RESEARCH Open Access Integro-differential inequality and stability of BAM FCNNs with time delays in the leakage terms and distributed delays Xinhua Zhang and Kelin Li * * Correspondence: lkl@suse.edu.cn School of Science, Sichuan University of Science & Engineering, Sichuan 643000, PR China Abstract In this paper, a class of impulsive bidirectional associative memory (BAM) fuzzy cellular neural networks (FCNNs) with time delays in the leakage terms and distributed delays is formulated and investigated. By establishing an integro-differential inequality with impulsive initial conditions and employing M-matrix theory, some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM FCNNs with time delays in the leakage terms and distributed delays are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on the delay kernel functions and system parameters. It is believed that these results are significant and useful for the design and applications of BAM FCNNs. An example is given to show the effectiveness of the results obtained here. Keywords: bidirectional associative memory, fuzzy cellular neural networks, impulses, distributed delays, global exponential stability 1 Introduction The bidirectional associative memory (BAM) neural network models were first introduced by Kosko [1]. It is a special class of recurrent neural networks that can store bipolar vector pairs. The BAM neural network is composed of neurons arranged in two layers, the X-layer and Y-layer. The neurons in one layer are fully interconnected to the neurons in the other layer. Thr ough iterations of forward and backward info rmation flows between the two layer, it performs a two-way associative search for stored bipolar vector pairs and generalize the single-layer autoassociative Hebbian correlation to a two-layer pattern- matched heteroassociative circuits. Therefore, this class of networks possesses good appli- cation prospects in some fields such as pattern recognition, signal and image process, and artificial intelligence [2]. In such applications, the stability of networks plays an important role; it is of significance and necessary to investigate the stability. It is well known, in both biological and artificial neural networks, the delays arise because of the processing of information. Time delays may lead to oscillation, divergence or instability which may be harmful to a system. Therefore, study of neural dynamics with consideration of the delayed problem becomes extremely important to manufacture high-quality neural net- works. In recent years, there have been many analytical results for BAM neural networks with various axonal signal transmission delays, for example, see [3-11] and references therein. I n addition, except various axonal s ignal transmissi on delays, time delay in the Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 © 2011 Zhang a nd Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unre stricted use, distribution, and reproduction in any medium, provided the original work is properly cited. leakage term also has great impact on the dynamics of neural networks. As pointed out by Gopalsamy [12,13], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. Recently, some authors have paid attention to stability analysis of neural networks with time delays in the leakage (or “forgetting”) terms [12-18]. Since FCNNs were introduced by Yang et. al [19,20], many researchers have done extensive works on this subject due to their extensive applications in classification of image processing and pattern recognition. Specially, in the past few years, the stability analysis