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Almost periodic solutions for SICNNs with time-varying delays in the leakage terms
Journal of Inequalities and Applications volume 2013, Article number: 494 (2013)
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 almost periodic solutions by using Lyapunov functional method and differential inequality techniques. We also provide numerical simulations to support the theoretical result.
where denotes the cell at the position of the lattice. The r-neighborhood is given as
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.
Recently, 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, periodic and almost periodic solutions of SICNNs with time-varying delays in the literature. We refer the reader to [4–9] and the references cited therein. Obviously, the first term in each of the right side of (1.1) corresponds to a stabilizing negative feedback of the system, which acts instantaneously without time delay; these terms are variously known as ‘forgetting’ or leakage terms (see, for instance, Kosko , 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 [13–19] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving leakage delays. However, to the best of our knowledge, few authors have considered the existence and exponential stability of almost periodic solutions of SICNNs with time-varying delays in the leakage terms. Motivated by the discussions above, in this paper, we consider the following SICNNs with time-varying leakage delays:
where , , , , and are almost periodic functions, and denote the leakage delay and transmission delay, respectively, the delay kernels are continuous and integrable, and is a bounded continuous function.
The main purpose of this paper is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (1.2). By applying 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 (1.2), which are new and complement previously known results. Moreover, an example with numerical simulations is also provided to illustrate the effectiveness of our results.
Throughout this paper, for , from the theory of almost periodic functions in [20, 21], it follows that for all , it is possible to find a real number , for any interval with length , there exists a number in this interval such that
for all , .
For , we define the norm . For the convenience, we shall introduce the notations
where is a bounded continuous function.
We also make the following assumptions.
(T1) There exist constants , , and such that
(T2) For , ,
and there exist positive constants and λ such that
The initial conditions associated with system (1.2) are of the form
where denotes a real-valued bounded continuous function defined on .
Let be continuous in t. is said to be almost periodic on R if for any , the set is relatively dense, i.e., for any , it is possible to find a real number , for any interval with length , there exists a number in this interval such that for all .
2 Preliminary results
The following lemmas will be useful to prove our main results in Section 3.
Lemma 2.1 Let (T1) and (T2) hold. Suppose that is a solution of system (1.2) with initial conditions
Proof Assume, by way of contradiction, that (2.2) does not hold. Then, there exist , and such that
It follows that
Consequently, in view of (2.5) and the fact (), we have
From system (1.2), we derive
Calculating the upper left derivative of , together with (2.4), (2.6), (2.7), (T1) and (T2), we obtain
It is a contradiction, and it shows that (2.2) holds. Then, using a similar argument as in the proof of (2.5) and (2.6), we can show that (2.3) holds. The proof of Lemma 2.1 is now completed. □
Remark 2.1 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 (1.2) with initial conditions (2.1) can be defined on .
Lemma 2.2 Suppose that (T1) and (T2) hold. Moreover, assume that is a solution of system (1.2) with initial function satisfying (2.1), and is bounded continuous on . Then for any , there exists , such that every interval contains at least one number δ, for which there exists satisfying
Proof For , set
By Lemma 2.1, the solution is bounded and
Thus, the right side of (1.2) is also bounded, which implies that is uniformly continuous on R. From (1.3), for any , there exists , such that every interval , , contains a δ, for which
Let be sufficiently large such that for , and denote . We obtain
Let be such an index that
Calculating the upper left derivative of along (2.11), we have
It is obvious that , and is non-decreasing. In particular,
Consequently, in view of (2.15) and the fact (), we have
Now, we consider two cases.
Case (i). If
Then, we claim that
Assume, by way of contradiction, that (2.18) does not hold. Then, there exists such that . Since
There must exist such that
which contradicts (2.17). This contradiction implies that (2.18) holds. It follows from (2.16) that there exists such that
Case (ii). If there is such a point that . Then, in view of (1.5), (2.9), (2.10), (2.13), (2.16), (T1) and (T2), we get
which yields that
For any , by the same approach used in the proof of (2.21) and (2.22), we have
On the other hand, if and , we can choose such that
which, together with (2.23), 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 2.2 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 (1.2).
Theorem 3.1 Suppose that (T1) and (T2) are satisfied. Then system (1.2) has exactly one almost periodic solution . Moreover, is globally exponentially stable.
Proof Let be a solution of system (1.2) with initial function satisfying (2.1), and is bounded continuous on .
where is any sequence of real numbers. By Lemma 2.1, the solution is bounded and
which implies that the right side of (1.2) is also bounded, and is a bounded function on R. Thus, is uniformly continuous on R. Then, from the almost periodicity of , , τ, and , we can select a sequence such that
for all , .
Since is uniformly bounded and equiuniformly continuous, by 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 R, and
Now, we prove that is a solution of (1.2). In fact, for any and , from (3.3), we have
which implies that
Therefore, is a solution of (1.2).
Secondly, we prove that is a almost periodic solution of (1.2). From Lemma 2.2, for any , there exists , such that every interval contains at least one number δ for which there exists satisfying
Then, for any fixed , we can find a sufficient large positive integer such that for any
Let , we obtain
which implies that is an almost periodic solution of (1.2).
Finally, we prove that is globally exponentially stable.
Let be the positive almost periodic solution of system (1.2) with initial value , and let be an arbitrary solution of system (1.2) with initial value , set . Then