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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 491268, 17 pages doi:10.1155/2009/491268 Research Article Global Exponential Stability of Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales Yongkun Li,1 Yuchun Hua,1 and Yu Fei2 Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn Received 18 April 2009; Accepted 14 July 2009 Recommended by Patricia J Y Wong Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional method, and some analysis techniques, sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg bidirectional associative memory BAM neural networks with distributed delays and impulses on time scales This is the first time applying the time-scale calculus theory to unify the discretetime and continuous-time Cohen-Grossberg BAM neural network with impulses under the same framework Copyright q 2009 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 Introduction In the recent years, bidirectional associative memory BAM neural networks and CohenGrossberg neural networks CGNNs with their various generalizations have attracted the attention of many mathematicians, physicists, and computer scientists see 1–17 due to their wide range of applications in, for example, pattern recognition, associative memory, and combinatorial optimization Particularly, as discussed in 18–20 , in the hardware implementation of the neural networks, when communication and response of neurons happens time delays may occur Actually, time delays are known to be a possible source of instability in many real-world systems in engineering, biology, and so forth see, e.g., 21 and references therein However, besides delay effect, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving fields such as medicine and biology, economics, mechanics, electronics, and telecommunications As artificial electronic systems, neural networks such as Hopfield neural networks, bidirectional neural networks, and recurrent neural networks Journal of Inequalities and Applications often are subject to impulsive perturbations which can affect dynamical behaviors of the systems just as time delays Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks As is well known, both continuous and discrete systems are very important in implementation and applications However, it is troublesome to study the stability for continuous and discrete systems, respectively Therefore, it is worth studying a new method, such as the time-scale theory, which can unify the continuous and discrete situations Motivated by the above discussions, the objective of this paper is to study the global exponential stability of the following Cohen-Grossberg bidirectional associative memory networks with impulses and time delays on time scales: ⎡ xiΔ t −ai xi t ⎣bi xi t m − pji fj yj t − τji j m − pji j Δxi tk Δ yj t Ik xi tk , −cj yj t i ∞ ⎤ hij s fj yj t − s Δs 1, 2, , n, k n − dj yj t ri ⎦, t ≥ 0, t / tk , t ∈ T, 1, 2, , 1.1 qij gi xi t − σij i n − qij i Δyj tk Jk yj tk , j ∞ kij s gi xi t − s