Almost periodic solutions for SICNNs with time-varying delays in the leakage terms
© Liu and Shao; licensee Springer. 2013
Received: 23 March 2013
Accepted: 23 September 2013
Published: 7 November 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.
Keywordsshunting inhibitory cellular neural networks almost periodic solution exponential stability time-varying delay leakage term
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.
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.
for all , .
where is a bounded continuous function.
We also make the following assumptions.
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.
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 .
Now, we consider two cases.
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 .
for all , .
Therefore, is a solution of (1.2).
which implies that is an almost periodic solution of (1.2).
Finally, we prove that is globally exponentially stable.