Pseudo Almost Periodic Sequence Solutions of Discrete Time Cellular Neural Networks

In this paper we discuss the existence and uniqueness of a k-pseudo almost periodic sequence solutions of a discrete time neural network. We give several sufficient conditions for the exponential and global attractivity of the solution.


Introduction
The theory of almost periodic functions was introduced by Harald Bohr during 1924-1926.In his study of Dirichlet series he developed the notion of a uniformly almost periodic functions.Later, Bochner extended Bohr's theory to general abstract spaces.Almost periodic functions have been widely treated by Favard, Levitan [1] and Besicovich [2] in their monographs.Amerio [3] extended certain results of Favard and Bochner to differential equations in abstract spaces.The concept of pseudo almost periodicity is a natural genelization of almost periodicity.The theory of pseudo almost periodicity was first treated by Zhang [4] around 1990.The existence and uniqueness of pseudo almost periodic solutions of differential equations have been of great interest to mathematicians in the past few decades.
A cellular neural network is a nonlinear dynamic circuit consisting of many processing units called cells arranged in two or three dimensional array.This is very useful in the areas of signal processing, image processing, pattern classification and associative memories.Hence the application of cellular networks is of great interest to many researchers ( [5][6][7][8] have dealt with the global exponential stability and the existence of a periodic solution of a cellular neural network with delays using the general Lyapunov functional).Many authors have established the almost periodic solutions of cellular neural networks ( [9,10] and references cited therein).The discrete analogues of continuous time cellular network models are very important for theoretical analysis as well as for implementation.
Thus it is essential to formulate a discrete time analogue of continuous time network.A most acceptable method is to discretize the continuous time network.For detailed analysis on the discretization method the reader may consult Mohamad [8], Stewart [11], Broomhead and Iserles [12].Huang et al. [13] have considered the following model of neural network with piecewise constant arguments The authors have proved the existence of an almost periodic sequence solution for the following discrete time analogue Huang, Xia and Wang in [14] have considered the following network model with the piecewise constant arguments in the following equation, where i = 1, 2, . . ., m.The authors in [14] have proved the existence and uniqueness of a k-almost periodic sequence solution of the discrete analogue of (1), and also shown the exponential attractivity of the solution.
In this paper we study the problem of existence, uniqueness and exponential attractivity of a k-pseudo almost periodic solution of the following differential equation, where x i (t) the potential of the cell i at time t, f i is the nonlinear output function, b ij and c ij denote the strengths of connectivity between the cells i and j at the instants t and t − τ ij , respectively.We have τ ij the time delay required in processing and transmitting a signal from j-th cell to the i-th cell.We denote the i-th component of an external input source from outside the network to the cell i by I i .

Preliminaries
We consider a continuous time neural network consisting of m interconnected cells described by the following system of delay differential equations for i ∈ {1, 2, . . ., m}, and t > 0. A discrete analogue of (3) can be written as, for i = 1, 2, . . ., m, and [•] denotes the greatest integer function and k > 0 the transmission step size.Throughout the paper we impose the following conditions, Assumptions: is relatively dense in R.That is there exists a positive number l ǫ such that any interval of the length l ǫ contains at least one point of T (f, ǫ).
The set of all almost periodic functions from R to R are denoted by AP .Denote Definition 2. A continuous function f : R → R is said to be pseudo almost periodic if it can be written as f = f 1 + f 2 , where f 1 ∈ AP and f 2 ∈ AP 0 .
The set of all such functions is denoted by P AP .Now we have a similar definition for pseudo almost periodic sequence.
That is there exists a positive integer l ǫ such that any interval with the length l ǫ contains at least one point of T (x, ǫ).
The class of all almost periodic sequences is denoted by AP S. The set of all such sequences is denoted by P AP S.
is relatively dense set in kZ.That is there exists a positive integer l ǫ such that any integer interval with length l ǫ contains at least one point of T (x, ǫ).
The set of all such sequences is denoted by AP S k .
Definition 8.A real sequence x : kZ → R is said to be k-pseudo almost periodic sequence if it can be written as x = x 1 + x 2 , where x 1 ∈ AP S k and x 2 ∈ AP 0 S k .
The set of all such sequences is denoted by P AP S k .Denote P AP S m k the set of all x = (x 1 , x 2 , . . ., x m ) in which every component is k-pseudo almost periodic sequence, that is x i ∈ P AP S k for i = 1, 2, . . ., m.
Using the discretization scheme one can have the following difference equations for equation ( 4), where i = 1, 2, . . ., m and n ∈ Z. Define the followings, ds, Fixing the k ij as k * and using above notations, equation ( 6) can be written as, for i = 1, 2, . . ., m, where x k i (n) = x i (nk).Denote: 8) is said to be globally attractive if for any other solution y(ν) = (y 1 (ν), . . ., y m (ν)) T of (8), we have The following Lemmas are easy to verify: ) are relatively dense, where If g i ∈ P AP S for i = 1, 2, then we can decompose g i in two components g i1 and g i2 , first one is almost periodic sequence and other is in AP 0 S. Lemma 2. Suppose that g i ∈ P AP S for i = 1, 2, then g i |kZ ∈ P AP S k , that is for all ǫ > 0, For x ∈ P AP S k , denoting x k (n) = x(nk) for all n ∈ Z, the following properties are true for sequence x k : • x ∈ P AP S k if and only if x k ∈ P AP S.
• f ∈ P AP , then f |Γ k ∈ P AP S k .
• Any x ∈ P AP S k is bounded.