Δs sj , t ≥ 0, t / tk , t ∈ T, 1, 2, , m, k 1, 2, , where T is a time scale; Ik , Jk : R → R are continuous, xi t , yj t are the states of the ith neuron from the neural field FX and the jth neuron from the neural field FY at time t, respectively; fj , gi denote the activation functions of the jth neuron from FY and the ith neuron from FX , respectively; ri and sj are constants, which denote the external inputs on the ith neuron from FX and the jth neuron from FY , respectively; τji and σij correspond to the transmission delays; xi t and cj yj t represent amplification functions; bi xi t and dj yj t are appropriately behaved functions such that the solutions of system 1.1 0 1 remain bounded; pji , pji , qij , and qij denote the connection strengths which correspond to the neuronal gains associated with the neuronal activations; Ii and Jj denote the external inputs xi tk −xi t− , Δyj tk yj tk − For each interval I of R, we denote that by IT I T, Δxi tk k − − 1, 2, , n, j yj tk are the impulses at moments tk , and xi tk , xi tk , yj tk , yj t− i k 1, 2, , m represent the right and left limits of xi tk and yj tk in the sense of time scales; < t1 < t2 < · · · < tk → ∞ is a strictly increasing sequence The system 1.1 is supplement with initial values given by xi s ϕi s , s ∈ −∞, T , i 1, 2, , n, yj s ψj s , s ∈ −∞, T , j 1, 2, , m, where ϕi , ψj are continuous real-valued functions defined on their respective domains 1.2 Journal of Inequalities and Applications As usual in the theory of impulsive differential equations, at the points of discontinuity tk of the solution t → x1 t , x2 t , , xn t , y1 t , y2 t , , ym t T we assume that xi tk xi t− , k yj tk yj t− , k xiΔ tk xiΔ t− , k Δ yj tk Δ yj t− , k 1.3 for i 1, 2, , n, j 1, 2, , m The organization of the rest of this paper is as follows In Section 2, we introduce some notations and definitions, and state some preliminary results which are needed in later sections In Section 3, by means of homeomorphism theory, we study the existence and uniqueness of the equilibrium point of system 1.1 In Section 4, by constructing a suitable Lyapunov function, we establish the exponential stability of the equilibrium of 1.1 In Section 5, we present an example to illustrate the feasibility and effectiveness of our results obtained in previous sections Preliminaries In this section, we will cite some definitions and lemmas which will be used in the proofs of our main results Let T be a nonempty closed subset time scale of 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}, μt σ t − t 2.1 A point t ∈ T is called left dense if t > inf T and ρ t t, left scattered if ρ t < t, right dense if t < sup T and σ t t, and right scattered if σ t > t If T has a left-scattered maximum m, then Tk T \ {m}; otherwise Tk T If T has a right-scattered minimum m, then Tk T \ {m}; otherwise Tk T A function f : T → R is right dense continuous provided that it is continuous at right dense point in T and its left-side limits exist at left-dense points in T If f is