Existence and Global Stability of a Periodic Solution for Discrete-Time Cellular Neural Networks

A novel sufficient condition is developed to obtain the discrete-time analogues of cellular neural network CNN with periodic coefficients in the three-dimensional space. Existence and global stability of a periodic solution for the discrete-time cellular neural network DT-CNN are analysed by utilizing continuation theorem of coincidence degree theory and Lyapunov stability theory, respectively. In addition, an illustrative numerical example is presented to verify the effectiveness of the proposed results.


Introduction
Cellular neural networks CNNs are the basis of both discrete-time cellular neural networks DT-CNNs 1 and the cellular neural networks universal machine CNNs-UM .The dynamical behaviour of Chua and Yang cellular neural network CY-CNN is given by the state equation where I, u, y, and x denotes input bias, input, output, and state variable of each cell, respectively.N r ij is the t-neighbourhood of cell C k, l as N r ij {C k, l | max{|k − i|, |l − j|} ≤ r}, i and j denote the position of the cell in the network, k and l denote the position of the neighbour cell relative to the cell in consideration.B is the nonlinear weights template matrices for input feedback and A is the corresponding template matrices for the outputs of neighbour cells.Non-linearity means that templates can change over time.
A large number of cellular neural networks CNNs models have appeared in the literature 2-4 , and these models differ in cell complexity, parameterization, cell dynamics, and network topology.Various generalizations of cellular neural networks have attracted attention of scientific community due to their promising potential for tasks of classification, associative memory, parallel computation 5-9 , pattern recognition, computer vision, and solving any optimization problem 10-13 .Such applications rely on the existence of equilibrium points and the qualitative properties of cellular neural networks.
Discrete-time cellular neural networks DT-CNNs have been studied both in theory and applications.Previous results introduced many properties of DT-CNN in the two dimensional plane.For instance, 14 has been successfully applied to investigate the discretetime analogues of cellular neural network CNN with variable coefficients in the twodimensional plane.However, three-dimensional structure is more accurate, specific, and closer to real structures of CNN.Based on the above discussion, this paper proposes some effective results of DT-CNN in the three-dimensional space.
Motivated by the constructing of continuous system 1.1 , the discrete analogue of the system 1.1 is considered as follows:

1.2
For any h > 0, the discrete-time analogues 1.2 converge to the continuous-time system 1.1 will be provided.Without loss of generality, 1.2 can be substituted in the DT-CNNs model:

1.3
Then, the spatial structure with respect to 1.3 is shown in Figure 1, where r C k ijh l ijh , and Θ will be denoted by the proof of Theorem 3.1 in Section 3.
The rest of the paper is organized as follows: in Section 2, system description and preliminaries are developed in detail and some definitions, assumptions, and lemmas are stated.Section 3 gives sufficient conditions for a periodic solution for DT-CNN in threedimensional space by utilizing continuation theorem of coincidence degree theory.Section 4 proposes global stability of a periodic solution for the DT-CNN.A numerical simulation is given to show correctness of our analysis in Section 5 and concluded in Section 6. x milh (n + 1) x milh (n + 1) x mimj (n + 1) x ihjh (n + 1) x lhlh (n + 1) x lhlh (n + 1) x lhmj (n + 1) Figure 1: Spatial structure with respect to 1.3 .

System Preliminaries and Description
Consider the following model which is equivalent to the 1.3 : ϕ ij n N h .Throughout the paper, the following definitions and lemmas will be introduced.Definition 2.1 Fredholm operator .Let X and Y be a Banach space, an operator L is called Fredholm operator if L is a bounded linear operator between X and Y whose kernel and cokernel are finite-dimensional and whose range is closed.Equivalently, an operator L : X → Y is Fredholm if it is invertible modulo compact operator, that is, if there exists a bounded linear operator S : Y → X such that Id X − SL, Id Y − LS are compact operators on X and Y, respectively, where Id X and Id Y are the identity operator.Definition 2.2 L-compact .An operator N will be called L-compact on Ω if the open bounded set QN Ω is bounded and K p I −Q N : Ω → X is compact, where K p is the inverse operator of N. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
The index of a Fredholm operator is ind L dim Ker L-codim Im L, then operator L will be called a Fredholm operator of index zero if dim Ker L codim Im L < ∞ and Im L is closed in Y. Then a following abstract equation in Banach space X is defined by Lx λNx.

