − xi∗ t La riM i aM ci i n xj t ∗ θ − xj t j n −Lab i μt La riM i ⎤⎫ ⎬ θ ⎦ ⎭ 4.10 e t, α xi t − xi∗ t i n n aM xj t i μ t e t, α ci ∗ θ − xj t θ , i 1j D V2 t Δ n n ≤ ∗ μ t − θ e t − θ, α aM ci xj t − xj t i i 1j n − 4.11 n μ t e t, α aM ci xj t i ∗ θ − xj t θ i 1j By assumption H8 , it concludes that D V t Δ D V1 t ≤ Δ D V2 t n Δ −Lab i La riM i e t, α xi t − xi∗ t μ t ∗ μ t − θ e t − θ, α aM ci xj t − xj t i i n n i 1j ≤ 4.12 n μt −Lab i La riM i i n μ t − θ e t − θ, t aM ci e t, α xi t − xi∗ t i j ≤ 0, t ∈ T , t / tk , k ∈ N 16 Journal of Inequalities and Applications Also, V tk V1 tk n V2 tk e tk , α xi tk − xi∗ tk i n n i 1j n ≤ tk ∗ μ s − θ e s − θ, α aM ci xj s − xj s Δs i tk θ 4.13 e tk , α xi tk − xi∗ tk i n n tk ∗ μ s − θ e s − θ, α aM ci xj s − xj s Δs i i 1j V tk , tk θ k ∈ N It follows that V t ≤ V for all t ∈ T On the other hand, we have V1 V n V2 e 0, α xi − xi∗ i n n i 1j n ≤ θ ⎧ ⎨ ∗ μ s − θ e s − θ, α aM ci xj s − xj s Δs i n j θ e 0, α i ⎩ n ≤M ⎫ ⎬ 4.14 μ s − θ e s − θ, α aM ci Δs sup xi s − xi∗ s i ⎭s∈ET sup φi s − xi∗ s , i s∈ET where ⎧ ⎨ M ⎛ max sup⎝e 0, α 1≤i≤n ⎩α∈E T n j θ ⎞⎫ ⎬ μ s − θ e s − θ, α aM cij Δs⎠ i ⎭ 4.15 It is obvious that n i e 0, α xi t − xi∗ t ≤V t ≤V ≤M n sup s∈ET i φi s − xi∗ s 4.16 Journal of Inequalities and Applications 17 So we can finally get n xi t − xi∗ t ≤M n e 0, α sup s∈ET i i φi s − xi∗ s M e 0, α φ − x∗ 4.17 Since M ≥ 1, from Definition 2.12, the ω/2-anti-periodic solution of system 1.1 is globally exponential stable This completes the proof An Example Example 5.1 Consider the following impulsive generalized neural networks: xΔ t A t, x t B t, x t Δx tk x tk − x t− k A t, u diag 10 t ∈ T, t / tk, F t, xt , I k x tk , t 5.1 tk , k ∈ Z, where u 100 B t, u u arctan |u|, 11 π ⎛ , F t, xt arctan |u| , π ⎞ ci t gj xjt ⎟ ⎜ ⎟ ⎜j ⎟, ⎜ ⎟ ⎜ ⎠ ⎝ ci t gj xjt 5.2 j gj 2×1 1000 sin u sin u ω when T , 2π, ci 2×1 0, 2π 1000 T sin t cos t , ∩ {tk : k ∈ N} Ik 2×2 500 −u − u −u − u , {t1 , t2 }, R, system 5.1 has at least one exponentially stable π-anti-periodic solution Proof By calculation, we have am 10, aM 11, am 11, am 12, La La 2/π, Lb Lb 2 2 1 1 M M 1/1000, c2 1/1000, LI LI LI LI 1/500, and μ1 μ2 1/100 1/100, c1 22 11 21 12 It is obvious that H1 – H4 , H6 – H8 , and H10 are satisfied Furthermore, we can easily calculate that E≈ 7.52 −12.74 −11.25 3.61 5.3 is a nonsingular M matrix, thus H5 is satisfied When T R, μ t Take 0.01, θ −1, we have that Ψ1 , t ≈ −0.04 < 0, Ψ2 , t ≈ −0.03 < 5.4 18 Journal of Inequalities and Applications Hence H10 holds By Theorems 3.1 and 4.1, system 5.1 has at least one exponentially stable π-anti-periodic solution This completes the proof Conclusions Using the time scales calculus theory, the coincidence degree theory, and the Lyapunov functional method, we obtain sufficient conditions for the existence and global exponential stability of anti-periodic solutions for a class of generalized neural 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