on FCNNs with various delays and fuzzy BAM neural networks with transmis- sion delays has been the highl ight in the neural network field, for example, see [21-27] and references therein. On the other hand, in respect of the comple xity, 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 such fields as medicine and biology, economics, me chanics, electronics and telecommunications. Many inte rest- ing results on impulsive effect have been gained, e.g., Refs. [28-37]. As artificial electronic systems, neural networks such as CNNs, bidirectional neural networks and recurrent neural networks 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 netwo rks. To the best of our knowledge, few authors have considered impulsive BAM FCNNs with time delays in th e leakage terms and distributed delays. Motivated by the above discussions, the objective of this paper is to formulate and study impulsive BAM FCNNs with time delays in the leakage terms and distributed delays. Under quite general conditions, some sufficient conditions ensuring the exis- tence, uniqueness and gl obal exponential stability of equilibrium point are obtained by the topological degree theory, properties of M-matrix, the integro-differential inequality with impulsive initial conditions and analysis technique. The paper is organized as follows. In Section 2, the new neural network model is for- mulated, and the necessary knowledge is provided. The existence and uniqueness of equilibrium point are presented in Section 3. In Section 4, we give some sufficient con- ditions of exponential stability of the impulsive BAM FCNNs with time delays in the leakage terms and distributed delays. An example is given to show the effectiveness of the results obtained here in Section 5. Finally, in Section 6, we give the conclusion. 2 Model description and preliminaries In this section, we will consider the model of impulsive BAM FCNNs with time delays in the leakage terms and distributed delays, it is described by the following functional dif- ferential equation: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ x i (t)=−a i x i (t − δ i )+ m  j=1 a ij g j (y j (t)) + m  j=1 ˜ a ij v j + I i + m ∧ j=1 α ij +∞  0 K ij (s)g j (y j (t − s))ds + m ∨ j=1 ˜α ij +∞  0 K ij (s)g j (y j (t − s))d s + m ∧ j=1 T ij v j + m ∨ j=1 H ij v j , t = t k x i (t + )=x i (t − )+P ik (x i (t − )), t = t k , k ∈ N = {1, 2, }, ˙ y j (t)=−b j y j (t − θ j )+ n  i=1 b ji f i (x i (t)) + n  i=1 ˜ b ji u i + J j + n ∧ i=1 β ji +∞  0 ¯ K ji (s)f i (x i (t − s))ds + n ∨ i=1 ˜ β ji +∞  0 ¯ K ij (s)f i (x i (t − s))ds + n ∧ i=1 ¯ T ji u i + n ∨ i=1 ¯ H ji u i , t = t k y j (t + )=y j (t − )+Q j k (y j (t − )), t = t k , k ∈ N = {1, 2, }, (1) Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 2 of 18 for i = 1, 2, , n, j = 1, 2, , m, t >0,wherex i (t)andy j (t) are the states of the ith neuron and the jth neuron at time t, respectively; δ i ≥ 0andθ j ≥ 0 denote the leakage delays, respectively; f i and g j denote the signal functions of the ith neuron and the jth neuron at time t, respectivel y; u i , v j and I i , J j denote inputs and bias of the ith neuron and the jth neur on, respect ively; a i >0,b j >0, a i j , ˜ a i j , α i j , ˜α i j , b j i , ˜ b j i , β j i , ˜ β ji are constants, a i and b j represent the rate with which the ith neuron and the jth neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively; a ij , b ji and ˜ a i j , ˜ b ji denote connection weights o f feed- back template and feedforward template, respectively; a ij , b ji and ˜ α i j , ˜ β j i denote connec- tion weights of the distrib uted