Almost automorphic solutions for shunting inhibitory cellular neural networks with time-varying delays

This paper is concerned with the shunting inhibitory cellular neural networks with time-varying delays. Under some suitable conditions, we establish some criteria on the existence and global exponential stability of the almost automorphic solutions of the networks. Numerical simulations are given to support the theoretical findings.

2013). However, to the best of our knowledge, there are very few papers published on the almost automorphic solutions of shunting inhibitory cellular neural networks with timevarying delays (Li and Yang 2014;Abbas et al. 2014).
Inspired by the discuss above, in this paper, we consider the following shunting inhibitory cellular neural networks with time-varying delays where C ij denotes the cell at the (i, j) position of the lattice. The r-neighborhood N r (i, j) of C ij is given as where i = 1, 2, . . . , m, j = 1, 2, . . . , n, N q (i, j) is similarly specified, x ij is the activity of the cell C ij , L ij (t) is the external input to C ij , the function a ij (t) > 0 represents the passive decay rate of the cell activity, C kl ij and B kl ij are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell C ij , and the activity functions f(.) and g (.) are continuous functions representing the output or firing rate of the cell C kl , and τ kl (t) ≥ 0 corresponds to the transmission delay, the kernel K ij is a piecewise continuous integrable function and satisfies It is easy to see that system (1) is equivalent to the form The main aim of this paper is to establish a set of sufficient conditions for the existence and exponential stability of almost automorphic solutions for model (3).
The remainder of the paper is organized as follows. In "Preliminary results", we introduce the basic properties of almost automorphic functions, some necessary notations, definitions and preliminaries which will be used later. In "Existence of almost automorphic solutions" , we present some sufficient conditions for the existence of almost automorphic solutions of (3). Some sufficient conditions on the global exponential stability of almost automorphic solutions of (3) are established in "Exponential stability of almost automorphic solutions". An example is given to illustrate the effectiveness of the obtained results in "Numerical example" . A brief conclusion is drawn in "Conclusions". (1)

Preliminary results
In this section, we would like to recall some basic definitions and lemmas related to the concept of almost automorphy which shall come into play later on.
Definition 2.1 (Bochner 1962) A continuous function f : R → R n is said to be almost automorphic if for every sequence of real numbers (s ′ n ) n∈N , there exists a subsequence (s n ) n∈N such that g(t) := lim n→∞ f (t + s n ) is well defined for each t ∈ R, and lim n→∞ g(t − s n ) = f (t) for each t ∈ R.
Remark 2.1 (Chérif 2014) Note that the function g in definition above is measurable but not necessarily continuous. Moreover, if g is continuous, then f is uniformly continuous. Besides, if the convergence above is uniform in t ∈ R, then f is almost periodic. Denote by AA(R, R n ) the collection of all almost automorphic functions, then where AP(R, R n ) and BC(R, R n ) are respectively the collection of all almost periodic functions and the set of bounded continuous functions from R to R n .
Definition 2.2 A function f ∈ C(R × R n , R n ) is said to be almost automorphic in t ∈ R for each x ∈ X if for every sequence of real numbers (s ′ n ) n∈N , there exists a subsequence (s n ) n∈N such that g(t, x) := lim n→∞ f (t + s n , x) is well defined for each t ∈ R , x ∈ R n and lim n→∞ g(t − s n , x) = f (t, x) for each t ∈ R, x ∈ R n . The collection of such functions will be denoted by AA(R × R n , R n ). (Diagana et al. 2008) Let f : R × R n → R n be an almost automorphic function in t ∈ R for each x ∈ R n and assume that f satisfies a Lipschitz condition in x uniformly in t ∈ R. Let ϕ : R → R n be an almost automorphic function. Then the function φ : t � → φ(t) = f (t, ϕ(t)) is almost automorphic.

Existence of almost automorphic solutions
In this section, we will establish sufficient conditions on the existence of almost automorphic solutions of (1). Denote Throughout this paper, we make the assumptions as follows.
Proof By the composition theorem of almost automorphic functions (N'Guérékata 2005), the functions ψ : s � → g(x kl (s)) belongs to AA(R, R) whenever x kl ∈ AA(R, R m+n ) . Now, let (s ′ n ) be a sequence of real numbers. By (H5), we can extract a subsequence (s n ) of (s Similarly we have for all t ∈ R, which implies that belongs to AA(R, R). The proof of Lemma 3.1 is completed.
Define the nonlinear operator by: for each ϕAA ∈ (R, R m+n ), Applying the Lebesgue DominatedConvergence Theorem, we have In a same way, we can prove that Thus the function (�ϕ) belong to AA(R, R). The proof of Lemma 3.2 is completed. Therefore, for any ϕ ∈ D and by (13), we see easily that Now we prove that is a self-mapping from D to D. In fact, for arbitrary ϕ ∈ D, it follows that for all t ∈ R.
which implies that (�ϕ) ∈ D. Next, we prove the mapping is a contraction mapping of D. In view of (H2), for any ϕ, ψ ∈ D, we have where (15) Then it follows from (H4) that is contracting operator in D. Thus there exists a unique almost automorphic solution x * ∈ D of (3) that is �(x * ) = x * . The proof of Theorem 3.1 is completed.

