Research Article | Open Access
Global Exponential Stability of Almost Periodic Solutions for SICNNs with Continuously Distributed Leakage Delays
Shunting inhibitory cellular neural networks (SICNNs) are considered with the introduction of continuously distributed delays in the leakage (or forgetting) terms. By using the Lyapunov functional method and differential inequality techniques, some sufficient conditions for the existence and exponential stability of almost periodic solutions are established. Our results complement with some recent ones.
It is well known that a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths; it is desired to model them by introducing continuously distributed delays over a certain duration of time [1–4]. In particular, shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays 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 constant represents the passive decay rate of the cell activity, and are the connection or coupling strengths of postsynaptic activity of the cell transmitted to the cell , the activity functions and are continuous functions representing the output or firing rate of the cell , and corresponds to the transmission delay.
Since SICNNs (1) have been introduced as a new cellular neural networks (CNNs) in Bouzerdout and Pinter in [5–7], it has been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, there have been extensive results on the problem of the existence and stability of the equilibrium point, periodic and almost periodic solutions of SICNNs with continuously distributed delays in the literature. We refer the reader to [8–12] and references cited therein.
As pointed out in Gopalsamy , the first term in each of the right side of (1) corresponds to a stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as “forgettin” or leakage terms (see, e.g., Kosko  and Haykin ). It is known from the literature on population dynamics and neural networks dynamics (see Gopalsamy ) that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system. Therefore, the authors of [17–21] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving leakage delays. Since leakage delays can have a destabilizing influence on the dynamical behaviors of neural networks and the incorporation of time delays in the leakage terms are usually not easy to handle, it necessary to investigate leakage delay effects on the stability of neural networks. On the other hand, as pointed out in [22, 23], periodically varying environment and almost periodically varying environment are foundations for the theory of nature selection. Compared with periodic effects, almost periodic effects are more frequent. Hence, the effects of the almost periodic environment on the evolutionary theory have been the object of intensive analysis by numerous authors, and some of these results can be found in [8, 9, 11] and references cited therein. However, to the best of our knowledge, few authors have considered the existence and exponential stability of almost periodic solutions of SICNNs with continuously distributed delays in the leakage terms. Motivated by the above discussions, in this present paper, we will consider the following SICNNs with continuously distributed leakage delays: where , , and are almost periodic functions, denotes transmission delay, the leakage delay kernels are continuous and integrable, respectively, and the delay kernels are continuous and integrable.
The main purpose of this paper is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (3). By applying the Lyapunov functional method and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the almost periodic solution for system (3), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results.
Throughout this paper, for , delay kernels and are continuous functions, and there exist constants and such that From the theory of almost periodic functions in [22, 23], it follows that, for any , it is possible to find a real number , for any interval with length , and there exists a number in this interval such that for all .
We set For any , we define the norm . We also assume that the following conditions and hold.
and are nonincreasing functions on , and there exist constants , , , and such that
For , and there exist positive constants and such that where , and .
The initial conditions associated with system (3) are of the form where denotes real-valued bounded continuous function defined on .
Definition 1 (see [22, 23]). Let be continuous in . 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 , and there exists a number in this interval such that , for all .
The remaining part of this paper is organized as follows. In Section 2, we will derive some new sufficient conditions for checking the existence of bounded solutions. In Section 3, we present some new sufficient conditions for the existence, uniqueness and exponential stability of the positive almost periodic solution of (3). In Section 4, we will give some examples and remarks to illustrate our results obtained in previous sections.
2. Preliminary Results
The following lemmas will be useful to prove our main results in Section 3.
Lemma 2. Let and hold. Suppose that is a solution of system (3) with initial conditions where . Then
Proof. Assume, by way of contradiction, that (12) does not hold. Then, there exist , , and such that
It follows that
Consequently, in view of (16) and the fact , we have
From system (3), we derive
Calculating the upper left derivative of , together with (14), (17), (18), , and , we obtain It is a contradiction and shows that (12) holds. Then, using a similar argument as in the proof of (16) and (17), we can show that (13) holds. The proof of Lemma 2 is now completed.
Remark 3. In view of the boundedness of this solution, from the theory of functional differential equations with infinite delay in , it follows that the solution of system (3) with initial conditions (11) can be defined on .
Lemma 4. Suppose that and hold. Moreover, assume that is a solution of system (3) with initial function satisfying (11), and is bounded continuous on . Then, for any , there exists , such that every interval contains at least one number for which there exists which satisfies
Proof. For , set
By Lemma 2, the solution is bounded and
Thus, the right side of (3) is also bounded, which implies that is uniformly continuous on . From (5), for any , there exists , such that every interval , contains a for which
Let be sufficiently large such that , for , and denote . We obtain which yields
Set where Let be such an index that Calculating the upper left derivative of along (25), we have
Let It is obvious that , and is nondecreasing. In particular, Consequently, in view of (31) and the fact , we have
Now, we consider two cases.
Case (i). If then, we claim that Assume, by way of contradiction, that (34) does not hold. Then, there exists , such that , since There must exist such that which contradicts (33). This contradiction implies that (34) holds. It follows from (32) that there exists such that
Case (ii). If there is such a point that , then, in view of (8), (22), (23), (29), (32), , and , we get which yields that
For any , by the same approach used in the proof of (39), we have
On the other hand, if and , we can choose such that which, together with (40), yields that Using a similar argument as in the proof of Case (i), we can show that which implies that
In summary, there must exist such that holds, for all . The proof of Lemma 4 is now complete.
3. Main Results
In this section, we establish some results for the existence, uniqueness, and exponential stability of the almost periodic solution of (3).
Theorem 5. Suppose that and are satisfied. Then system (3) has exactly one almost periodic solution . Moreover, is globally exponentially stable.
Proof. Let be a solution of system (3) with initial function satisfying (11), and is bounded continuous on .
Set where is any sequence of real numbers. By Lemma 2, the solution is bounded and which implies that the right side of (3) is also bounded, and is a bounded function on . Thus, is uniformly continuous on . Then, from the almost periodicity of , and , we can select a sequence such that for all , .
Since is uniformly bounded and equiuniformly continuous, by the Arzala-Ascoli Lemma and diagonal selection principle, we can choose a subsequence of , such that (for convenience, we still denote by ) uniformly converges to a continuous function on any compact set of , and
Now, we prove that is a solution of (3). In fact, for any and , from (47), we have which implies that