continuous at each right dense point and each left-dense point, then f is said to be a continuous function on T The set of continuous functions f : T → R will be denoted by C T For y : T → R and t ∈ Tk , we define the delta derivative of y t , yΔ 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 y σ t −y s − yΔ t σ t − s < ε|σ t − s| 2.2 for all s ∈ U If y is continuous, then y is right dense continuous, and y is delta differentiable at t, then y is continuous at t y t , then we define the delta integral by Let y be right dense continuous If yΔ t t a y s Δs Y t −Y a 2.3 Journal of Inequalities and Applications Definition 2.1 see 22 For each t ∈ T, let N be a neighborhood of t, then, for V ∈ Crd T × Rn , R , define D V Δ t, x t to mean that, given ε > 0, there exists a right neighborhood Nε ⊂ N of t such that − V s, x σ t μ t, s V σ t ,x σ t − μ t, s f t, x t < D V Δ t, x t for each s ∈ Nε , s > t, where μ t, s ≡ σ t − s If t is rd and V t, x t reduce to Definition 2.2 see 23 If a ∈ T, sup T we define the improper integral by ∞ 2.4 is continuous at t, this V σ t , x σ t − V t, x σ t σ t −t D V Δ t, x t ε 2.5 ∞, and f is right dense continuous on a, ∞ , then f t Δt a b lim b→∞ a f t Δt 2.6 provided that this limit exists, and we say that the improper integral converges in this case If this limit does not exist, then we say that the improper integral diverges A function r : T → R is called regressive if μ t r t /0 2.7 for all t ∈ Tk If r is regressive function, then the generalized exponential function er is defined by t er t, s exp ξμ τ r τ Δτ for s, t ∈ T, 2.8 s with the cylinder transformation ξh z ⎧ ⎪ Log hz ⎨ , h ⎪ ⎩z, if h / 0, if h 2.9 Let p, q : T → R be two regressive functions, then we define p⊕q : p q μpq, p q: p⊕ q , p: p μp Then the generalized exponential function has the following properties 2.10 Journal of Inequalities and Applications Lemma 2.3 see 24 Assume that p, q : T → R are two regressive functions, then i e0 t, s ≡ and ep t, t ≡ ii ep σ t , s iii ep t, s / ep t, σ s iv 1/ep t, s v ep t, s e p μ t p t ep t, s μ s p s t, s 1/ ep s, t e vi ep t, s ep s, r ep t, r vii ep t, s eq t, s s, t p ep⊕q t, s viii ep t, s /eq t, s ep t, s q ∗ ∗ ∗ ∗ ∗ ∗ x1 , x2 , , xn , y1 , y2 , , ym T of system Definition 2.4 The equilibrium point u∗ 1.1 is said to be exponentially stable if there exists a positive constant α such that for every δ ∈ T, there exists N N δ ≥ such that the solution u t x1 t , x2 t , , xn t , y1 t , y2 t , , ym t T of 1.1 with initial value ϕ1 s , ϕ2 s , , ϕn s , ψ1 s , ψ2 s , , ψm s T satisfies ⎡ u − u∗ ≤ Ne−α t, δ ⎣ n max i δ∈ −∞,0 Lemma 2.5 see 25 If H x ∈ C Rn i H x is injective on Rn ii H → ∞ as x → m m ϕi δ − xi∗ max j , Rn m δ∈ −∞,0 T ∗ ψj δ − yj ⎦ 2.11 satisfies the following conditions: , ∞, then H x is a homeomorphism of Rn For z T ⎤ m m onto itself x1 , x2 , , xn , y1 , y2 , , ym T n z ∈ Rn |xi | i m , we define the norm as m yj 2.12 j Throughout this paper, we assume that H1 , cj ∈ C T, R , and satisfy < ≤ x ≤ , < cj ≤ cj x ≤ cj , ∀x ∈ R, i 1, 