Pseudo almost periodic solutions
Proof.Because a i is pseudo almost periodic, one have a i = a i1 + a i2 where a i1 ∈ AP and a i2 ∈ AP 0 .For any τ ∈ Z, we have a i1 is almost periodic so given ǫ > 0, i = 1, 2, . . ., m, we have the set As we know that a i2 ∈ AP 0 , we have Also because a i2 is bounded, we get Because I i ∈ P AP S, thus we have I i = I i1 + I i2 , where I i1 ∈ AP S and I i2 ∈ AP 0 S.For I i1 ∈ AP S, we have for any t ∈ Z, because of almost periodicity of I i1 (t) and a i1 (t).Now we show that F i2 ∈ AP 0 S.
Thus we get the following Hence, we conclude that F i ∈ P AP S. By similar argument one can show that D ij and E ij are pseudo almost periodic.
Proof.This Lemma is a direct consequence of Theorem 3.1 of [14].
A brief summary is as follows: One can easily observe that the relation where Considering the following difference equations where xk i (0) = x k i (0) one get the following estimate Consider the following difference equations Lemma 5.Under assumption (A1), there exists a k-pseudo almost periodic sequence solution of (13).
Proof.Using the induction argument, one obtain Consider the sequence Since Thus the sequence xk i (n) is well defined.It is easy to verify that

Hence the sequence xk
Thus xk i is a k almost periodic sequence if I i and a i are k almost periodic sequences.That is for I i1 and a i1 this is an almost periodic sequence, denote it by xk i1 .Consider Now we can easily observe that Thus above inequality becomes For I i2 we have −nk One can easily observe that every finite sum of these integral when divided by 2n and passing limit as n → ∞ is zero.Thus we have Therefore we get that xi = xi1 + xi2 is a k-pseudo almost periodic sequence solution of equation ( 13).
Theorem 1. Suppose assumptions (A1), (A2) holds.There exists a unique k-pseudo almost periodic sequence solution of (8) which is globally attractive, if Proof.Denote a metric d : Now define a mapping F : P AP S m k → P AP S m k by F x = y, where such that F i x = y i and y i = {y k i (n)}.Define where xk i is k-pseudo almost periodic sequence solution of (13).Using Lemma 3 and assumption (A2), one can observe that F maps k-pseudo almost periodic sequences into k-pseudo almost periodic sequences.Now denote Define the set Also we have, Thus we conclude that F x ∈ BA * k .For x, y ∈ BA * k , we have Hence F is a contraction.It follows that equation ( 8) has a unique k pseudo almost periodic sequence x which satisfies Let y be any sequence satisfying equation (8).Consider Taking modulus both side one have By induction we have Hence Thus x is a unique k-pseudo almost periodic sequence solution of (8) which is globally attractive.
Theorem 2. Suppose the assumptions (A1), (A2) holds.There exists a unique k-pseudo almost periodic sequence solution of (8) which is exponentially attractive, if Proof.Let x be the unique k-pseudo almost periodic sequence solution and y be a arbitrary sequence solution of (8).Denote Φ(n) = x k (n) − y k (n), we get ), then we have Using induction we have, Taking the norm both side we have, Defining Assuming that we have and V (0) = z(0) for n > 0. It is easy to note that Thus we have Thus y converges exponentially to the unique k-pseudo almost periodic sequence solution x if For k = 1 we can easily observe that the discrete analogue of (3) is as follows, From Theorem 1 and Theorem 2 it is easy to generalize the results for pseudo almost periodic sequence solution.
Theorem 3. Suppose (A1) and (A2) holds.There exists a unique pseudo almost periodic sequence solution of (24) which is globally attractive, if Theorem 4. Suppose (A1) and (A2) holds.There exists a unique pseudo almost periodic sequence solution of (24) which is exponentially attractive, if

Example
We consider the following pseudo almost periodic cellular neural network, where It is easy to verify that these functions are pseudo almost periodic.Consider the following nonlinear activation functions f i (x) = f (x) = tanh(x), i = 1, 2. The discrete time analogues are x 1 (n + 1)k = C 1 (n)x 1 (nk) + D 12 tanh x 2 (nk) x 2 (n + 1)k = C 2 (n)x 2 (nk) + D 21 tanh x 1 (nk) We have our L i = 1, i = 1, 2, and the constants are as follows,

Definition 4 .Definition 5 .
A real sequence x : Z → R is said to be in AP 0 S if lim A real sequence x : Z → R is said to be pseudo almost periodic sequence if it can be written as x = x 1 + x 2 , where x 1 ∈ AP S and x 2 ∈ AP 0 S.
s) ds ∈ P AP S k .
s)| ds + . . . .As we know that I i2 ∈ AP 0 S and bounded, we get lim