Global Exponential Stability of a Unique Almost Periodic Solution for Neutral-Type Cohen-Grossberg Neural Networks with Time Delays

This paper is concerned with the problem of the existence, uniqueness, and global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks with time delays. Based on fixed point theory and Lyapunov functional, several sufficient conditions are established for the existence, uniqueness, and global exponential stability of almost periodic solution for the above system. Finally, an example and numerical simulations are given to illustrate the feasibility and effectiveness of our main results.


Introduction
In recent years, the Cohen-Grossberg neural networks [1] have been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, optimization, and signal and image processing.It is well known that studies on Cohen-Grossberg neural networks not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior and almost periodic oscillatory properties.In applications, if the various constituent components of the temporally nonuniform environment is with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions.Therefore, if we consider the effects of the environmental factors, almost periodicity is sometimes more realistic and more general than periodicity.In addition, the (almost) periodic oscillatory behavior is of great interest; it has been found that applications in learning theory [2], which is motivated by the fact that learning usually requires repetition.And in pattern recognition or associative memory, the existence of stable almost periodic encoded patterns (including equilibria or periodic orbits) is an important feature [3,4].Hence, it is of prime importance to study the existence and stability of (almost) periodic solutions for Cohen-Grossberg neural networks with delays.We refer the reader to [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein.
On the other hand, owing to the complicated dynamic properties of the neural cells in the real world, the existing neural network models in many cases can not characterize the properties of a neural reaction process precisely.It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions.This new type of neural networks is called neutral neural networks or neural networks of neutral type.The motivation for us to study neural networks of neutral type comes from three aspects.First, based on biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process.Second, in view of electronics, it has been shown that neutral phenomena exist in large-scale integrated (LSI) circuits.Last, the key point is that cerebra can be considered as a super LSI circuit with chemical reactivity, which reasonably implies that the neutral dynamic behaviors should be included in neural dynamic systems [19].To the best of our knowledge, the problem of global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks (see [20][21][22][23][24]) has not been fully investigated in the literature.
In this papers we consider the following neutral-type Cohen-Grossberg neural networks with time delays: where   denotes the state variable associated to the th neuron,   represents an amplification function,   is an appropriately behaved function;   ,   , and   denote the normal and the delayed activation function;   ,   , and V  denote the connection strengths of the th neuron on the th neuron at time , respectively;   ≥ 0 and ]  ≥ 0 are the delays caused during the switching and transmission processes;   denotes external input to the th neuron at time , ,  = 1, 2, . . ., .
We list some assumptions which will be used in this paper.
The organization of this paper is as follows.In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections.In Sections 3 and 4, by using a fixed point theorem and constructing suitable Lyapunov functional, we obtain some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of almost periodic solution of system (1).Finally, an example and numerical simulations are given to illustrate that our results are feasible.

Preliminaries
First of all, we shall transform system (1) and state some notations, which will be used in later sections.
By ( 2 ), ( 3 ), and the definition of ℎ −1  , we obtain that   (, ℎ −1  (  )) is strictly monotone increasing about   .Hence,   (,   ) is unique for any   and continuous about   ; moreover, From the definition of ℎ −1  , using Lagrange mean-value theorem, one gets The existence and global exponential stability of almost periodic solution for system (1) are equivalent to the existence and global exponential stability of almost periodic solution for system (5), respectively.So, we investigate the existence and global exponential stability of almost periodic solution for system (5).
Let  := max 1≤≤ { +  , ] +  }.The initial conditions associated with system (5) are of the form Now, let us state the following definitions and lemmas, which will be useful in proving our main result.
Definition 1 (see [25]). ∈ (R, R  ) is called almost periodic, if for any  > 0, it is possible to find a real number  = () > 0, for any interval with length (), there exists a number  = () in this interval such that ‖(+)−()‖ < , for all  ∈ R. The collection of those functions is denoted by (R, R  ).
Then X is a Banach space with the norm ‖ ⋅ ‖ X .
For given positive constant , define an open bounded subset Ω  in X by By Lemmas 3 and 4, system (5) has a unique almost periodic solution which can be expressed as follows: where where  = 1, 2, . . ., , for all  ∈ R.
Let the map  : X → X be defined by where  = 1, 2, . . ., .Therefore, Ω  is a family of uniformly bounded and equicontinuous subsets.Using Arzela-Ascoli theorem, we obtain that Ω  is relatively compact.Hence,  is completely continuous.The proof of this lemma is complete.

Existence of Almost Periodic Solution
In this section, we study the existence of almost periodic solutions of system (1).

Global Exponential Stability of Almost Periodic Solution
Theorem 9. Assume that ( where  0 is defined as that in Theorem 8. Then system (1) has a unique almost periodic solution, which is globally exponentially stable.
By ( 6 ), there exists a small enough positive constant  such that Define where In view of system (5), we have Thus, the almost periodic solution of system ( 5) is globally exponentially stable.That is, the corresponding almost periodic solution of system (1) is globally exponentially stable.

An Example and Numerical Simulations
Example 1.Consider the following neutral-type Cohen-Grossberg neural networks: where  = 1, 2, () = () = () = 0.1, Then system (45) has a unique almost periodic solution, which is globally exponentially stable.It is easy to verify that all the conditions of Theorem 9 are satisfied and that the result follows from Theorem 9. Numerical simulations show that system (45) has a unique almost periodic solution, which is globally exponentially stable (see Figures 1, 2, and 3).This completes the proof.
In [28][29][30], the authors studied the existence and global exponential stability of almost periodic solution for a kind of Cohen-Grossberg neural networks without neutral delays based on the exponential dichotomy with fixed point theorem.Therefore, in some sense, we generalize the results in [28][29][30].

Conclusions
Compared with periodic effects, almost periodic effects are more frequent.In recent years, the study on the almost periodic behavior of neural networks has been the object of intensive analysis by numerous authors, and some of these results can be found in the literature.However, the neural networks of neutral type have not been fully investigated in the literature.In this paper, we give some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks.Numerical simulation is given to illustrate the feasibility of our main result.By using the method in this paper, we could study the existence, uniqueness, and global exponential stability of almost periodic solution of other (impulsive) neural networks (on time scales).We leave these for our future work.

Figure 3 :
Figure 3: Exponential stability of state variable  2 of Example 1.