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We study a class of impulsive neural networks with mixed time delays and generalized activation functions. The mixed delays include time-varying transmission delay, bounded time-varying distributed delay, and discrete constant delay in the leakage term. By using the contraction mapping theorem, we obtain a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. In addition, a delay-independent sufficient condition for existence of an equilibrium point and some delay-dependent sufficient conditions for stability are derived, respectively, by using topological degree theory and Lyapunov-Krasovskii functional method. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays. Moreover, the proposed stability criteria are given in terms of linear matrix inequalities (LMI), which can be conveniently checked by the MATLAB toolbox. Finally, an example is given to show the effectiveness and less conservativeness of the obtained results.

As we know, time delay in a system is a common phenomenon that describes the fact that the future state of the system depends not only on the present state but also on the past state and is always unavoidably encountered in many fields such as automatic control, biological chemistry, physical engineer, and neural networks [

Recently, a special type of time delay, namely, leakage delay (or forgetting delay), is identified and investigated due to its existence in many real systems. In 2007, Gopalsamy [

On the other hand, besides delay, impulses are also likely to exist in neural networks. In implementation of electronic networks, the state is subject to instantaneous perturbations and experiences abrupt change at certain moments, which may be caused by switching phenomenon, frequency change, or other sudden noises; that is, it does exhibit impulsive effects; see [

Motivated by aforementioned discussion, in this paper, we consider a class of impulsive neural networks with mixed time delays and generalized activation functions which could be different from each other. The mixed time delays include time-varying transmission delay, bounded time-varying distributed delay, and constant delay in the leakage term. Firstly, by using the contraction mapping theorem, we obtain a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Secondly, we present a delay-independent sufficient condition to guarantee the existence of an equilibrium point by using topological degree theory. Thirdly, some sufficient conditions which are dependent on the leakage delay, time-varying transmission delay, and distributed delay have been derived to guarantee the global asymptotic stability of the equilibrium point by using a new Lyapunov-Krasovskii functional and some analysis technique. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays, which are more effective and less conservative than other existing literatures [

Consider the following impulsive neural networks model:

Throughout this paper, we make the following assumptions.

The delay kernels

where

The neuron activation functions

for any

The impulse times

We will consider model (

Assume that

Generally speaking, the topological degree of

Given any real matrix

Given any real matrices

Supposing that

For presentation convenience, in the following, we denote

In this section, by using the contraction mapping theorem, we give a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for models (

Assume that the assumptions

Transform the global existence and uniqueness of solution of the models (

First we show that

Next we show that

Finally we show that

Continuing in this manner, we construct

In previous sections, we have showed the global existence and uniqueness of solution for models (

Assume that the assumption

From (

It should be noted that Theorem

Let

Under the conditions in Theorem

Consider the following Lyapunov-Krasovskii functional as

For arbitrary

Firstly, it follows from (

Finally, we can prove that

When there is no leakage delay, that is,

Under the conditions in Theorem

In [

In [

Recently, more researchers have begun to take into account the effect of time-varying leakage delay, which has essential difference from constant leakage delay, on dynamics of models. We would like to think that it may lead to more technical difficulties. How to improve the dynamics, especially the stability properties of neural networks with (time-varying) leakage delay, may be an interesting problem and requires further research.

In this section, an example is given to demonstrate the effectiveness of our results.

Consider the following recurrent neural networks model:

In this case, we know that

In this paper, we have investigated a class of impulsive neural networks with mixed time delays and generalized activation functions. Firstly, by using the contraction mapping theorem, we have given a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Then, a delay-independent sufficient condition for existence of the equilibrium point and some delay-dependent sufficient conditions for stability have been derived, respectively, by using topological degree theory and suitable Lyapunov-Krasovskii functional. The obtained results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability of time-varying delay. Finally, an example has been given to show the effectiveness and less conservativeness of the obtained results. In the future, we will do some further research on impulsive neural network models with leakage time-varying delay and continuously distributed delay.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by the National Natural Science Foundation of China Grant no. 11301304.