fuzzy feedback MIN template and the distributed fuzzy feedback MAX template, respectively; T i j , ¯ T ji and H i j , ˜ H ji are elements of fuzzy feedfor- ward MIN template and fuzzy feedforward MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively; K ij (s)and ¯ K j i (s ) corre- spond to the delay kernel functions, respectively. t k is called impulsive moment and satisfies 0 <t 1 <t 2 < , lim k →+∞ t k =+ ∞ ; x i (t − k ) and x i (t + k ) denote the left-hand and right- hand limits at t k , respectively; P ik and Q jk show impulsive perturbations of the ith neu- ron and jth neuron at time t k , respectively. We always assume x i (t + k )=x i (t k ) and y j (t + k )=y j (t k ) , k Î N . The initial conditions are given by  x i (t )=φ i (t ), −∞ ≤ t ≤ 0 , y j (t )=ϕ j (t ), −∞ ≤ t ≤ 0 , where j i (t),  j (t)(i =1,2, ,n; j =1,2, ,m) are bounded and continuous on (-∞,0], respectively. If the impulsive operators P ik (x i )=0,Q jk (y j )=0,i =1,2, ,n, j = 1, 2, , m, k Î N, then system (1) may reduce to the following model: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ x i (t )=−a i x i (t − δ i )+ m  j=1 a ij g j (y j (t )) + m  j=1 ˜ a ij v j + I i + m ∧ j=1 α ij +∞  0 K ij (s)g j (y j (t − s))ds + m ∨ j=1 ˜α ij +∞  0 K ij (s)g j (y j (t − s))d s + m ∧ j=1 T ij v j + m ∨ j=1 H ij v j , ˙ y j (t )=−b j y j (t − θ j )+ n  i=1 b ji f i (x i (t )) + n  i=1 ˜ b ji u i + J j + n ∧ i=1 β ji +∞  0 ¯ K ji (s)f i (x i (t − s))ds + n ∨ i=1 ˜ β ji +∞  0 ¯ K ij (s)f i (x i (t − s))ds + n ∧ i=1 ¯ T ji u i + n ∨ i=1 ¯ H ji u i . (2) System (2) is called the continuous system of model (1). Throughout this paper, we make the following assumptions: Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 3 of 18 (H1) For neuron activation functions f i and g j (i = 1, 2, , n; j =1,2, ,m), there exist two positive diagonal matrices F =diag(F 1 , F 2 , ,F n )andG =diag(G 1 , G 2 , , G m ) such that F i =sup x= y     f i (x) − f i (y) x − y     , G j =sup x= y     g j (x) − g j (y) x − y     for all x, y Î R (x ≠ y). (H2) The delay kernels K ij : [0, +∞) ® R and ¯ K j i :[0,+∞) → R are real-valued piece- wise continuous, and there exists δ > 0 such that k ij (λ)=  +∞ 0 e λs |K ij (s)|ds, ¯ k ji (λ)=  +∞ 0 e λs | ¯ K ji (s)|d s Are continuous for l Î [0,δ), i = 1,2, , n, j = 1,2, , m. (H3) Let ¯ P k ( x ) = x + P k ( x ) and ¯ Q k ( y ) = y + Q k ( y ) be Lipschitz continuous in R n and R m , respectively, that is, there exist nonnegative diagnose matrices Γ k = diag(g 1k , g 2k , , g nk ) and ¯ Γ k =diag ( ¯γ 1k , ¯γ 2k , , ¯γ mk ) such that | ¯ P k (x) − ¯ P k (y)|≤Γ k |x − y|, for all x, y ∈ R n , k ∈ N , | ¯ Q k ( u ) − ¯ Q k ( v ) |≤ ¯ Γ k |u − v|, for all u, v ∈ R m , k ∈ N , where ¯ P k (x)=( ¯ P 1k (x 1 ), ¯ P 2k (x 2 ), , ¯ P nk (x n )) T , ¯ Q k (x)=( ¯ Q 1k (y 1 ), ¯ Q 2k (y 2 ), , ¯ Q mk (y m )) T , P k (x)=(P 1k (x 1 ), P 2k (x 2 ), , P nk (x n )) T , Q k ( y ) = ( Q 1k ( y 1 ) , Q 2k ( y 2 ) , , Q mk ( y m )) T . To begin with, we introduce some notation and recall some basic definitions. PC[J, R l ]={z(t): J ® R l |z(t) is continuous at t ≠ t k , z (t + k )=z(t k ) , and z (t − k ) exists for t, t k Î J, k Î N}, where J ⊂ R is an interval, l Î N. PC ={ψ:(-∞,0]® R l | ψ(s)isbounded,andψ(s + )=ψ(s)fors Î (-∞,0),ψ(s - )exists for s Î (-∞, 0], j(s - )=j(s) for all but at most a finite number of points s Î (-∞, 0]}. For an m × n matrix A,|A| denotes the absolute value matrix given by |A|=(|a ij |) m ×n . For A =(a ij ) m × n , B =(b ij ) m × n Î R m × n , A ≥ B (A>B) means that each pair of corresponding elements of A and B such that the inequality