Existence and Global Exponential Stability of Equilibrium for Impulsive Cellular Neural Network Models with Piecewise Alternately Advanced and Retarded Argument

and Applied Analysis 3 l, m(k + 1) − l). Then, the identification functionm[(t + l)/m] is equal to mk. If t ∈ I k = [mk − l,mk), then m[(t + l)/m] ≥ t and IDEPCA (4a)-(4b) is an equation with advanced argument. Similarly, if t ∈ I k = (mk,m(k + 1) − l) then m[(t + l)/m] < t and IDEPCA (4a)-(4b) is an equation with retarded argument. Consequently, IDEPCA (4a)-(4b) changes the type of deviation of the argument during the process. In other words, the IDEPCA (4a)-(4b) is of alternately advanced and retarded type. For any solution x(t) = 1 . . . , n T of IDEPCA (4a)-(4b), the model can be summarized as follows:


Introduction
Chua and Yang [1] proposed a novel class of informationprocessing systems called cellular neural networks (CNNs) in 1988.Like neural networks, it is a large-scale nonlinear analog circuit which processes signals in real time.Like cellular automata [2] it is made of a massive aggregate of regularly spaced circuit clones, called cells, which communicate with each other directly only through its nearest neighbors.Each cell is made of a linear capacitor, a nonlinear voltage-controlled current source, and a few resistive linear circuit elements.The key features of neural networks are asynchronous parallel processing and global interaction of network elements.Impressive applications of neural networks have been proposed for various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.For the circuit diagram and connection pattern implementing the CNN, one can refer to [1].The CNN can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems [3].However, it is necessary to solve some dynamic image processing and pattern recognition problems by using delayed cellular neural networks (DCNN) [4][5][6].The study of the stability of CNN and DCNN is known to be an important problem in theory and applications.
On the other hand, in real world, many evolutionary processes are characterized by abrupt changes at certain time.These changes are known to be impulsive phenomena, which are included in many fields such as physics, chemistry, population dynamics, and optimal control.Fundamental theory of impulsive differential equations has been developed in [7].Furthermore, researches of impulsive differential equations have been received much interesting in recent years [8][9][10][11][12][13][14][15][16][17][18].Meanwhile, several kinds of neural networks with impulse have been investigated.In particular, Xu and Yang established the delay differential inequalities with impulsive initial conditions; some new sufficient conditions for global exponential stability of impulsive delay model were obtained [15,16].
Most neural networks can be classified into two types, continuous or discrete.However, many real-world systems and natural processes cannot be categorized into one of them.They display characteristics both continuous and discrete styles.For instance, some biological neural networks in biology, bursting rhythm models in pathology, and optimal control models in economics are characterized by abrupt changes of state.These are the familiar impulsive phenomena.
It is well known that applications of CNN depend crucially on the dynamical behavior of the networks.In these applications, stability and convergence of neural networks are prerequisites.However, in the design of neural networks one is interested not only in the uniform asymptotic stability but also in the global exponential stability, which guarantees a neural network to converge fast enough in order to achieve fast response.In addition, in the analysis of dynamical neural networks for parallel computation and optimization, to increase the rate of convergence to the equilibrium point of the networks and reduce the neural computing time, it is necessary to ensure a desired exponential convergence rate of the networks' trajectories, starting from arbitrary initial states to the equilibrium point which corresponds to the optimal solution.Thus, from the mathematical and engineering points of view, it is required that the neural networks have a unique equilibrium point which is globally exponentially stable.Therefore, the problem of stability analysis has received great attention and many results on this topic have been reported in the literature.See, for instance, [4,9,13,[19][20][21][22][23][24][25][26][27] and references cited therein.
1.1.Piecewise Constant Impulsive Systems.Differential equations with piecewise constant argument (in short DEPCA) are first considered by Shah and Wiener [28] and Cooke and Wiener [29] in the 80s and have been developed by many authors.Applications of DEPCAs are discussed in [30].Theory and practice of DEPCA of general type, have been discussed extensively in [31][32][33][34][35][36][37].Piecewise constant systems exist in widely expanded areas such as biomedicine, chemistry, mechanical engineering, and physics.The systematical studies with mathematical models involving piecewise constant arguments were initiated for solving some biomedical problems.These kinds of equations are similar in structure to those found in certain sequential-continuous models of disease dynamics.In [38], the following system of equations describing the dynamics of the disease for generation  = 1, 2, . . . is investigated: while  (1) (1) =  0 ,  (1) (1 where  is the death rate and  is the horizontal transmission factor.These types of models are special cases of the general form which arise naturally in a number of models of epidemic.DEPCAs usually describe hybrid dynamical systems (a combination of continuous and discrete) and so combine properties of both differential and difference equations.
Impulsive differential equations with discontinuous argument are proposed as an open problem by Wiener [30] in 1994, namely, the impulsive differential equations with piecewise constant argument: IDEPCA.As we know, impulsive differential equations with piecewise constant arguments (in short IDEPCA) are studied in a few papers [8,39,40].

