Abstract
This paper is concerned with the shunting inhibitory cellular neural networks (SICNNs) with time-varying delays in the leakage (or forgetting) terms. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of pseudo almost periodic solutions by using Lyapunov functional method and differential inequality techniques. We also provide numerical simulations to support the theoretical result.
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 [1–7] 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 where is similarly specified, is the activity of the cell , is the external input to , the function 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 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 [13–19] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving leakage delays. Recently, Liu and Shao [20] considered the following SICNNs with time-varying leakage delays: where , , 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 [21–24]). 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.
For convenience, we denote by () the set of all -dimensional real vectors (real numbers). We will use For any , we let denote the absolute-value vector given by and define . A matrix or vector means that all entries of are greater than or equal to zero. can be defined similarly. For matrices or vectors and , (resp. ) means that (resp. ). For the convenience, we will introduce the notations: where is a bounded continuous function.
The initial conditions associated with system (3) are of the form: where and are real-valued bounded continuous functions defined on .
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.
2. Preliminary Results
In this section, we will first recall some basic definitions and lemmas which are used in what follows.
In this paper, denotes the set of bounded continued functions from to . Note that is a Banach space where denotes the sup norm .
Definition 1 (see [25, 26]). Let . is said to be almost periodic on if, for any , the set is relatively dense; that is, for any , it is possible to find a real number ; for any interval with length , there exists a number in this interval such that , .
We denote by the set of the almost periodic functions from to . 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 as follows:
A function is called pseudo almost periodic if it can be expressed as
where and . The collection of such functions will be denoted by . The functions 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 a Banach space and is a proper subspace of since the function 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].
Lemma 2 (see [25, page 57]). If and is its almost periodic component, then we have Therefore .
Lemma 3 (see [25, page 140]). Suppose that both functions and its derivative are in . That is, and , where and . Then the functions and are continuous differentiable so that
Lemma 4. Let equipped with the induced norm defined by , and then is a Banach space.
Proof. Suppose that is a Cauchy sequence in , and then for any , there exists , such that
By the definition of pseudo almost periodic function, let
From Lemma 3, we obtain
On combining (11) with Lemma 2, we deduce that, are Cauchy sequence, so that are also Cauchy sequence.
Firstly, we show that there exists such that uniformly converges to , as .
Note that is Cauchy sequence in . , , such that
So for fixed , it is easy to see is Cauchy number sequence. Thus, the limits of exist as and let . In (14), let , and we have
Thus, uniformly converges to , as . Moreover, from the Theorem 1.9 [26, page 5], we obtain . Similarly, we also obtain that there exist and , such that
which lead to
where and “” means uniform convergence.
Next, we claim that . Together with (16) and the facts that
we have
Hence . Let , , then , and , as .
Finally, we reveal . For , it follows that
In view of the uniform convergence of and , let for (20), and we get
which implies that
In summary, in view of (15), (16), and (22), we obtain that the Cauchy sequence satisfies
and . This yields that is a Banach space. The proof is completed.
Remark 5. Let equipped with the induced norm defined by . It follows from Lemma 4 that is a Banach space.
Definition 6 (see [19, 20]). Let and be a continuous matrix defined on . The linear system is said to admit an exponential dichotomy on 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 . 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 and Then the linear system admits an exponential dichotomy on .
3. Existence of Pseudo Almost Periodic Solutions
In this section, we establish sufficient conditions on the existence of pseudo almost periodic solutions of (3).
For , is an almost periodic function, , and are pseudo almost periodic functions. We also make the following assumptions which will be used later.
We also make the following assumptions.(S1)There exist constants , , , and such that (S2)For , the delay kernels are continuous, and are integrable on for a certain positive constant .(S3)Let Moreover, there exists a constant such that where
Lemma 9. Assume that assumptions (S1) and (S2) hold. Then, for , the function belongs to , where .
Proof. Let . Obviously, (S1) implies that is a uniformly continuous function on . By using Corollary 5.4 in [25, page 58], we immediately obtain the following: where and . Then, for any , it is possible to find a real number ; 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.
Theorem 10. Let (S1), (S2), and (S3) hold. Then, there exists at least one continuously differentiable pseudo almost periodic solution of system (3).
Proof. Let . Obviously, the boundedness of and (S1) imply that and are uniformly continuous functions on for . Set . By Theorem 5.3 in [25, page 58] and Definition 5.7 in [25, page 59], we can obtain that and is continuous in and uniformly in for all compact subset of . This, together with and Theorem 5.11 in [25, page 60], implies that
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 , , and it follows from Lemma 8 that the linear system,
admits an exponential dichotomy on . Thus, by Lemma 7, we obtain that the system (42) has exactly one pseudo almost periodic solution:
From (S1), (S2), and the Corollary 5.6 in [25, page 59], we get
which is a pseudo almost periodic function. Therefore, . Let . Then,
Set
If , then
Now, we define a mapping by setting
We next prove that the mapping is a contraction mapping of the .
First we show that, for any , .
Note that
It follows that
that is, .
Second, we show that is a contract operator.
In fact, in view of (44), (48), (S1), (S2), and (S3), for , we have
which yields
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.
4. 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).
Definition 11. Let be the pseudo almost periodic solution of system (3). If there exist constants and such that, for every solution of system (3) with any initial value satisfying (6), where . Then is said to be globally exponentially stable.
Theorem 12. 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.
Proof. By Theorem 10, (3) has at least one continuously differentiable pseudo almost periodic solution such that
Suppose that is an arbitrary solution of (1) associated with initial value satisfying (6). Let . Then
Define continuous functions and by setting
where , , . Then, from (S3), we have
which, together with the continuity of and , implies that we can choose a constant such that
where
Let be a constant such that
which, together with (60), yields
Consequently, for any , it is obvious that
In the following, we will show that
Otherwise, there must exist and such that
Note that
Multiplying both sides of (68) by and integrating on , we get
Thus, with the help of (67), we have
which, together with (64) and (67), implies that
From (60), (61) and (67)–(72) yield
which contradicts (72). Hence, (66) holds. Letting , we have from (66) that
which implies
This completes the proof.
5. An Example
In this section, we give an example with numerical simulation to demonstrate the results obtained in previous sections.
Example 13. Consider the following SICNNs with time-varying delays in the leakage terms:Set clearly, where . Then, It follows that system (56) satisfies all the conditions in Theorems 10 and 12. Hence, system (76) has exactly one pseudo almost periodic solution. Moreover, the pseudo almost periodic solution is globally exponentially stable. The fact is verified by the numerical simulation in Figures 1, 2, and 3 and there are three different initial values which are , , , , , , , , ; , , , , , , , , and , , , , , , , , , respectively.
Remark 14. By using the inequality analysis technique, in [19, 20], the authors obtained the existence of almost periodic solution of SICNNs with leakage delays, but they did not give the existence and global exponential convergence for the pseudo almost periodic solution. Since [1–9] only dealt with SICNNs without leakage delays, [14–18, 21–24] give no opinions about the problem of pseudo almost periodic solutions for SICNNs with leakage delays. One can observe that all the results in these literatures and the references therein cannot be applicable to prove the existence and exponential stability of pseudo almost periodic solutions for SICNNs (56).
Conflict of Interests
The authors declare no conflict of interests. They also declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this present paper.
Acknowledgments
The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P. R. China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P. R. China (Grant no. Z201122436).