2.2
Let L : Dom L ⊂ X → Y be linear operator, and N : X → Y be a continuous operator.If L is a Fredholm operator of index zero, there must exist continuous projectors P : X → X and Q : Y → Y, such that:

2.3
In other words, L| Dom L∩Ker P : Dom L ∩ Ker P → Im L is invertible, and the inverse of the operator L is denoted by K p .Lemma 2. 3 Gaines and Mawhin 15 .Let X be a Banach space, L be a Fredholm operator of index zero, and let N : Ω → X be L-compact on Ω, Ω ⊂ X, where Ω is an open bounded set, suppose: i Lx / λNx, for any x, λ ∈ ∂Ω ∩ Dom L × 0, 1 ; ii QNx / 0, for any x ∈ ∂Ω ∩ Ker L; iii deg JQN, Ω ∩ Ker L, 0 / 0.

2.4
Then Lx Nx has at least one solution in Dom L ∩ Ω. Proof.Assuming a and b are some certain non-negative vectors, β is a positive constant, then Thus, the proof of Lemma 2.4 is completed.
Assumption 2.5.A k ijh l ijh , B k ijh l ijh , I ij i 1, . . ., m i , j 1, . . ., m j , h 1, . . ., m h are Nperiodic sequence of Z 0 .For the sake of convenience, we use the following notations: For each operator P : N r ijh → N r ijh and any s s u, v, w , t t i, j, h ∈ N r ijh , such that: where o ijh is the spherical centre of N r ijh with a radius length r, m ijh max{|m i |, |m j |, |m h |}, then it is easy to obtain:

Existence of a Periodic Solution with respect to 2.1
In many cases, many proposed results are not ideal and therefore it is necessary to formulate a novel and effective result for DT-CNN in the three-dimensional space.Can we obtain the result about the existence and stability of a periodic solution for DT-CNN in threedimensional space?This is the topic we wish to address in this paper.The aim of the present work is to develop a strategy to determine the existence and global stability of a periodic solution with respect to 2.1 in the three-dimensional space.Consequently, we processed with the following result.

has at least one N-periodic solution.
Proof.In this section, by means of using Mawhin's continuation theorem of coincidence degree theory, we will study the existence of at least one periodic solution with respect to 2.1 , for convenience, some following notations will be used: . ., m j }, and y N ⊂ X Y be the subspace of all N-periodic sequence; equip it with the norm x 2 1 is a Cauchy sequence in N r ijh and o ijh is the spherical centre of N r ijh , d x, y max{|x − y| : x ∈ X}.By utilizing the meaning of N r ijh and Bolzano-Weierstrass theorem Each bounded sequence in R n has a convergent subsequence, here R m i m j ⊂ R n , dim R m i m j < ∞ , it is easy to know that X, • is a Banach space. where x n , for all n ∈ I N .Then we will learn that dim Ker L codim Im L < ∞, it is easy to prove that L is a bounded linear operator, P and Q are two continuous operators such that Ker

A k m i m j h l m i m j h y k m i m j h l m i m j h t B k m i m j h l m i m j h u k m i m j h l m i m j
where δ β h , n, N is a constant, which is only depended on variables h, n, and N.
Obviously, employing the Lebesgue's convergence theorem, we can easily learn that QN Ω is bounded, K p I − Q N Ω is compact for any open bounded set Ω ⊂ X ⊂ N r ijh by using Ascoli-Arzela's theorem A subset F of C X is compact if and only if it is closed, bounded and equi-continuous .Thus, N is L-compact on a closed set Ω with any open bounded set Ω ⊂ X ⊂ N r ijh .
Suppose that x n x 11 n , . . ., x 1m j n , . . ., x m i 1 n , . . ., x m i m j n T ∈ X is a solution with respect to 2.1 , for certain λ ∈ 0, 1 .Then the following equation can be derived by 2.2 :

3.7
Then, the following results can be derived by utilizing 3.7 : where Therefore, the solution with respect to 2.1 is bounded for certain λ ∈ 0, 1 .In other words,

3.9
Then the open bounded set Ω is presented as follows:
In Figure 2, the nonlinear weights template matrices B and the boundary of Ω are shown, respectively.Then for any two dimensional plane of any spherical neighbourhood is denoted.Thus, for any x ∈ ∂Ω ∩ Ker L ∂Ω ∩ R m i m j , Ker L {x {x n } ∈ y N ⊂ X : x n c ∈ R m i m j , n ∈ I N }, it is easy to learn that x is a constant vector in R m i m j with x Θ; Thus, we have

QNx
where Thus for any x ∂Ω ∩ Ker L, QNx / 0, this proves the condition ii in Lemma 2. 3.

A k ijh l ijh y k ijh l ijh n B k ijh l ijh u k ijh l ijh
I ij n 0.

3.14
Equivalently, 3.14 can be written as the following form:

3.15
Combining 3.12 and 3.15 , the following results are obtained: Thus, the following result is derived by calculating the 3.16 :

has at least one N-periodic solution.
Proof.Similar to the proof of Theorem 3.1, so it is omitted.