Model Description.
First, let us give a general description of the mathematical model of ICNNs with piecewise alternately advanced and retarded argument: where where  = diag( 1 , . . .,   ),  = (  ) × , and  = (  ) × are constant matrices and  = ( 1 , . . .,   ) is a constant vector.Moreover, the functions  : R   → R  ,  : R   → R  satisfy (  /  ) = (  /  ) = 0 when  ̸ = .To the best of our knowledge, cellular neural network with piecewise constant argument has been developed by few authors, for example, Huang et al.Reference [41] considered first the following cellular neural network with piecewise constant delay: where [⋅] signifies the greatest integer function.Some sufficient conditions of existence and attractivity of almost periodic sequence solution were given for the corresponding discrete-time analogue: In 2010, Akhmet and Yılmaz [8] where () =   if   <  <  +1 ,  ∈ N,  ∈ R + , is an identification function and   > 0,  ∈ N, is a sequence of real numbers.Several sufficient conditions are obtained for the existence and stability of a unique -periodic solution.
In this paper, we for the first time study the dynamic behavior of impulsive cellular neural network models that combine the properties of impulsive differential equations and discrete-time difference equations, that is, the following impulsive cellular neural network with piecewise alternately advanced and retarded argument: The purpose of this paper is to derive some new and simple sufficient conditions for the existence and uniqueness of solutions of the ICNNs with IDEPCA system (5a)-(5b), which is globally exponentially stable.This paper is organized as follows.In Section 2, we establish several criteria for the existence and uniqueness of a unique equilibrium of the ICNNs with IDEPCA system and the equivalence lemma for (5a)-(5b).Here, a new IDEPCA Gronwall-type inequality is very useful.In Section 3, we derive some sufficient conditions which ensure that a unique equilibrium of the ICNNs with IDEPCA system (5a)-(5b) is globally exponentially stable.
In Section 4, two illustrative examples and the numerical simulations are given to demonstrate the effectiveness of our results.The conclusions are drawn in Section 5.

Existence and Uniqueness Theorems
In this section, sufficient conditions that govern the network parameters and the activation functions are established for the existence of a unique equilibrium state of the impulsive cellular neural network models (5a)-(5b).

Preliminaries and Definition.
In this section, we will focus our attention on some preliminary results which will be used in the existence and uniqueness of solutions of the ICNNs with IDEPCA system (5a)-(5b).
For the sake of convenience, two of the standing assumptions are formulated below.
First, we prove the existence and uniqueness of solutions of IDEPCA system (5a)-(5b).A natural extension of the original definition of a solution of DEPCA [28][29][30]42] allows us to define a solution of IDEPCA system.To study nonlinear IDEPCA system, we will use the approach based on the construction of an equivalent integral equation.Let us give the following proposition.

) in the sense of Definition 1 if and only if it is a solution of the integral equation
In particular, one has the following integral equations: for  = 1, . . ., ,  ∈ R + , The proof of Proposition 2 is almost identical to the verification in [7] with slight changes which are caused by the piecewise constant argument.
In the next, we give the following lemma about IDEPCA integral inequality of Gronwall type, which is one of the most important auxiliary results of the present paper.Lemma 3. Let  : R → [0, ∞) be a function such that  is continuous with possible points of discontinuity of the first kind at  =  − ,  ∈ N, and  1 ,  2 are nonnegative real constants satisfying Suppose that for  ≥  the inequality holds.Then for  ≥ ,  () ≤  () Proof.Call V() the right member of (16).So V() = (),  ≤ V, and V is a piecewise differentiable and nondecreasing function and, by (16), it satisfies ∈ N and for any  ≥  with ,  ∈ With  =  and  =  −  in (21) for  ∈   , since V is a nondecreasing function, we get Considering the particular case  =   and taking V(  ) = (  ) and  ≤ V, by ( 15) and ( 22), estimate (19) follows.Take now in ( 21)  ∈   and  =  −  to obtain because V is a nondecreasing function.Now, we can apply the classical Gronwall's Lemma to get By the impulsive effect (20), we have From ( 25), recursively we obtain using V() = () and ( 19); then we give (17) and (18).The proof is complete.This IDEPCA inequality of Gronwall type seems to be new.
In the following theorem, we obtain sufficient conditions for the existence of a unique equilibrium,  * = ( for any two vectors , V ∈ R  implying that the mapping  : R  → R  is a global contraction on R  endowed with the supremum norm.Hence, there is a unique fixed point  * ∈ R  that satisfies ( * ) =  * (i.e.,   ( * ) =  *  for  = 1, . . ., ).This point defines the unique equilibrium state of the impulsive cellular neural network models (5a)-(5b).The proof is now complete.

