Discrete Dynamics in Nature and Society

1. Introduction

As we know, the mathematical model of neural network consists of four basic components: an input vector, a set of synaptic weights, summing function with an activation, or transfer function, and an output. From the view point of mathematics, an artificial neural network corresponds to a nonlinear transformation of some inputs into certain outputs. Due to their promising potential for tasks of classification, associative memory, parallel computation and solving optimization problems, neural networks architectures have been extensively researched and developed [1–25]. Most of neural models can be classified as either continuous-time or discrete-time ones. For relative works, we can refer to [20, 24, 26–28].

However, besides the delay effect, an impulsive effect likewise exists in a wide variety of evolutionary process, in which states are changed abruptly at certain moments of time [5, 29]. As is well known, stability is one of the major problems encountered in applications, and has attracted considerable attention due to its important role in applications. However, under impulsive perturbation, an equilibrium point does not exist in many physical systems, especially, in nonlinear dynamical systems. Therefore, an interesting subject is to discuss the invariant sets and the attracting sets of impulsive systems. Some significant progress has been made in the techniques and methods of determining the invariant sets and attracting sets for delay difference equations with discrete variables, delay differential equations, and impulsive functional differential equations [30–37]. Unfortunately, the corresponding problems for discrete-time neural networks with impulses and delays have not been considered. Motivated by the above-mentioned papers and discussion, we here make a first attempt to arrive at results on the invariant sets and attracting sets of discrete-time neural networks with impulses and delays.

2. Preliminaries

In this paper, we consider the following discrete-time networks under impulsive effect:

where are real constants, ; , are positive real numbers such that is an impulsive sequence such that and are real-valued functions.

By a solution of (2.1), we mean a piecewise continues real-valued function defined on the interval which satisfies (2.1) for all .

In the sequel, by we will denote the set of all continuous real-valued function defined on the interval , which satisfies the compatibility condition:

By the method of steps, one can easily see that, for any given initial function , there exists a unique solution of (2.1) which satisfies the initial condition:

this function will be called the solution of the initial problem (2.1)–(2.3).

For convenience, we rewrite (2.1) and (2.3) into the following vector form

where , , , , , , and , in which

In what follows, we will introduce some notations and basic definitions.

Let be the space of -dimensional real column vectors and let denote the set of real matrices. denotes an identical matrix with appropriate dimensions. For or means that each pair of corresponding elements of and satisfies the inequality . Particularly, is called a nonnegative matrix if and is denoted by and is called a positive vector if . denotes the spectral radius of .

denotes the space of continuous mappings from the topological space to the topological space .

and exist for for and except for points , where is an interval, and denote the right limit and left limit of function , respectively. Especially, let .

Definition 2.1. The set is called a positive invariant set of (2.4), if for any initial value , one has the solution for .

Definition 2.2. The set is called a global attracting set of (2.4), if for any initial value , the solution converges to as That is, where In particular, is called asymptotically stable.

Following [33], we split the matrices into two parts, respectively,

with ,, ,,,

Then the first equation of (2.4) can be rewritten as

Now take the symmetric transformation . From (2.7), it follows that

where

Set

By virtue of (2.8) and (2.7), we have

Set , in view of the impulsive part of (2.4), we also have , , and so we have

where

Lemma 2.3 (see [34]). Suppose that and , then there exists a positive vector such that
For and , one denotes

By Lemma 2.3, we have the following result.

Lemma 2.4. is nonempty, and for any scalars and vectors , one has

Lemma 2.5. Assume that satisfy where ,
If , then there exists a positive vector such that
where is a constant and defined as for the given .

Proof. Since and , by Lemma 2.3, there exists a positive vector such that .
Set then we have
Due to there must exist a , such that
For , , there exists a positive constant such that
where
By Lemma 2.4, . Without loss of generality, we can find a such that

Set substituting this into (2.15), we have
By (2.23), we get that
Next, we will prove for any ,
To this end, we consider an arbitrary number , we claim that Otherwise, by the continuity of , there must exist a and index such that Then, by using (2.24) and (2.28), from (2.22), we obtain which is a contradiction. Hence, (2.27) holds for all numbers . It follows immediately that (2.26) is always satisfied, which can easily be led to (2.16). This completes the proof.

3. Main Results

For convenience, we introduce the following assumptions.

Theorem 3.1. Assume that hold. Then there exists a positive vector such that is a positive invariant set of (2.4), where .

Proof. From and , we can claim that for any , where , and satisfied , ,
So, by using (2.10) and (2.11) and taking into account (3.3), we get
respectively.
By assumptions , and Lemma 2.3, there exists a positive vector such that

Since and are positive constant vectors, by (3.6), there must exist two scalars , such that
respectively.
Set
by Lemma 2.4, clearly, and
Next, we will prove, for any , that is, ,

In order to prove (3.11), we first prove, for any ,

If (3.12) is false, by the piecewise continuous nature of , there must exist a and an index such that

Denoting , we get
This is a contradiction and hence (3.12) holds. From the fact that (3.12) is fulfilled for any , it follows immediately that (3.11) is always satisfied.
On the other hand, by using (3.5), (3.10), and (3.11), we obtain that
Therefore, we can claim that In a similar way to the proof of (3.11), we can proof that (3.16) implies Hence, by the induction principle, we conclude that which implies holds for any , that is, for any . This is completes the proof of the theorem.

Remark 3.2. In fact, under the assumptions of Theorem 3.1, the must exist, for example, since and imply and , respectively, so we may take as the follows: