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Almost periodic solutions for neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales
Advances in Difference Equations volume 2014, Article number: 178 (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.
Motivated by above, in this paper, we propose the following neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scale :
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.
If , then system (1.1) is reduced to the following continuous-time neutral delay Hopfield neural network:
and if , then system (1.1) is reduced to the discrete-time neutral delay Hopfield neural network
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.
For convenience, for any almost periodic function defined on , we define , .
The initial condition associated with system (1.1) is of the form
where denotes a real-value bounded Δ-differentiable function defined on and .
Throughout this paper, we assume that:
(H1) with , , , , , , , , , and are all almost periodic functions on , , , , , for , .
(H2) There exist positive constants , , , such that for ,
where and .
(H3) For , the delay kernels are continuous and integrable with
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.
Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by
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 and , we define the delta derivative of , , to be the number (if it exists) with the property that for a given , there exists a neighborhood U of t such that
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.
Let y be right-dense continuous. If , then we define the delta integral by
A function is called regressive if
for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .
If r is a regressive function, then the generalized exponential function is defined by
with the cylinder transformation
Let be two regressive functions, we define
Then the generalized exponential function has the following properties.
Definition 2.1 
Let be two regressive functions, define
Lemma 2.1 
Assume that are two regressive functions, then
if , then .
Definition 2.2 
Assume that is a function and let . Then we define to be the number (provided it exists) with the property that given any , there is a neighborhood U of t (i.e., for some ) such that
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 
A time scale is called an almost periodic time scale if
Definition 2.4 
Let be an almost periodic time scale. A function is called an almost periodic function if the ε-translation set of
is a relatively dense set in for all ; that is, for any given , there exists a constant such that each interval of length contains a such that
τ is called the ε-translation number of f and is called the inclusion length of .
Definition 2.5 
Let be an rd-continuous matrix on , the linear system
is said to admit an exponential dichotomy on if there exist positive constants k, α, projection P and the fundamental solution matrix of (2.1), satisfying
where is a matrix norm on (say, for example, if , then we can take ).
Consider the following almost periodic system:
where is an almost periodic matrix function, is an almost periodic vector function.
Lemma 2.2 
If the linear system (2.1) admits exponential dichotomy, then system (2.2) has a unique almost periodic solution
where is the fundamental solution matrix of (2.1).
Lemma 2.3 
Let be an almost periodic function on , where , , , and , then the linear system
admits an exponential dichotomy on .
One can easily prove the following.
Lemma 2.4 Suppose that is an rd-continuous function and is a positive rd-continuous function satisfying . Let
where , then
3 Existence of almost periodic solutions
Let , and
For , if we define induced modulus , where
and , then is a Banach space.
Theorem 3.1 Assume that (H1)-(H3) and
hold, then there exists exactly one almost periodic solution of system (1.1) in the region , where
Proof Rewrite (1.1) in the form
For any , we consider the following system:
Since , it follows from Lemma 2.2 and Lemma 2.3 that system (3.1) has a unique almost periodic solution which can be expressed as follows:
Now, we define a mapping by , .
By the definition of , we have
Hence, for any , one has
Next, we will show that . In fact, for any , we have
Thus, we obtain
which implies , so the mapping T is a self-mapping from to .
Finally, we prove that T is a contraction mapping. Taking , we have that
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
Definition 4.1 The almost periodic solution of system (1.1) with initial value is said to be globally exponentially stable. If there exist positive constants λ with and such that every solution of system (1.1) with initial value satisfies
Theorem 4.1 Assume that (H1)-(H4) hold, then system (1.1) has a unique almost periodic solution which is globally exponentially stable.
Proof From Theorem 3.1, we see that system (1.1) has at least one almost periodic solution . Suppose that is an arbitrary solution. Set , , then it follows from system (1.1) that
where and for ,
From (H2) we have that for ,
The initial condition of (4.1) is
Let and be defined by
By (H3), for , we get
Since , are continuous on and , , as , so there exist such that and for , for , .
By choosing , we have , , . So, we can choose a positive constant such that
which implies that
where . Let
by (H3) we have . Thus
Rewrite (4.1) in the form
Multiplying the both sides of (4.4) by and integrating over , we get
It is easy to see that
We claim that
To prove (4.6), we first show that for any , the following inequality holds:
If (4.7) is not true, then there must be some and some such that
Therefore, there must exist a constant such that
By (4.5), (4.8), (4.9) and (H1)-(H3), we obtain
By Lemma 2.4 and (4.5), we have, for ,
Thus, it follows from (4.8), (4.9) and (4.11) that