a ij ≥ b ij (a ij >b ij ). Definition 1 A function (x, y) T :(-∞,+∞) ® R n+m is said to be the special solution of system (1) with initial conditions x ( s ) = φ ( s ) , y ( s ) = ϕ ( s ) s ∈ ( −∞,0] , if the following two conditions are satisfied (i) (x, y) T is piecewise continuous with first kind disc ontinuity at the points t k , k Î K. Moreover,(x, y) T is right continuous at each discontinuity point. (ii) (x, y) T satisfies model (1) for t ≥ 0, and x(s)=j(s), y(s)=(s) for s Î (-∞, 0]. Especially, a point (x*, y*) T Î R n+m is called an equilibrium point of model (1),if(x (t), y(t)) T =(x*, y*) T is a solution of (1). Throughout this paper, we always assume th at the impulsive jumps P k and Q k satisfy (referring to [28-37]) Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 4 of 18 P k ( x ∗ ) =0 and Q k ( y ∗ ) =0, k ∈ N , i.e., ¯ P k ( x ∗ ) = x ∗ and ¯ Q ( y ∗ ) = y ∗ , k ∈ N , (3) where (x*, y*) T is the equilibrium point of continuous systems (2). That is, if (x*, y*) T is an equilibrium point of continuous system (2), then (x*, y*) T is also the equilibrium of impulsive system (1). Definition 2 The equilibrium point (x*, y*) T of model (1) is said to be globally expo- nentially stable, if there exist constants l >0and M ≥ 1 such that ||x ( t ) − x ∗ || + ||y ( t ) − y ∗ || ≤ M ( ||φ − x ∗ || + ||ϕ − y ∗ || ) e −λ t for all t ≥ 0, where (x(t), y(t)) T is any s olution of system (1) with initial value (j(s), (s)) T and | |x(t) − x ∗ || = n  i=1 |x i (t ) − x ∗ i |, ||y(t) − y ∗ || = m  j=1 |y j (t ) − y ∗ j |, | |φ − x ∗ || =sup −∞<s≤0 n  i=1 |φ i (s) − x ∗ i |, ||ϕ − y ∗ || =sup −∞<s≤0 m  j =1 |ϕ j (s) − y ∗ j | . Definition 3 A real matrix D =(d ij ) n × n is said to be a nonsingular M-matrix if d ij ≤ 0, i, j = 1, 2, , n, i ≠ j, and all successive principal minors of D are positive. Lemma 1 [38]Let D =(d ij ) n × n with d ij ≤ 0(i ≠ j), then the following state ments are true: (i) D is a nonsingular M-matrix if and only if D is inverse-positive, that is, D -1 exists and D -1 is a nonnegative matrix. (ii) D is a nonsingular M-matrix if and only if there exists a positive vector ξ =(ξ 1 , ξ 2 , , ξ n ) T such that Dξ >0. Lemma 2 [20]For any positive integer n, let h j : R ® Rbeafunction(j = 1, 2, , n), then we have | n ∧ j=1 α j h j (u j ) − n ∧ j=1 α j h j (v j )|≤ n  j=1 |α j |·|h j (u j ) − h j (v j )| , | n ∨ j=1 α j h j (u j ) − n ∨ j=1 α j h j (v j )|≤ n  j =1 |α j |·|h j (u j ) − h j (v j )| for all a =(a 1 , a 2 , , a n ) T , u =(u 1 , u 2 , , u n ) T , v =(v 1 , v 2 , , v n ) T Î R n . 3 Existence and uniqueness of equilibrium point In this section, we will proof the existence and uniqueness of equilibrium point of model (1). For the sake of simplification, let ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˜ I i = m  j=1 ˜ a ij v j + I i + m ∧ j=1 T ij v j + m ∨ j=1 H ij v j , i =1,2, , n, ˜ J j = n  i =1 ˜ b ji u i + J j + n ∧ i=1 ¯ T ji u i + n ∨ i=1 ¯ H ji u i , j =1,2, , m , Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 5 of 18 then model (2) is reduced to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ x i (t )=−a i x i (t − δ i )+ m  j=1 a ij g j (y j (t )) + m ∧ j=1 α ij +∞  0 K ij (s)g j (y j (t − s))d s + m ∨ j=1 ˜α ij +∞  0 K ij (s)g j (y j (t − s))ds + ˜ I i , ˙ y j (t )=−b j y j (t − θ j )+ n  i=1 b ji f i (u i (t )) + n ∧ i=1 β ji +∞  0 ¯ K ji (s)f i (x i (t − s))ds + n ∨ i=1 ˜ β ji +∞  0 ¯ K ji (s)f i (u i (t − s))ds + ˜ J j . (4) It is evident that the dynamical characteristics of model (2) are as same as of model (4). Theorem 1 Under assumptions (H1) and (H2),system(1) has one unique equili- brium point, if the following condition holds, (C1) there exist vectors ξ =(ξ 1 , ξ 2 , , ξ n ) T >0,h =(h 1 , h 2 , , h m ) T >0and positive number l >0such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (λ − a i e λδ i )ξ i + m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (λ)  G j η j < 0, i =1,2, , n , (λ − b j e λθ j )η j + n  i =1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (λ)  F i ξ i < 0. j =1,2, , m. Proof. Let h ( x 1 , , x n , y 1 , , y m ) = ( h 1 , , h n , h 1 , , h m ) T , where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h i = a i x i − m  j=1 a ij g j (y j ) − m ∧ j=1 α ij k ij (0)g j (y j ) − m ∨ j=1 ˜α ij k ij (0)g j (y j ) − ˜ I i , h j = b j y j − n  i=1 b ji f i (x i ) − n ∧ i=1 β ji ¯ k ji (0)f i (x i ) − n ∨ i=1 ˜ β ji ¯ k ji (0)f i (x i ) − ˜ J j for i = 1, 2, , n; j = 1, 2, , m. Obviously, from assumption (H2), the equilibrium points of model (4) are the solutions of system of equations:  h i =0,i =1,2, , n, h j =0,j =1,2, , m . (5) Define the following homotopic mapping: H(x 1 , , x n , y 1 , , y m )=θh(x 1 , , x n , y 1 , , y m ) + (1 - θ)(x 1 , , x n , y 1 , , y m ) T , where θ Î [0, 1]. Let H k (k = 1, 2, , n + m) denote the kth component of H(x 1 , , x n , y 1 , , y m ), then from assumption (H1) and Lemma 2, we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |H i |≥[1 + θ (a i − 1)]|x i |−θ  m j=1  |a ij | +(|α ij | + |˜α ij |)k ij (0)  G j |y j | −θ  m j=1  |a ij | +(|α ij | + |˜α ij |)k ij (0)  |g j (0)|−θ| ˜ I i |, |H n+j |≥[1 + θ(b j − 1)]|y j |−θ  n i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  F i |x i | −θ  n i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  |f i (0)|−θ| ˜ J j | (6) Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 6 of 18 for i = 1, 2, , n, j = 1, 2, , m. Denote ¯ H =(|H 1 |, |H 2 |, , |H n+m |) T , z =(|x 1 |, , |x n |, |y 1 |, , |y m |) T , C =diag(a 1 , , a n , b 1 , , b m ), L =diag(F 1 , , F n , G 1 , , G m ), P =(| ˜ I 1 |, , | ˜ I n , |, | ˜ J 1 |, , | ˜ J m |) T Q =(|f 1 (0)|, , |f n (0)|, |g 1 (0)|, , |g m (0)|) T , A =  |a ij | +(|α ij | + |˜α ij |)k ij (0)  n×m , B =  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  m×n , T =  0 A B 0  , ω =(ξ 1 , , ξ n , η 1 , , η m ) T > 0. Then, the matrix form of (6) is ¯ H ≥ [E + θ ( C − E ) ]z − θ TLz − θ ( P + TQ ) = ( 1 − θ ) z + θ [ ( C − TL ) z − ( P + TQ ) ] . Since condition (C1) holds, and k ij (l), ¯ k j i (λ ) are continuous on [0, δ ), when l =0in (C1), we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −a i ξ i + m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (0)  G j η j < 0, i =1,2, , n , −b j η j + n  i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  F i ξ i < 0. j =1,2, , m. or in matrix form, ( −C + TL ) ω<0 . (7) From Lemma 1, we know that C - TL is a nonsingular M-mat rix, so (C - TL) -1 is a nonnegative matrix. Let Γ =  z =(x 1 , , x n , y 1 , , y m ) T |z ≤ ω +(C − TL) −1 (P + TQ)  , then Γ is nonempty, and from (6), for any z =(x 1 , , x n , y 1 , , y m ) T Î∂Γ, we have ¯ H ≥ (1 − θ)z + θ(C − TL)[z − (C − TL) −1 (P + TQ)] = ( 1 − θ ) [ω + ( C − TL ) −1 ( P + TQ ) ]+θ ( C − TL ) ω>0, θ ∈ [0, 1] . Therefore, for any (x 1 , , x n , y 1 , ,y m ) T Î ∂Γ and θ Î [0, 1], we have H ≠ 0. From homotopy invariance theorem [39], we get deg ( h, Γ ,0 ) =deg ( H, Γ ,0 ) =1 , by topological degree theory, we know that (5) has at least one solution in Γ. That is, model (4) has at least an equilibrium point. Now, we show that the solution of the system of Equations (5) is unique. Assume that (x ∗ 1 , , x ∗ n , y ∗ 1 , , y ∗ m ) T and ( ˆ x 1 , , ˆ x n , ˆ y 1 , , ˆ y m ) T are two solutions of the system of Equations (5), then Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 7 of 18 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a i (x ∗ i − ˆ x i )= m  j=1 a ij [g j (y ∗ j )+g j ( ˆ y j )] +  m ∧ j=1 α ij k ij (0)g j (y ∗ j ) − m ∧ j=1 α ij k ij (0)g j ( ˆ y j )  +  m ∨ j=1 ˜α ij k ij (0)g j (y ∗ j ) − m ∨ j=1 ˜α ij k ij (0)g j ( ˆ y j )  , b j (y ∗ j − ˆ y j )= n  i=1 b ji [f i (x ∗ i ) − f i ( ˆ x i )] +  n ∧ i=1 β ji ¯ k ji (0)f i (x ∗ i ) − n ∧ i=1 β ji ¯ k ji (0)f i ( ˆ x i )  +  n ∨ i=1 ˜ β ji ¯ k ji (0)f i (x ∗ i ) − n ∨ i=1 ˜ β ji ¯ k ji (0)f i ( ˆ x i )  , it follows that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a i |x ∗ i − ˆ x i |≤ m  j=1 |a ij ||g j (y ∗ j )+g j ( ˆ y j )| +| m ∧ j=1 α ij k ij (0)g j (y ∗ j ) − m ∧ j=1 α ij k ij (0)g j ( ˆ y j )| +| m ∨ j=1 ˜α ij k ij (0)g j (y ∗ j ) − m ∨ j=1 ˜α ij k ij (0)g j ( ˆ y j )| , b j |y ∗ j − ˆ y j |≤ n  i=1 |b ji ||f i (x ∗ i ) − f i ( ˆ x i )| +| n ∧ i=1 β ji ¯ k ji (0)f i (x ∗ i ) − n ∧ i=1 β ji ¯ k ji (0)f i ( ˆ x i )| +| n ∨ i =1 ˜ β ji ¯ k ji (0)f i (x ∗ i ) − n ∨ i =1 ˜ β ji ¯ k ji (0)f i ( ˆ x i )|. By using of Lemma 2 and hypothesis (H1), we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a i |x ∗ i − ˆ x i |− m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (0)  G j |y ∗ j − ˆ y j |≤0 , b j |y ∗ j − ˆ y j |− n  i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  F i |x ∗ i − ˆ x i |≤0. (8) Let Z =diag ( |x ∗ 1 − ˆ x 1 |, , |x ∗ n − ˆ x n |, |y ∗ 1 − ˆ y 1 |, , |y ∗ m − ˆ y m | ) , then the mat rix form of (8) is (C -TL)Z ≤ 0. Since C - TL is a nonsingular M-matrix, (C - TL) -1 ≥ 0, thus Z ≤ 0, accordingly, Z = 0, i.e., x ∗ i = ˆ x i , y ∗ j = ˆ y j (i =1,2, , n, j =1,2, , m ) . This shows that model (4) has one unique equilibrium point. According to (3), this implies that system (1) has one unique equilibrium point. The proof is completed. Corollary 1 Under assumptions (H1) and (H2),system(1) has one unique equili- brium point if C - TL is a nonsingular M-matrix. Proof. Since that C - TL is a nonsingular M-matrix, from Lemma 1, there exists a vector ω =(ξ 1 , ξ n , h 1 , , h m ) T > 0 such that (CTL) ω >0,or(-C + TL) ω <0. It fol- lows that Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 8 of 18 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −a i ξ i + m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (0)  G j η j < 0, i =1,2, , n, −b j η j + n  i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (0)  F i ξ i < 0, j =1,2, , m . From the continuity of k ij (l) and ¯ k j i (λ ) , it is easy to know that there exists l > 0 such that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (λ − a i e λδ i )ξ i + m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (λ)  G j η j < 0, i =1,2, , n, (λ − b j e λθ j )η j + n  i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (λ)  F i ξ i < 0, j =1,2, , m . That is, condition (C1) holds. This completes the proof. 4 Exponential stability and exponential convergence rate In this section, we will discuss the global exponential stability of system (1) and give an estimation of exponential convergence rate. Lemma 3 Let a < b ≤ +∞, and u(t)=(u 1 (t), , u n (t)) T Î PC[[a, b), R n ] and v(t)=(v 1 (t), , v m (t)) T Î PC[[a, b), R m ] satisfy the following integro-differential inequalities with the initial conditions u(s) Î PC[(-∞, 0], R n ] and v(s) Î PC[(-∞, 0], R m ]: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ D + u i (t ) ≤−r i u i (t − δ i )+ m  j=1 p ij v j (t )+ m  j=1 q ij +∞  0 |K ij (s)|v j (t − s)ds , D + v j (t ) ≤− ¯ r j v j (t − θ j )+ n  i=1 ¯ p ji u i (t )+ n  i=1 ¯ q ji +∞  0 | ¯ K ji (s)|u i (t − s)ds (9) for i = 1, 2, , n, j = 1, 2, , m, where r i >0,p ij >0,q ij >0, ¯ r j > 0 , ¯ p j i > 0 , ¯ q j i > 0 , i = 1, 2, ,n, j = 1, 2, , m. If the initial conditions satisfy  u(s) ≤ κξe −λ(s−a) , s ∈ (−∞, a] , v(s) ≤ κηe −λ(s−a) , s ∈ (−∞, a], (10) in which l >0,ξ =(ξ 1 , ξ 2 , , ξ n ) T >0and h =(h 1 , h 2 , , h m ) T >0satisfy ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (λ − r i e λδ i )ξ i + m  j=1 (p ij + q ij k ij (λ))η j < 0, i =1,2, , n, (λ − ¯ r j e λθ j )η j + n  i =1 ( ¯ p ji + ¯ q ji ¯ k ji (λ))ξ i < 0, j =1,2, , m . (11) Then  u(t ) ≤ κξe −λ ( t−a ) , t ∈ [a, b) , v(t) ≤ κηe −λ(t−a) , t ∈ [a, b). Proof. For i Î {1, 2, , n}, j Î {1, 2, , m} and arbitrary ε > 0, set z i (t)=( + ε) ξ i e -l (t - a) , ¯ z j (t )=(κ + ε)η j e −λ(t−a ) , we prove that Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 9 of 18  u i (t ) ≤ z i (t )=(κ + ε)ξ i e −λ(t−a) , t ∈ [a, b), i =1,2, , n, v j (t ) ≤¯z j (t )=(κ + ε) η j e −λ(t−a) , t ∈ [a, b), j =1,2, , m . (12) If this is not true, no loss of generality, suppose that there exist i 0 and t* Î [a, b) such that u i 0 (t ∗ )=z i 0 (t ∗ ), D + u i 0 (t ∗ ) ≥˙z i 0 (t ∗ ), u i (t ) ≤ z i (t ), v j (t ) ≤¯z j (t ) (13) for t Î [a, t*], i = 1, 2, , n, j = 1, 2, , m. However, from (9) and (12), we get D + u i 0 (t ∗ ) ≤−r i 0 u i 0 (t ∗ − δ i 0 )+ m  j=1 p i 0 j v j (t ∗ )+ m  j=1 q i 0 j +∞  0 |K i 0 j (s)|v j (t ∗ − s)d s ≤−r i 0 (κ + ε)ξ i 0 e −λ(t ∗ −δ i 0 −a) + m  j=1 p i 0 j η j (κ + ε)η j e −λ(t ∗ −a) + m  j=1 q i 0 j (κ + ε)η j e −λ(t ∗ −a) +∞  0 e λs |K i 0 j (s)|ds =[−r i 0 ξ i 0 e λδ i 0 + m  j =1 (p i 0 j + q i 0 j k i 0 j (λ))η j ](κ + ε)e −λ(t ∗ −a) . Since(11)holds,itfollowsthat −r i 0 ξ i 0 e λδ i 0 +  m j =1 (p i 0 j + q i 0 j k i 0 j (λ))η j < −λξ i 0 < 0 . Therefore, we have D + u i 0 (t ∗ ) < −λξ i 0 (κ + ε)e −λ(t ∗ −a) = ˙z i 0 (t ∗ ) , which contradicts the inequality D + u i 0 (t ∗ ) ≥˙z i 0 (t ∗ ) in (13). Thus (12) holds for all t Î [a, b). Letting ε ® 0, we have  u i (t ) ≤ κξ i e −λ(t−a) , t ∈ [a, b), i =1,2, , n, v j (t ) ≤ κη j e −λ(t−a) , t ∈ [a, b), j =1,2, , m . The proof is completed. Remark 1. Lemma 3 is a generalization of the famous Halanay inequality. Theorem 2 Under assumptions (H1)-(H3), if the following conditions hold, (C1) there exist vectors ξ =(ξ 1 , ξ 2 , , ξ n ) T >0,h =(h 1 , h 2 , , h m ) T >0and positive number l >0such that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (λ − a i e λδ i )ξ i + m  j=1  |a ij | +(|α ij | + |˜α ij |)k ij (λ)  G j η j < 0, i =1,2, , n, (λ − b j e λθ j )η j + n  i=1  |b ji | +(|β ji | + | ˜ β ji |) ¯ k ji (λ)  F i ξ i < 0, j =1,2, , m ; (C2) μ =sup k ∈ N { ln μ k t k −t k−1 } < λ , where μ k =max 1≤i≤n,1≤ j ≤m {1, γ ik , ¯γ jk } , k Î N, then system (1) has exactly one globally exponentially stable equilibrium point, and its exponential convergence rate equals l - μ. Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 10 of 18 [...]... is somewhat artificial, the possible application of our theoretical theory is clearly expressed Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 15 of 18 Example Consider the following impulsive BAM FCNNs with time delays in the leakage terms and distributed delays: ⎧ 2 2 ⎪ ⎪˙ ⎪ xi (t) = −ai xi (t − δi ) +... holds From Theorem 2, the unique equilibrium point (1, 1, 1, 1) T of system (27) is globally exponentially stable, and its exponential convergence rate is about 0.1368 The numerical simulation is shown in Figure 1 and 2 6 Conclusions In this paper, a class of impulsive BAM FCNNs with time delays in