Almost periodic solutions for neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales
© Li et al.; licensee Springer 2014
Received: 15 March 2014
Accepted: 30 June 2014
Published: 22 July 2014
In this paper, a class of neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales is considered. By utilizing the exponential dichotomy of linear dynamic equations on time scales, Banach’s fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results even when the time scale .
The dynamical properties for delayed Hopfield neural networks have been extensively studied since they can be applied into pattern recognition, image processing, speed detection of moving objects, optimization problems and many other fields. Besides, due to the finite speed of information processing, the existence of time delays frequently causes oscillation, divergence, or instability in neural networks. Therefore, it is of prime importance to consider the delay effects on the stability of neural networks. Up to now, neural networks with various types of delay have been widely investigated by many authors [1–20].
However, so far, very little attention has been paid to neural networks with time delay in the leakage (or ‘forgetting’) term [21–35]. Such time delays in the leakage terms are difficult to handle and have been rarely considered in the literature. In fact, the leakage term has a great impact on the dynamical behavior of neural networks. Also, recently, another type of time delays, namely, neutral-type time delays which always appear in the study of automatic control, population dynamics and vibrating masses attached to an elastic bar, etc., has drawn much research attention. So far there have been only a few papers that have taken neutral-type phenomenon into account in delayed neural networks [33–43].
In fact, both continuous and discrete systems are very important in implementation and applications. But it is troublesome to study the existence of almost periodic solutions for continuous and discrete systems respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations (see [44–50]).
To the best of our knowledge, up to now, there have been no papers published on the existence and stability of almost periodic solutions to neutral-type delay neural networks with time-varying delays in the leakage term on time scales. Thus, it is important and, in effect, necessary to study the existence of almost periodic solutions for neutral-type neural networks with time-varying delay in the leakage term on time scales.
where is an almost periodic time scale that will be defined in the next section, denotes the potential (or voltage) of cell i at time t, represents the rate at which the i th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t, , , and represent the delayed strengths of connectivity and neutral delayed strengths of connectivity between cell i and j at time t, respectively, and are the kernel functions determining the distributed delays, , , and are the activation functions in system (1.1), is an external input on the i th unit at time t, and correspond to the transmission delays of the i th unit along the axon of the j th unit at time t.
where and . When , , , , Bai  and Xiao  studied the almost periodicity of (1.2), respectively. However, even when , , the almost periodicity to (1.3), the discrete-time analogue of (1.2), has been not studied yet.
where denotes a real-value bounded Δ-differentiable function defined on and .
Throughout this paper, we assume that:
where and .
Our main purpose of this paper is to study the existence and global exponential stability of the almost periodic solution to (1.1). Our results of this paper are completely new and complementary to the previously known results even when the time scale . The organization of the rest of this paper is as follows. In Section 2, we introduce some definitions and make some preparations for later sections. In Section 3 and Section 4, by utilizing Banach’s fixed point theorem and the theory of calculus on time scales, we present some sufficient conditions which guarantee the existence of a unique globally exponentially stable almost periodic solution for system (1.1). In Section 5, we present examples to illustrate the feasibility and effectiveness of our results obtained in previous sections. We draw a conclusion in Section 6.
In this section, we shall first recall some basic definitions and lemmas which will be useful for the proof of our main results.
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum m, then ; otherwise . If has a right-scattered minimum m, then ; otherwise .
A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on .
for all .
If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.
for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .
Then the generalized exponential function has the following properties.
Definition 2.1 
Lemma 2.1 
if , then .
Definition 2.2 
for all . We call the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable (or, in short, differentiable) on provided exists for all . The function is then called the (delta) derivative of f on .
Definition 2.3 
Definition 2.4 
τ is called the ε-translation number of f and is called the inclusion length of .
Definition 2.5 
where is an almost periodic matrix function, is an almost periodic vector function.
Lemma 2.2 
where is the fundamental solution matrix of (2.1).
Lemma 2.3 
One can easily prove the following.
3 Existence of almost periodic solutions
Theorem 3.1 Assume that (H1)-(H3) and
Now, we define a mapping by , .
which implies , so the mapping T is a self-mapping from to .
Noticing that , it means that T is a contraction mapping. Thus, there exists a unique fixed point such that . Then system (1.1) has a unique almost periodic solution in the region . This completes the proof. □
4 Exponential stability of the almost periodic solution
Theorem 4.1 Assume that (H1)-(H4) hold, then system (1.1) has a unique almost periodic solution which is globally exponentially stable.
Since , are continuous on and , , as , so there exist such that and for , for , .
which contradicts (4.8), and so (4.7) holds. Letting , then (4.6) holds. Hence, the almost periodic solution of system (1.1) is globally exponentially stable. This completes the proof. □
Remark 4.1 When , , , Theorem 3.1 and Theorem 4.1 are reduced to Theorem 2.3 and Theorem 3.1 in , respectively.
Remark 4.2 According to Theorem 3.1 and Theorem 4.1, we see that the existence and exponential stability of almost periodic solutions for system (1.1) only depend on time delays (the delays in the leakage term) and do not depend on time delays and .