Globally Stability of a Periodic Solution with respect to 2.1
The existence of a periodic solution for the system 2.1 is derived in the Theorem 3.1.Then global stability of a periodic solution with respect to 2.1 in the three-dimensional space is presented in the following.Theorem 4.1.Suppose that Assumptions 2.5 and 2.6 hold, and the following condition holds: where i 1, 2, . . ., m i , C y kl ,m j and m h are positive constants, then the periodic solution with respect to 2.1 is global stability.
Proof.It follows from the Theorem 3.1 that 2.1 has at least a periodic solution, without loss of generality, the periodic solution can be described by: Then we can define the following formula: Now, we show that the a periodic solution x * n is globally stable, and the following inequality is obtained by utilizing 2.1 and 4.3 :

4.4
We design the following Lyapunov-type sequence V n by Then, we can calculate the ΔV n by combining 2.1 and 4.5 :

4.6
Thus, it is easy to obtain V n ≤ V 0 by the meaning of the 4.6 , and furthermore, where d s Obviously, from the proof of Theorem 4.1, the globally stable of a periodic solution with respect to 2.1 is derived.Then, existence and global stability of a periodic solution for DT-CNNs are obtained by utilizing the conditions of the proposed theorems in an arbitrary diameter plane of a convex space.Thus the proof of Theorem 4.1 is completed.

Numerical Simulation
In this section, we give an example to show the effectiveness and improvement of the derived results.Consider the following continuous cellular neural networks:  3.
From Figure 3, it is easy to know that a 1/4 -periodic solution of the continuous cellular neural networks is globally stable.Compared to the system 5.1 , we design the discrete-time analogue of the continuous cellular neural network as follows: x 0 x 11 0 , x 12 0 , x 21 0 , x 22 0 , x 31 0 , x 32 0 T The derived results of this paper are verified by the following steps.
1 According to the illustrations of the neighbourhood distance r for cell C k, l C k ijh l ijh which is given by N r ijh function, and by 3.9 and 3.10 , the exact values of distance r and Ω are illustrated as:

5.5
Thus, the subset Ω of function N r ijh is derived by the following: 2 We will verify the condition of Theorem 3.1 if we want to utilize Theorem 4.1.After strictly calculating the condition of Theorem 3.1, it is easy to obtain that the function g n , n ∈ I N ; therefore, the condition of the Theorem 3.1 is critically satisfied as well.
3 According to 4.1 , the condition of the Theorem 4.1 will be derived as follows: Then state trajectories of neurons x 11 , x 12 , x 21 , x 22 , x 31 , x 32 are shown in Figures 4 and 5.
From Figures 4 and 5, we can learn that all the periodic solution converges to a unique a 1/4h-periodic solution, then the DT-CNN 2.1 has a globally stable 1/4h-periodic solution.Thus, all conditions of Theorems 3.1 and 4.1 are strictly satisfied; therefore all conditions of proposed theorems are critically verified.

Conclusions
Existence and global stability are important dynamical properties in CNN.In this paper, we consider the discrete-time analogues of CNN with periodic coefficients and obtain some new

Lemma 2 . 4 .
If a and b are some certain nonnegative vectors, then there exists a positive constant β, such that ab ≤ β/2 a 2 1/2β b 2 .

Figure 2 :
Figure 2: Input B-template and the boundary of Ω.
, Figure 3: −4.836, 8.863, −3.683, 6.386, −4.836 T .Then, state trajectories of x ij i 1, 2, 3, j 1, 2 are denoted in Figure 5.386, −4.836, 8.863, −3.683, 6.386, −4.836 T , i21 −0.08 0.06 cos 8πnh , i 1, State trajectories of the neurons x 11 and x 12 , h =1/4.State trajectories of the neurons x 21 and x 22 , h =1/4.State trajectories of the neurons x 31 and x 32 , h =1/4.State trajectories of neurons x 11 , x 12 , x 21 , x 22 , x 31 , x 32 h 1/4 .resultsfor the DT-CNN in the three-dimensional space.Comparisons between our results and the previous results have also been made.And it has been demonstrated that our criteria are more general and effective than those reported in the literature.State trajectories of the neurons x 31 and x 32 , h =1/2.State trajectories of the neurons x 21 and x 22 , h =1/2.State trajectories of the neurons x 11 and x 12 , h =1/2.State trajectories of neurons x 11 , x 12 , x 21 , x 22 , x 31 , x 32 h 1/2 .Program of Higher Education of China under Grant 20100092110020, and Scientific Research Foundation of Graduate School of Southeast University under Grant YBJJ1215.