Research Article | Open Access
Existence and Global Exponential Stability of Almost Periodic Solutions for SICNNs with Nonlinear Behaved Functions and Mixed Delays
By using the Leray-Schauder fixed point theorem and differential inequality techniques, several new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with discrete and distributed delays. The model in this paper possesses two characters: nonlinear behaved functions and all coefficients are time varying. Hence, our model is general and applicable to many known models. Moreover, our main results are also general and can be easily deduced to many simple cases, including some existing results. An example and its simulation are employed to illustrate our feasible results.
Consider the following shunting inhibitory cellular neural networks (SICNNs) with discrete and distributed delays (mixed delays):
where , denotes the cell at the position of the lattice at time , the -neighborhood of is
is the activity of the cell , is the external input to , represents an appropriately behaved function of the cell at time ; and are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell depending upon discrete delays and distributed delays, respectively; the activation functions and are continuous representing the output or firing rate of the cell ; represent axonal signal transmission delays; are all continuous almost periodic functions.
Since Bouzerdoum and Pinter described SICNNs as a new cellular neural networks [1–3], SICNNs have been extensively studied and found many important applications in different areas such as psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. There have been some results on the existence of periodic and almost solutions for SICNNs with discrete or distributed delays (distributed delay was first introduced in ) [5–21]. We find that all the behaved functions in the models in [5–21] are linear. Actually, may be nonlinear. Moreover, we find that the models in [5–20] are special cases of (1.1). For example, let , then in , in , in , are constants and in , are constants, and in [10, 11]. To the best of our knowledge, few authors have considered the existence and global exponential stability of almost periodic solutions for SICNNs with nonlinear behaved functions, periodic coefficients and mixed delays. Obviously, (1.1) is general and is worth to continue to investigate its dynamical properties such as existence and global exponential stability of almost periodic solutions.
The main purpose of this paper is to get sufficient conditions on the existence and global exponential stability of almost periodic solutions for SICNNs (1.1) by using the Leray-Schauder fixed point theorem and differential inequality techniques. Our results are general and possess infinitely adjustable real parameters and can be deduced to many simple results, including some existing results as special cases. Therefore, our results provide a wider application criteria for neural networks.
The remaining part of this paper is organized as follows. We first state some useful definitions and lemmas in Section 2. In Section 3, we study the existence of almost periodic solutions of system (1.1) by using the Leray-Schauder's fixed point theorem. In Section 4, by using Lemma 2.7, we will derive sufficient conditions for the global exponential stability of the almost periodic solution of system (1.1). A useful corollary is also obtained. An illustrative example and its simulation are given in Section 5.
For convenience, we denote
Definition 2.1. The continuous function is called almost periodic on , if for any , it is possible to find a real number such that, for any interval with length , there exists a number in this interval such that , for any .
Definition 2.2 (see [22, page 21]). Let be a Banach space, an open subset in and . For , is called uniformly almost periodic about , if for any and any compact subset , there exists a real number such that, for any interval with length , there exists a number in this interval such that , for any .
The initial condition of (1.1) is of the form
where are continuous almost periodic solutions.
Definition 2.3. Let be an almost periodic solution of (1.1) with initial value . If there exist constants and such that for every solution of (1.1) with initial value where . Then is said to be globally exponentially stable.
Lemma 2.4 (see [22, page 136]). Suppose that is uniformly continuous on , where is any compact set on , and that there exists a nonsingular matrix , such that (a)there exists a constant satisfies , for all , (b)the eigenvalues , of satisfy , where is a constant, and there are negative eigenvalues, nonnegative eigenvalues, (c)all the eigenvalues of the following symmetric matrix: satisfy where are constants.Then, for any fixed and any satisfying , the linear differential equation admits an exponential dichotomy on : where is fundamental solution matrix of (2.6) satisfying , is the identity matrix, is a constant projection, the constant has no relationship with .
Lemma 2.5 (see [22, page 139]). Suppose that the matrix function and the -dimensional vector function are uniformly almost periodic on , and that there exists real symmetric nonsingular matrix satisfying the conditions (a)–(c) in Lemma 2.4, then, has a unique almost periodic solution , where is almost periodic function and
Lemma 2.6 (Leray-Schauder). Let be a Banach space, and let the operator be completely continuous. If the set is bounded, then has a fixed point in , where
Lemma 2.7 (see ). Let , , the following inequality holds where is some constants, , and .
