Existence and exponential stability of anti-periodic solutions for HCNNs with time-varying leakage delays
© Zhang; licensee Springer 2013
Received: 27 March 2013
Accepted: 21 May 2013
Published: 10 June 2013
This paper is concerned with a class of high-order cellular neural networks (HCNNs) model with time-varying delays in the leakage terms. By using the Lyapunov functional method and differential inequality techniques, we establish sufficient conditions on the existence and exponential stability of anti-periodic solutions for the model. Our results complement some recent ones.
in which n corresponds to the number of units in a neural network, corresponds to the state vector of the i th unit at the time t, represents the rate with which the i th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, , and are the first- and second-order connection weights of the neural network, corresponds to the time-varying leakage delays, , and correspond to the transmission delays, and correspond to the transmission delay kernels, denotes the external inputs at time t, , and are the activation functions of signal transmission.
where denotes a real-valued bounded continuous function defined on .
Under some suitable conditions on coefficients of (1.1), the author in  derived some new sufficient conditions ensuring that all solutions of system (1.1) converge exponentially to the anti-periodic solution, but the result leaves room for improvement. In fact, in the proof of Lemma 2.1, the expression was used, and the author replaced by the right-hand side of equation (1.1). The case that is possible, so the integration should be handled as , for can be replaced by the right-hand side of equation (1.1), but for cannot be replaced by the right-hand side of equation (1.1). A similar error also occurs in Lemma 2.2 of . For this reason, the course of proof in Lemmas 2.1 and 2.2 is not true. Motivated by this, we shall give a new proof to ensure the existence and exponential stability of the anti-periodic solutions for system (1.1). Moreover, an example is also provided to illustrate the effectiveness of our results.
where , and are real-valued bounded continuous functions defined on R.
In order to investigate the anti-periodic solution of HCNNs (1.1), we also give some usual assumptions.
where , .
2 Preliminary lemmas and main results
for all , .
This contradicts with and hence (2.3) is proved. This completes the proof. □
Remark 2.1 In view of the boundedness of this solution in Lemma 2.1, from the theory of functional differential equations with infinite delay in , it follows that the solution of system (1.1) with initial conditions satisfying (2.1) can be defined on .
for all and . The proof of Lemma 2.2 is completed. □
Remark 2.2 If is the T-anti-periodic solution of system (1.1), it follows from Lemma 2.2 that is globally exponentially stable.
Theorem 2.1 Suppose that (H1) and (H2) are satisfied. Then system (1.1) has exactly one T-anti-periodic solution . Moreover, is globally exponentially stable.
Proof The proof proceeds in the same way as in Theorem 3.1 in . □
3 Example and remark
In this section, some examples and remarks are provided to demonstrate the effectiveness of our results.
which implies that system (3.1) satisfies all the conditions in Theorem 2.1. Hence, system (3.1) has exactly one π-anti-periodic solution. Moreover, the π-anti-periodic solution is globally exponentially stable.
This implies that the results of this paper are new and complement the corresponding ones in .
This work was supported by the Scientific Research Fund of Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ3007) and the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant No. Y6110436).
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