Global Exponential Stability of Equilibrium
The existence and stability of a unique equilibrium state are usually a requirement in the design of cellular neural network models for various applications, particularly when there are destabilizing agents such as retarded arguments and impulses.However, even if the unique stable state exists, these agents may affect the convergence speed of the network, which in turn can downgrade the performance of the network in applications that demand fast computation in real-time mode.Thus, exponential stability is usually desirable for an impulsive network, and sufficient conditions for the global exponential stability of the unique equilibrium state  * of the ICNNs with IDEPCA system (5a)-(5b) are obtained in this section.
It is clear that the stability of the zero solution of ( 36) is equivalent to that of the equilibrium  * of the ICNNs with IDEPCA system (4a)-(4b).Therefore, we restrict our discussion to the stability of the zero solution of (36).
First of all, we give the following definition and lemma, which will be used in the proof of the stability of the zero solution for the ICNNs with IDEPCA system.( Notice that (L) implies that for any finite  ∈ [, ∞).Hence, by Lemma 3 of IDEPCA Gronwall's inequality implies Then, we have and the statement (41) follows.
The following result will show sufficient conditions for the global exponential stability of the unique equilibrium of the ICNNs with IDEPCA system (5a)-(5b).Theorem 6,(42) and

Theorem 9. If the assumptions of
are satisfied, then the unique equilibrium  * of the ICNNs with IDEPCA system (5a)-(5b) is globally exponentially stable.
Remark 10.To the best of the author's knowledge, this is the first time we investigate impulsive cellular neural network models with piecewise alternately advanced and retarded argument in equilibrium case.Sufficient conditions are gained for the existence and exponential stability of a unique equilibrium of the ICNNs with IDEPCA system.And our results can be extended to a unique equilibrium of the CNNs with DEPCA system.See Corollaries 11-12.Our results about exponential stability of a unique equilibrium of the ICNNs with IDEPCA system may give some insight into the application of neural networks.
As immediate corollaries of Lemma 8 and Theorem 9, the following results without impulsive effects are true.Remark 13.In [41], authors investigated discrete-time cellular neural network without impulsive effects in almost periodic case.Simple sufficient conditions are gained for a unique almost periodic sequence solution which is globally attractive.When  = 1,  = 0, this conclusion of Corollary 12 cannot be derived by applying the corresponding stability result for cellular neural networks given in the literature [41] with   ,   ,   , and   being constant coefficients.

Examples and Simulations
In ) where  * 1 = 1.943,  * 2 = 1.57.One can check that the point  * = ( * 1 ,  * 2 )  satisfies the algebraic system approximately.And it is clear that   ( *  ) = 0 for  = 1, 2. By simple calculation, we can see that  * = 0.9, ) , By Theorem 6, we know that the ICNNs with IDEPCA system (61) have a unique equilibrium.(63) From Theorem 9, the ICNNs with IDEPCA system (61) have the unique equilibrium  * which is globally asymptotically stable and all other solutions of the IDEPCA system (61) converge exponentially to it as  → ∞.The numerical simulations, showing the convergence of the unique equilibrium  * of the ICNNs with and without impulses (61), are given in Figures 2(a) and 2(b).

Conclusions
This is the first time that impulsive differential equations with alternately advanced and retarded argument have been applied to the model of cellular neural network models, and this paper has provided sufficient conditions guaranteeing the existence, uniqueness, and global exponential stability of the unique equilibrium of the impulsive cellular neural network models for the considered system based on a new IDEPCA integral inequality of Gronwall type and fixed point theorem.
In addition, our method gives new ideas not only from the modeling point of view but also from that of theoretical opportunities since the impulsive cellular neural network model equation involves piecewise constant arguments of both advanced and delayed types.The obtained results could be useful in the design and applications of impulsive cellular neural network models.Furthermore, the examples with numerical simulations are given to show the effectiveness of the proposed method and results.