the leakage terms and distributed delays has been formulated and investigated Some new criteria on the existence,... supported by the Scientific Research Fund of Sichuan Provincial Education Department under Grant 09ZC057 Authors’ contributions ZX designed and performed all the steps of proof in this research and also wrote the paper KL participated in the design of the study and helped to draft and revise manuscript All authors read and approved the final manuscript Competing interests The authors declare that they have... American Mathematical Society, Providence (1964) doi:10.1186/1029-242X-2011-43 Cite this article as: Zhang and Li: Integro-differential inequality and stability of BAM FCNNs with time delays in the leakage terms and distributed delays Journal of Inequalities and Applications 2011 2011:43 Page 18 of 18 ... uniqueness and global exponential stability of equilibrium point for the networks have been derived by using M-matrix theory and the impulsive delay integro-differential inequality Our stability criteria are delay-dependent and impulse-dependent The neuronal output activation functions and the impulsive operators only need to are Lipschitz continuous, but need not to be bounded and monotonically increasing... 16 Li, X, Cao, J: Delay-dependent stability of neural networks of neutral type with time delay in the leakage term Nonlinearity 23, 1709–1726 (2010) doi:10.1088/0951-7715/23/7/010 17 Balasubramaniam, P, Kalpana, M, Rakkiyappan, R: Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays Math Comput Model 53, 839–853... of the state variable x(t) with time impulses 30 35 Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 17 of 18 Y 1 1.4 Y 2 1.2 1 0.8 0.6 0.4 0.2 0 5 10 15 20 Time: t 25 30 35 Figure 2 Behavior of the state variable y(t) with time impulses convenient The effectiveness of our results has been demonstrated by the. .. min1≤i≤n,1≤j≤m {ξi ,ηj } then we have ||x(t) − x∗ || + ||y(t) − y∗ || ≤ M ||φ − x∗ || + ||ϕ − y∗ || e−(λ−μ)t The proof is completed Remark 2 In Theorem 2, the parameters μk and μ depend on the impulsive disturbance of system (1), and l is actually an estimate of exponential convergence rate of continuous system (2), which depends on the delay kernel functions and system parameters In order to obtain... solutions of BAM neural networks with continuously distributed delays in the leakage terms Nonlinear Anal Real World Appl 11, 2141–2151 (2010) doi:10.1016/j.nonrwa.2009.06.004 15 Li, X, Fu, X, Balasubramaniam, P, Rakkiyappan, R: Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations Nonlinear Anal Real World Appl 11, 4092–4108... precise estimate of the exponential convergence rate of system (1) (or system (2)), we suggest the following optimization problem: (OP) max λ, s.t (C1) holds Zhang and Li Journal of Inequalities and Applications 2011, 2011:43 http://www.journalofinequalitiesandapplications.com/content/2011/1/43 Page 14 of 18 Obviously, for continuous system (2), we can immediately obtain the following corollaries Corollary . Integro-differential inequality and stability of BAM FCNNs with time delays in the leakage terms and distributed delays. Journal of Inequalities and Applications 2011 2011:43. Zhang and Li Journal of Inequalities. Access Integro-differential inequality and stability of BAM FCNNs with time delays in the leakage terms and distributed delays Xinhua Zhang and Kelin Li * * Correspondence: lkl@suse.edu.cn School of. conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM FCNNs with time delays in the leakage terms and distributed delays are obtained. In particular,