2, , n, j 1, 2, , m; H2 the activation functions fj , gi ∈ C R, R and there exist positive constants Mj , Ni such that fj x − fj y for all x, y ∈ R, i ≤ Mj x − y , 1, , n, j 1, , m; gi x − gi y ≤ Ni x − y , 2.13 Journal of Inequalities and Applications dj 0, i H3 bi , dj ∈ C R, R , bi positive constants ηi , ωj such that bi x − b i y ≥ ηi , x−y 1, 2, , n, j dj x − d j y ≥ ωj , x−y 1, 2, , m, and there exist ∀x / y; 2.14 H4 the kernels hji and kij defined on 0, ∞ T are nonnegative continuous integral func∞ ∞ ∞ ∞ tions such that hji s Δs 1, shji s Δs < ∞, kij s Δs 1, skij s Δs < ∞ Existence and Uniqueness of the Equilibrium In this section, using homeomorphism theory, we will study the existence and uniqueness of the equilibrium point of system 1.1 ∗ ∗ ∗ ∗ ∗ ∗ An equilibrium point of 1.1 is a constant vector x1 , x2 , , xn , y1 , y2 , , ym T ∈ Rn m which satisfies the system ⎡ ⎤ m xi∗ ⎣bi xi∗ − pji ri ⎦ ∗ pji fj yj 0, i 1, 2, , n, j 3.1 ∗ cj yj n ∗ dj yj − qij qij gi xi∗ 0, j j sj 1, 2, , m, 1, 2, , m i where the impulsive jumps Ik · , Jk · satisfy Ik xi∗ 0, i 1, 2, , n, ∗ Jk yj 0, 3.2 From the assumptions H1 and H4 , it follows that bi xi∗ m pji ∗ pji fj yj ri , i 1, 2, , n, j 3.3 ∗ dj yj n qij qij gi xi∗ sj , j 1, 2, , m i −1 Noting that if bi−1 · , dj · exist and activation functions fj · and gj · are bounded, then the existence of an equilibrium point of system 1.1 is easily obtained from Brouwer’s fixed point theorem We can refer to 2–8 Journal of Inequalities and Applications Theorem 3.1 Assume that H2 and H3 hold Suppose further that for each i 1, 2, , m, the following inequalities are satisfied: m ηi > qij n qij Ni , pji ωj > j 1, 2, , n, j pji Mj 3.4 i Then there exists a unique equilibrium point of system 1.1 Proof Consider a mapping Φ : Rn Φi z m → Rn m bi xi − pji m defined by pji fj yj ri , i 1, 2, , n, j Φi z dj yj − n 3.5 qij qij gi xi sj , j 1, 2, , m, i where z x1 , x2 , , xn , y1 , y2 , , ym T ∈ Rn m , Φ z Φ1 z , , Φn z , , Φn m z T ∈ Rn m First, we want to show that Φ is an injective mapping on Rn m By contradiction, Φ z , where z suppose that there exists a distinct z, z ∈ Rn m such that Φ z x1 , x2 , , xn , y1 , y2 , , ym T ∈ Rn m and z x1 , x2 , xn , y , y2 , , y m T ∈ Rn m Then it follows from 3.5 that bi xi − bi xi m pji pji qij qij fj yj − fj yj , i 1, 2, , n, j dj yj − dj yj n 3.6 gi xi − gj xi , j 1, 2, , m i In view of H2 - H3 and 3.6 , we have n ηi |xi − xi | ≤ i m n m pji pji Mj yj − y j , i 1j ωj yj − y j ≤ j m n 3.7 qij qij Ni |xi − xi | j 1i Thus, we can obtain n i ⎡ ⎣ηi − m j ⎤ qij qij Ni ⎦|xi − xi | m j ωj − n pji pji Mj yj − y j ≤ 3.8 i It follows from 3.4 and 3.8 that |xi − xi | and |yj − y j | 0, i 1, 2, , n, j 1, 2, , m That is z z, which leads to a contradiction Therefore, Φ is an injective on Rn m Journal of Inequalities and Applications Then we will prove Φ is a homeomorphism on Rn Φ z − Φ , where m bi xi − Φi z pji m For convenience, we let Φ z fj yj − fj , pji i 1, 2, , n, j Φn j dj yj − z n 3.9 qij gi xi − gi , qij j 1, 2, , m i We assert