Global Exponential Stability of Pseudo Almost Periodic Solutions for SICNNs with Time-Varying Leakage Delays

and Applied Analysis 3 Lemma 3 (see [25, page 140]). Suppose that both functions f and its derivative f󸀠 are in PAP(R,R). That is, f = g + φ and f 󸀠 = α + β, where g, α ∈ AP(R,R) and φ, β ∈ PAP 0 (R,R). Then the functions g andφ are continuous differentiable so that g 󸀠 = α, φ 󸀠 = β. (10) Lemma 4. Let B∗ = {f | f, f󸀠 ∈ PAP(R,R)} equipped with the induced norm defined by ‖f‖ B ∗ = max{‖f‖ ∞ , ‖f 󸀠 ‖ ∞ } = max{sup t∈R supt∈R|f 󸀠 (t)|}, and then B∗ is a Banach space. Proof. Suppose that {f p } +∞ p=1 is a Cauchy sequence in B, and then for any ε > 0, there existsN(ε) > 0, such that 󵄩 󵄩 󵄩 󵄩 󵄩 f p − f q 󵄩 󵄩 󵄩 󵄩 󵄩B ∗ =max{sup t∈R 󵄨 󵄨 󵄨 󵄨 󵄨 f p (t) −f q (t) 󵄨 󵄨 󵄨 󵄨 󵄨 , sup t∈R 󵄨 󵄨 󵄨 󵄨 󵄨 f p 󸀠 (t) − f q 󸀠 (t) 󵄨 󵄨 󵄨 󵄨 󵄨 }< ε, ∀p, q ≥ N (ε) . (11) By the definition of pseudo almost periodic function, let f p = g p + φ p , where g p ∈ AP (R,R) , φ p ∈ PAP 0 (R,R) , p = 1, 2, . . . . (12) From Lemma 3, we obtain f 󸀠 p = g 󸀠 p + φ 󸀠 p , where g󸀠 p ∈ AP (R,R) , φ 󸀠 p ∈ PAP 0 (R,R) , p = 1, 2, . . . . (13) On combining (11) with Lemma 2, we deduce that, {g p } +∞ p=1 , {g 󸀠 p } +∞ p=1 ⊂ AP(R,R) are Cauchy sequence, so that {φ p } +∞ p=1 , {φ 󸀠 p } +∞ p=1 ⊂ PAP 0 (R,R) are also Cauchy sequence. Firstly, we show that there exists g ∈ AP(R,R) such that g p uniformly converges to g, as p → +∞. Note that {g p } is Cauchy sequence in AP(R,R). for all ε > 0, ∃N(ε), such that for all p, q ≥ N(ε) 󵄨 󵄨 󵄨 󵄨 󵄨 g p (t) − g q (t) 󵄨 󵄨 󵄨 󵄨 󵄨 < ε, ∀t ∈ R. (14) So for fixed t ∈ R, it is easy to see {g p (t)} +∞ p=1 is Cauchy number sequence. Thus, the limits of g p (t) exist as p → +∞ and let g(t) = lim p→+∞ g p (t). In (14), let q → +∞, and we have 󵄨 󵄨 󵄨 󵄨 󵄨 g (t) − g p (t) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ε, ∀t ∈ R, p ≥ N (ε) . (15) Thus, g n uniformly converges to g, as p → +∞. Moreover, from the Theorem 1.9 [26, page 5], we obtain g ∈ AP(R,R). Similarly, we also obtain that there exist g∗ ∈ AP(R,R) and φ, φ ∗ ∈ BC(R,R), such that 󵄨 󵄨 󵄨 󵄨 󵄨 g ∗ (t) − g 󸀠 p (t) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ε, 󵄨 󵄨 󵄨 󵄨 󵄨 φ (t) − φ p (t) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ε, 󵄨 󵄨 󵄨 󵄨 󵄨 φ ∗ (t) − φ 󸀠 p (t) 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ε, ∀t ∈ R, p ≥ N (ε) , (16) which lead to g 󸀠 p 󳨐⇒ g ∗ , φ p 󳨐⇒ φ, φ 󸀠 p 󳨐⇒ φ ∗ , (17) where p → +∞ and “⇒” means uniform convergence. Next, we claim that φ, φ∗ ∈ PAP 0 (R). Together with (16) and the facts that lim r→+∞ 1