Furthermore, throughout this paper, we assume that(H1) is continuous almost periodic about the first argument and, there exists a positive continuous almost periodic function such that , , and , (H2)there exist nonnegative constants and , , such that (H3)the delay kernels are continuous, integrable and there are positive constants such that (H4)there exists a constant such that (H5)the following inequality holds: (H6) there are nonnegative constants such that for all .
3. Existence of Almost Periodic Solutions
Let , , be constants. Make the following transformation:
then (1.1) can be reformulated as
System (3.2) can be rewritten as
where , is between and , . By , we know that is strictly monotone increasing about . Hence, is unique for any . Obviously, is continuous almost periodic about the first argument and .
Take is an almost periodic function, , . Then is a Banach space with the norm
For for all , we consider the following auxiliary equation:
From , we know that
are uniformly almost periodic functions on . Since are positive continuous almost periodic functions, there exists a positive constant such that
Hence, the conditions in Lemma 2.5 are satisfied (take ).
Set a mapping by setting
Lemma 3.1. Suppose that (H1)–( H5) hold. Then is completely continuous.
Proof. Under our assumptions, it is clear that the operator is continuous. Next, we show that is compact.
For any constant , let . Then, for any , we have
where , . Hence, is uniformly bounded.
By the definition of , we get
Since are uniformly almost periodic functions on , there exists a positive constant such that Hence, where , . So, is a family of uniformly bounded and equicontinuous subsets. By using the Arzela-Ascoli theorem, is compact. Therefore, is completely continuous. This completes the proof.
Theorem 3.2. Suppose that hold. Let , be constants. Then system (1.1) has an almost periodic solution with , where
Proof. Let . From Lemma 3.1, we get that is completely continuous. Consider the following operator equation: If is a solution of (3.15), we obtain This and imply that In view of Lemma 2.6, we obtain that has a fixed point with . From (3.5) and (3.8), we know that satisfies (3.3) and (3.2). Hence, system (3.2) has an almost periodic solution with . It follows from (3.1) that is one almost periodic solution of (1.1) with This completes the proof.
4. Stability of Almost Periodic solution
In this section, we prove that, under suitable conditions, the almost periodic solution obtained in Theorem 3.2 is globally exponentially stable.
Theorem 4.1. Assume that and hold and (H7)there are constants , , , , , such that where, , . Then (1.1) has unique almost periodic solution, which is globally exponentially stable.
Proof. Obviously, that holds implies that holds. By Theorem 3.2, there exists an almost periodic solution of (1.1) with initial value and . Suppose that is an arbitrary solution of (1.1) with initial value . Set . Then, from (1.1), we have
Clearly, , , , are continuous functions about and almost periodic about on .
From , we get
In addition to and for , we obtain that , , are strictly monotone increasing functions about . Therefore, for any , , there is unique real number such that In view of (4.4), we have . Let , we obtain Now, we choose a positive constant such that It is obvious that where is defined as that in Definition 2.3.
Let In view of (4.2) and (2.11), for we obtain
We now prove that the following statement is true:
Contrarily, there must exist some , , and , such that for all Together with (4.9), (4.11), and (4.7), we obtain which is a contradiction. Hence, (4.10) holds. It follows that Let , then By means of Definition 2.3, the almost periodic solution of (1.1) is globally exponentially stable. This completes the proof.
Since Theorem 3.2 is general, the expression of is complex. In order to check the applicability of the results easily, we give the following corollary.
Corollary 4.2. Assume that and hold. Furthermore, if one of the following conditions holds (A1) there are constants , , , , , such that (A2) there are constants , , , , such that (A3) there are constants , , , , such that
Then (1.1) has exactly one almost periodic solution, which is globally exponentially stable.
Proof. Let , , , and , , , in , respectively, then turns to .
Let and in , respectively, then we get and . This completes the proof.
Remark 4.3. Corollary 4.2 can still be simplified into many simple results, which include the results in [7–11]. For example, authors of  considered a special case of our model (1.1) and obtained sufficient conditions of existence and exponential stability of almost periodic solutions for their model. Although the results of  and ours are completely different, we claim that the results in  are also special cases of the results in this paper. The model considered in  is