that Φ → ∞ as z → ∞ Otherwise there is a sequence {zv } such that zv → v v v v v v x1 , x2 , , xn , y1 , y2 , , ym T ∈ Rn m ∞ and Φ zv is bounded as v → ∞, where zv Noting that n n bi xiv − Φi zv sgn xiv i i ≤ n m sgn xiv pji pji v fj yj − fj j m pji v pji Mj yj , i 1j m sgn v yj v dj yj − Φn j m v sgn z j j ≤ m 3.10 n v yj qij qij − gi gi xiv i n qij qij Ni xiv , j 1i we have n sgn xiv m bi xiv − Φi zv i v dj yj − Φn v sgn yj j zv j ≤ n m pji pji v Mj yj i 1j n m 3.11 qij qij Ni xiv j 1i On the other hand, we have n sgn xiv m bi xiv − Φi zv i v dj yj − Φn v sgn yj j zv j ≥ n i ηi xiv − n i Φi z v m j v ωj yj − m j 3.12 Φn j v z Journal of Inequalities and Applications It follows from 3.11 and 3.12 that ⎛ n Θ⎝ ⎞ m xiv i v yj ⎠≤ j n m Φi zv i Φn j zv , 3.13 j where Θ ⎧ ⎨ ⎧ ⎨ η − ⎩1≤i≤n⎩ i m qij qij Ni j ⎫ ⎬ n , ωj − ⎭ 1≤j≤m pji pji Mj i ⎫ ⎬ ⎭ > 3.14 That is zv ≤ Φ zv Θ 3.15 , which contradicts our choice of {zv } Hence, Φ satisfies Φ → ∞ as z → ∞ By Lemma 2.5, Φ is a homeomorphism on Rn m and there exists a unique point z∗ ∗ ∗ ∗ ∗ ∗ ∗ From the definition of Φ, we know that x1 , x2 , , xn , y1 , y2 , , ym T such that Φ z∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ T z x1 , x2 , , xn , y1 , y2 , , ym is the unique equilibrium point of 1.1 Global Exponential Stability of the Equilibrium In this section, we will construct some suitable Lyapunov functions to derive the sufficient conditions which ensure that the equilibrium of 1.1 is globally exponentially stable Theorem 4.1 Assume that (H1 )–( H4 ) hold, suppose further that H5 for each i 1, 2, , n, j m ηi > cj qij 1, 2, , m, the following inequalities are satisfied: qij n Ni , cj ωj > j Jjk yj tk pji pji Mj 4.1 i H6 the impulsive operators Iik xi t Iik xi tk and Jjk yj t −γik xi tk − xi∗ , ∗ −γjk yj tk − yj , satisfy < γik < 2, i < γjk < 2, j 1, , n, k ∈ Z , 1, , m, k ∈ Z 4.2 Then the unique equilibrium point of system 1.1 is globally exponentially stable Proof According to Theorem 3.1, we know that 1.1 has a unique equilibrium point z∗ ∗ ∗ ∗ ∗ ∗ ∗ and Jj yj x1 , x2 , , xn , y1 , , ym T In view of H6 , it is easy to see that Ii xi∗ 10 Journal of Inequalities and Applications Suppose that z t x1 t , x2 t , , xn t , y1 t , y2 t , , ym t T is an arbitrary solution of ∗ xi t − xi∗ , vj t yj t − yj , t ≥ 0, then system 1.1 can be rewritten as 1.1 Let ui t ⎡ uΔ t i −ai ui t ⎣bi ui t m − pji fj vj t − τji j m − pji j −cj vj t ⎤ hji s fj vj t − s Δs − ri ⎦, 1, 2, , n, t > 0, t / tk , t ∈ T, i Δ vj t ∞ − dj vj t n 4.3 qij gi ui t − σij i − n qij i j where, for i 1, 2, , n, j ui t cj vj t dj vj t Also, for all t dj vj t 1, 2, , m, t > 0, t / tk , t ∈ T, ui t xi∗ , bi ui t bi ui t cj vj t ∗ yj , fi vi t fj vj t ∗ ∗ yj − fj yj , ∗ ∗ yj − bi yj , gi ui t g i ui t xi∗ − bi xi∗ 1, 2, , n, j xi tk − xi∗ − γik vj tk kij s gi ui t − s Δs − sj , 1, 2, , m, tk , k ∈ Z , i ui tk ∞ − γjk 4.4 1, 2, , m, xi tk xi tk − xi∗ ∗ yj tk − yj xi∗ − bi xi∗ , yj tk ∗ yj tk − yj Ik xi tk − xi∗ ≤ xi tk − xi∗ ≤ |ui tk |, Ik yj tk ∗ − yj ∗ ≤ yj tk − yj ≤ vj