Introduction
In the last three decades, shunting inhibitory cellular neural networks (SICNNs) have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing.Hence, they have been the object of intensive analysis by numerous authors in recent years.In particular, there have been extensive results on the problem of the existence and stability of the equilibrium point and periodic and almost periodic solutions of SICNNs with time-varying delays in the literature.We refer the reader to [1][2][3][4][5][6][7] and the references cited therein.
It is well known that SICNNs have been introduced as new cellular neural networks (CNNs) in Bouzerdoum et al. in [1,8,9], which can be described by where   denotes the cell at the (, ) position of the lattice.
The -neighborhood   (, ) of   is given as   (, ) = {  : max (| − | ,      −      ) ≤ , where   (, ) is similarly specified,   is the activity of the cell   ,   () is the external input to   , the function   () > 0 represents the passive decay rate of the cell activity,    () and    () are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell   , and the activity functions (⋅) and (⋅) are continuous functions representing the output or firing rate of the cell   , and   () ≥ 0 corresponds to the transmission delay.
Obviously, the first term in each of the right side of (1) corresponds to stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as "forgettin" or leakage terms (see, for instance, Kosko [10], Haykin [11]).It is known from the literature on population dynamics and neural networks dynamics (see Gopalsamy [12]) that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system.Therefore, the authors of [13][14][15][16][17][18][19] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving ()  (  ( − ))    () +   () , where  = 1, 2, . . ., ,  = 1, 2, . . ., ,   , : R → [0 + ∞) denotes the leakage delay.By using Lyapunov functional method and differential inequality techniques, in [20], some sufficient conditions have been established to guarantee that all solutions of (1) converge exponentially to the almost periodic solution.Moreover, it is well known that the global exponential convergence behavior of solutions plays a key role in characterizing the behavior of dynamical system since the exponential convergent rate can be unveiled (see [21][22][23][24]).However, to the best of our knowledge, few authors have considered the exponential convergence on the pseudo almost periodic solution for (1).Motivated by the above discussions, in this paper, we will establish the existence and uniqueness of pseudo almost periodic solution of (1) by using the exponential dichotomy theory and contraction mapping fixed point theorem.Meanwhile, we also will give the conditions to guarantee that all solutions and their derivatives of solutions for (1) converge exponentially to the pseudo almost periodic solution and its derivative, respectively.
The initial conditions associated with system (3) are of the form: ∈  := {11, . . ., 1, 21, . . ., 2, . . ., 1, . . ., } , (6) where   (⋅) and    (⋅) are real-valued bounded continuous functions defined on (−∞, 0]. The paper is organized as follows.Section 2 includes some lemmas and definitions, which can be used to check the existence of almost periodic solutions of (3).In Section 3, we present some new sufficient conditions for the existence of the continuously differentiable pseudo almost periodic solution of (3).In Section 4, we establish sufficient conditions on the global exponential stability of pseudo almost periodic solutions of (3).At last, an example and its numerical simulation are given to illustrate the effectiveness of the obtained results.

Preliminary Results
In this section, we will first recall some basic definitions and lemmas which are used in what follows.
We denote by AP(R, R  ) the set of the almost periodic functions from R to R  .Besides, the concept of pseudo almost periodicity (pap) was introduced by Zhang in the early nineties.It is a natural generalization of the classical almost periodicity.Precisely, define the class of functions PAP 0 (R, R) as follows: A function  ∈ BC(R, R  ) is called pseudo almost periodic if it can be expressed as where ℎ ∈ AP(R, R  ) and  ∈ PAP 0 (R, R  ).The collection of such functions will be denoted by PAP(R, R  ).The functions ℎ and  in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function .The decomposition given in definition above is unique.Observe that is pseudo almost periodic function but not almost periodic.It should be mentioned that pseudo almost periodic functions possess many interesting properties; we shall need only a few of them and for the proofs we shall refer to [25].
So for fixed  ∈ R, it is easy to see {  ()} +∞ =1 is Cauchy number sequence.Thus, the limits of   () exist as  → +∞ and let () = lim  → +∞   ().In (14) In view of the uniform convergence of   and    , let  → +∞ for (20), and we get which implies that In summary, in view of ( 15), (16) Definition 6 (see [19,20]).Let  ∈ R  and () be a  ×  continuous matrix defined on R. The linear system is said to admit an exponential dichotomy on R if there exist positive constants , , and projection  and the fundamental solution matrix () of ( 24) satisfying Lemma 7 (see [19]).Assume that () is an almost periodic matrix function and () ∈ PAP(R, R  ).If the linear system (24) admits an exponential dichotomy, then pseudo almost periodic system has a unique pseudo almost periodic solution (), and Lemma 8 (see [19,20]).Let   () be an almost periodic function on R and Then the linear system admits an exponential dichotomy on R.

Existence of Pseudo Almost Periodic Solutions
In this section, we establish sufficient conditions on the existence of pseudo almost periodic solutions of (3).For ,  ∈ ,   : R → (0, +∞) is an almost periodic function,   ,   : R → [0, +∞), and   ,    ,    : R → R are pseudo almost periodic functions.We also make the following assumptions which will be used later.
We also make the following assumptions.
Moreover, there exists a constant  such that where Lemma 9. Assume that assumptions (S 1 ) and (S 2 ) hold.Then, for (⋅) ∈ PAP(R, R), the function Proof.Let  ∈ PAP(R, R).Obviously, (S 1 ) implies that  is a uniformly continuous function on R. By using Corollary 5.4 in [25, page 58], we immediately obtain the following: where  1 ∈ AP(R, R) and  2 ∈ PAP 0 (R, R).Then, 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 It follows that Thus, which yield The proof of Lemma 9 is completed.
Again from Corollary 5.4 in [25, page 58], we have which, together with Lemma 9, implies For any  ∈ , we consider the pseudo almost periodic solution   () of nonlinear pseudo almost periodic differential equations Then, notice that [  ] > 0,  ∈ , and it follows from Lemma 8 that the linear system, admits an exponential dichotomy on R. Thus, by Lemma 7, we obtain that the system (42) has exactly one pseudo almost periodic solution: From (S 1 ), (S 2 ), and the Corollary 5.6 in [25, page 59], we get We next prove that the mapping  is a contraction mapping of the  * * .First we show that, for any  ∈ which implies that the mapping  :  * * →  * * is a contraction mapping.Therefore, using Theorem 0.3.1 of [27], we obtain that the mapping  possesses a unique fixed point By ( 42) and (44),  * satisfies (42).So (3) has at least one continuously differentiable pseudo almost periodic solution  * .The proof of Theorem 10 is now completed.

Exponential Stability of the Pseudo Almost Periodic Solution
In this section, we will discuss the exponential stability of the pseudo almost periodic solution of system (3).

An Example
In this section, we give an example with numerical simulation to demonstrate the results obtained in previous sections.

Figure 3 :
Figure 3: Numerical solutions of system (76) for different initial values.
*.This yields that  * is a Banach space.The proof is completed.
Suppose that all conditions in Theorem 10 are satisfied.Then system (3) has at least one pseudo almost periodic solution  * ().Moreover,  * () is globally exponentially stable.