Exponential stability of almost automorphic solutions
In this section, we will obtain the exponential stability of the almost automorphic solutions of system (1).

Suppose the contrary. Let us denote
Clearly t ij > 0 and for all −τ ≤ t ≤ t ij . Further, one has z ij (t) ≤ M + ε. Let us denote t ij s = min ij∈ t ij . It follows that 0 < t ij s < +∞. and for all −τ ≤ t ≤ t ij s . Note that Since x ij (.) and x * ij (.) are solutions of (3), we get

It follows that
Then ϒ ij (ν) ≥ 0 which contradicts the fact that ϒ ij (ν) < 0. Thus we obtain that Note that ||x(t) − x * ij (t)|| = max ij∈� |x ij (t)x * ij (t)|, then letting ε → 0, we obtain which means that the almost automorphic solution of (3) is globally exponentially stable. The proof of Theorem 4.2 is completed.  (1) with C ij (t) = 0 on time scales. In addition, there are many papers that have investigated almost periodic solutions or convergence behavior of the special form or a more general form of model (1). We refer the reader to (Zhao and Zhang 2008;Cai et al. 2008;Huang and Cao 2003;Ding et al. 2008;Huang 2006, 2007;Liu 2007Liu , 2009aFan and Shao 2010;Shao et al. 2009;Xia et al. 2007;Zhou et al. 2006b;Liu and Ding 2014;Li and Wang 2012;Meng and Li 2008;Li and Huang 2008). In this paper, we consider the almost automorphic solutions of (1), which complement with some previous studies in (Shao 2008;Peng and Huang 2009;Zhao et al. 2010;Peng and Wang 2013;Zhou et al. 2006a;Zhao and Zhang 2008;Cai et al. 2008;Huang and Cao 2003;Ding et al. 2008;Liu and Huang 2007;Liu 2007Liu , 2009aFan and Shao 2010;Liu and Huang 2006;Shao et al. 2009;Xia et al. 2007;Zhou et al. 2006b;Liu and Ding 2014;Li and Wang 2012;Meng and Li 2008;Li and Huang 2008).
Remark 4.2 In Li and Yang (2014), authors considered the almost automorphic solutions for neutral type neural networks with delays in leakage on time ccales, in Abbas et al. (2014), authors considered the almost automorphic solutions for neural networks with impulses. All the methods can not be applied to this paper to obtained our results in this paper. Therefore our results are completely new.
for all t > 0.
for all t > 0.

Numerical example
In this section, we will give an example to illustrate the feasibility and effectiveness of our main results obtained in previous sections.

Conclusions
In this paper, we consider a class of shunting inhibitory cellular neural networks with time-varying delays. Some sufficient conditions for the existence and exponential stability of almost automorphic solutions for the shunting inhibitory cellular neural networks γ = max ij∈� sup t∈R C kl ∈N 1 (i,j) |C kl ij (t)|L f + M u C kl ∈N 1 (i,j) |B kl ij (t)|L g a − ≤ 0.0014 + 0.0016 2 = 0.0015 < 1, ||L|| ∞ a − (1 − γ ) = 0.005 1(1 − 0.0015) = 10 17 < 1,  = 1, 2), where the red line stands for x 11 , the magenta line stands for x 12 ,, the blue line stands for x 21 and the green line stands for x 22 with time-varying delays have been established. It is shown that the time delay has no effect on the existence of almost automorphic solutions for system (1) but has important effect on the global exponential stability of almost automorphic solutions for system (1). To the best of our knowledge, it is the first time to deal with the almost automorphic solution for the shunting inhibitory cellular neural networks with time-varying delays. Moreover, our criteria are easy to check and apply in practice and are of prime importance and great interest in many application fields and the designs of networks. Our results complement with some previous ones. The method of this paper can be applied directly to many other neural networks, such as BAM neural networks, Hopfield neural networks and so on.