tk 4.5 Journal of Inequalities and Applications 11 Hence by H1 and H2 , we have m D |ui t |Δ ≤ −ai ηi |ui t | pji Mj vj t − τji j m pji ∞ Mj Δ hji s vj t − s Δs, i 1, 2, , n, j D vj t 4.6 n ≤ −cj vj t qij Ni ui t − σij cj i n qij Ni cj i Also, for i ∞ 4.7 kij s |ui t − s |Δs, j 1, 2, , m 1, 2, , n, xi tk − xi∗ tk xi tk − xi∗ tk − Iik xi∗ tk Iik xi tk − γik xi tk − xi∗ tk , k∈Z , 4.8 thus xi tk − xi∗ tk − γik xi tk − xi∗ tk ≤ xi tk − xi∗ tk , 1, , n, k ∈ Z i 4.9 Similarly, we have ∗ yj tk − yj tk ∗ ∗ − γjk yj tk − yj tk ≤ yj tk − ∗ yj 4.10 tk , j 1, 2, , m, k ∈ Z Let Gi and G∗ be defined by j Gi εi ηi − εi − m cj qij Ni eεi σ t , t − σij j − m ∞ cj qij Ni G∗ ξj j kij s eεi σ t , t − s Δs, i 1, 2, , n, j 4.11 cj ωj − ξj − n pji Mj eξj σ t , t − τji i − n i 1 pji Mj ∞ hji s eξj σ t , t − s Δs, j 1, 2, , m, 12 Journal of Inequalities and Applications respectively, where εi , ξj ∈ 0, ∞ By H5 , we have m ηi − Gi cj qij qij Ni > 0, i 1, 2, , n, j G∗ j 4.12 n cj ωj − pji pji Mj > 0, j 1, 2, , m i Since Gi , G∗ are continuous on 0, ∞ and Gi εi → −∞, G∗ ξj → −∞, as εi → ∞, ξj → j j ∗ ∗ 0, G∗ ξj and Gi εi > 0, for εi ∈ 0, εi∗ , G∗ ξj > ∞, there exist εi∗ , ξj > such that Gi εi∗ j j ∗ ∗ 0, for ξj ∈ 0, ξj By choosing α min1≤i≤n,1≤j≤m {εi∗ , ξj }, we obtain Gi α m ηi − α − cj qij Ni eα σ t , t − σij j − m ∞ cj qij Ni ≥ 0, G∗ j α i kij s eα σ t , t − s Δs j 1, 2, , n, n cj ωj − α − pji 4.13 Mj eα σ t , t − τji i n − ∞ pji Mj ≥ 0, j hji s eα σ t , t − s Δs i 1, 2, , m, Denote μi t νj t t ∈ R, i eα t, δ |ui t |, eα t, δ vj t , t ∈ R, j where δ ∈ −∞, T For t > 0, t / tk , k ∈ Z , i 4.6 – 4.15 , we can obtain D μΔ t i αeα t, δ |ui t | ≤ αeα t, δ |ui t | ⎡ × ⎣−ai ηi |ui t | 1, 2, , n, 1, 2, , m, 1, 2, , n, j eα σ t , δ m pji Mj vj t − τji j m j 1 pji Mj ∞ 4.15 1, 2, , m, it follows from eα σ t , δ D |ui t |Δ 4.14 ⎤ hji s vj t − s Δs⎦ Journal of Inequalities and Applications 13 m ≤ − ηi − α μi t pji Mj eα σ t , t − τji νj t − τji j m ∞ pji Mj hji s eα σ t , t − s νj t − s Δs, j n Δ D νj t ≤ − cj ωj − α νj t qij Ni eα σ t , t − σij μi t − σij cj i n ∞ qij Ni ci kij s eα σ t , t − s μi t − s Δs i 4.16 Also, μi tk ≤ μi tk , νj tk ≤ νj tk , i 1, 2, , n, j 1, 2, , m, k ∈ Z 4.17 Define a Lyapunov function ⎡ n m ⎣μi t V t pji Mj eα σ t , t − τji i j m ∞ pji Mj n νj t cj j hji s eα σ t , t − s n qij Ni eα σ t , t − σij qij Ni i ⎤ t νj z ΔzΔs⎦ t−s i cj νj s Δs t−τji j m t ∞ kij s eα σ t , t − s t 4.18 μi s Δs t−σij t μi z ΔzΔs t−s And we note that V t > for t > and V > Calculating the Δ-derivatives of V , we get D VΔ t ≤ ⎡ n m ⎣− a ηi − η μi t i i pji Mj eα σ t , t − τji νj t j m ∞ pji Mj hji s eα σ t , t − s νj t Δs⎦ j m ⎤ − cj ωj − η νj t j n cj qij Ni eα σ t , t − σij μi t i n cj i 1 qij Ni ∞ kij s eα σ t , t − s μi t Δs 14 Journal of Inequalities and Applications ⎡ n − m ⎣a ηi − η − i i cj qij Ni eη σ t , t − σij j − m ∞ cj qij Ni m n cj ωj − η − j pji Mj eη σ t , t − τji i − n pji Mj i − n kij s eα σ t , t − s Δs⎦μi t j − ⎤ Gi η μi t − i m ∞ hji s eα σ t , t − s Δs νj t G∗ η νj t j j ≤ 0, t > 0, t / tk , t ∈ T, k ∈ Z 4.19 Also, ⎡ n m ⎣μi t k V tk i ∞ pji Mj j m n j hji s eα σ tk , tk − s tk −s tk tk −σij i n ∞ qij Ni cj i ⎡ m ⎣μi tk pji i Mj eα σ tk , tk − τji m pji Mj j m n νj tk cj j hji s eα σ tk , tk − s 0 qij Ni eα σ tk , tk − σij i n cj qij Ni i V tk , ∞ k∈Z ∞ kij s eα σ tk , tk − s νj z ΔzΔs⎦ μi s Δs tk kij s eα σ tk , tk − s j νj s Δs ⎤ tk qij Ni eα σ tk , tk − σij νj tk ≤ tk −τji j m n tk pji Mj eα σ tk , tk − τji tk −s μi z ΔzΔs 4.20 tk tk −τji νj s Δs ⎤ tk tk −s νj z ΔzΔs⎦ tk tk −σij tk tk −s μi s Δs μi z ΔzΔs Journal of Inequalities and Applications 15 It follows that V t ≤ V for t > and hence, for t > 0, we can obtain n m νj t ≤ μi t i j ⎡ n m ⎣μi i pji Mj eα σ , −τji j m ∞ pji Mj n n qij Ni i ∞ νj z ΔzΔs⎦ 0−s 4.21 μi s Δs 0−σij i cj ⎤ hji s eα σ , − s qij Ni eα σ , − σij νj j νj s Δs 0−τji j m kij s eα σ , − s 0 μi z ΔzΔs 0−s In view of 4.14 - 4.15 and the previous inequality, we have n m xi t − xi∗ t i ⎡ ≤e α t, δ ⎣ ∗ yj t − yj t j ⎛ n m ⎝1 i ci qij Ni eα σ , −σij σij j m ci qij Ni i m n kij s eα σ , −s sΔs pji ∞ Mj i ≤ Ne α t, δ ⎣ n max i T ϕi δ − xi∗ δ ⎤ i n ⎡ max δ∈ −∞,0 pji Mj eα σ , −τji τji j ∞ δ∈ −∞,0 T hji s eα σ , −s sΔs ϕi δ − xi∗ δ T δ ⎦ ⎤ m max j ψj δ − max δ∈ −∞,0 ∗ yj δ∈ −∞,0 T ∗ ψj δ − yj δ ⎦, 4.22 where ⎧ ⎨ N max ⎩ m ci qij Ni eα σ , −σij σij j m ci qij Ni j n pji Mj i The proof is complete ∞ kij s eα σ , −s sΔs, ∞ n i hji s eα σ , −s sΔs ≥ pji Mj eα σ , −τji τji 4.23 16 Journal of Inequalities and Applications An Example In this section, we give an example to illustrate our results Consider the following Cohen-Grossberg BAM neural networks system with distributed delays and impulses: Δ x1 t − 1 cos x1 t × 5x1 t − sin 2y1 t − − ∞ −s e sin 2y1 t − s Δs r1 , t > 0, t / tk , t ∈ T, Δx1 tk yΔ t I1 x1 tk , − k 1, 2, , 5.1 sin y1 t × 3y1 t − cos 2x1 t − − ∞ −s e cos 2x1 t − s Δs s1 , t > 0, t / tk , t ∈ T, Δy1 tk J1 y1 tk , k 1, 2, , where T R, γ1k 1/2 sin k , γ1k 2/3 cos 2k, k ∈ Z A simple computation 0 0 shows that a c 2/3, a c 4/3, M1 N1 1, η1 5, ω1 3, p11 p11 q11 q11 1/4, < γ1k , γ1k < It is easy to check that all conditions of Theorems 3.1 and 4.1 are satisfied Hence, 5.1 has a unique equilibrium point, which is globally exponentially stable Conclusion Using the time-scale calculus theory, the homeomorphism theory and the Lyapunov functional method, some sufficient conditions are obtained to ensure the existence and the global exponential stability of the unique equilibrium point of Cohen-Grossberg BAM neural networks with distributed delays and impulses on time scales This is the first time applying the time-scale calculus theory to unify and improve impulsive Cohen-Grossberg BAM neural networks with distributed delays on time scales under the same framework The sufficient conditions we obtained can easily be checked in practice by simple algebraic methods Acknowledgments This work was supported by the National Natural Sciences Foundation of People’s Republic of China and the Natural Sciences 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