Journal of Applied Mathematics

1. Introduction

Neural networks and their various generalizations have been successfully employed in many areas such as pattern recognition, cognitive modeling, adaptive control, and combinatorial optimization [1–7]. Hopfield neural networks (HNNs), as some forms of recurrent artificial neural networks, have been widely studied in recent years [8–12]. The earlier HNNs model proposed by Hopfield [13, 14] was based on the theory of analog circuit consisting of capacitors, resistors, and amplifiers and can be formulated as a system of ordinary equations. Time delays are inevitable in the interactions of neurons in biological and artificial neural networks. The existence of delays is frequently a source of instability for neural networks [9, 10, 15–19].

Over the past few decades, the stability of HNNs with time delays has attracted considerable attention in the literature [20–23]. One of the most investigated problems in the study of HNNs is global exponential stability of the equilibrium point. An equilibrium point of HNNs is globally exponentially stable, if the domain of attraction of the equilibrium point is the whole space and the convergence is in real time.

It is worth noting that although the signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that the distributed delays should be incorporated in the model [24]. Discrete delays and distributed delays attract the attention of many scholars and have been widely studied [17, 19, 25]. To the best of our knowledge, the stability problem for system with both discrete and distributed delays has been a challenging issue, mainly due to the mathematical difficulties in dealing with discrete and distributed delays simultaneously. In 2002, Wang and Xu [26] presented a new neural network model with S-type distributed time delays and demonstrated that S-type distributed time delays include discrete or continuously distributed time delays, but it is not true in the opposite way. In the following years, S-type distributed time delayed neural network models have raised great interest [12, 27–29].

Compared with traditional Hopfield neural networks, the high-order Hopfield type neural networks (HOHNNs) [11, 12, 30–34] have the advantages of stronger approximation properties, faster convergence rate, greater storage capacity, and higher fault tolerance. Therefore, it is of considerable interest to explore the theoretical foundations and practical applications of HOHNNs.

Motivated by the aforementioned discussion, we studied the problem of global exponential robust stability of HOHNNs with S-type distributed time delays. By employing differential inequality technique and a new Lyapunov functional method, some criteria for the global exponential robust stability of the high-order neural networks with S-type distributed time delays have been established, which are easy to be verified with a wider adaptive scope. Meanwhile, the systems in [10, 12, 26, 31] are some special cases of the HOHNNs with S-type distributed time delays.

2. Model Description and Preliminaries

We consider the following HOHNNs with S-type distributed time delays: where , , , , , and have the same meanings as those in [28], , are the first- and second-order synaptic weights of the system (1) (see [12]), and are Lebesgue-Stieltjes integrable, where and are nondecreasing bounded variation functions which satisfy

In this paper the superscript “” presents the transpose and denotes a set of continuous bounded functions.

For system (1), the initial condition is

If there is an equilibrium point of system (1) with conditions (3), we can rewrite system (1) as the following equivalent form:

From [12], we know that the following system (5) is equivalent to system (4): where .

Definition 1. The equilibrium point of system (1) is called globally exponentially robustly stable, if, for any , there exist scalars and such that the solution to system (1) with initial condition , , satisfies

Let , , , , , , , = , = , = , = = , = , = = be Hadamard product of two matrices,

We assume throughout that the neuron activation functions , , , satisfy the following conditions. () Consider  () Consider  () Consider

3. Main Results

Theorem 2. The equilibrium of system (1) is globally exponentially robustly stable, if system (1) satisfies , , and .

Proof. Part 1: Existence of the Equilibrium Point. Let
It is obvious that the solutions to (11) are the equilibrium points of system (1).
Let us define homotopic mapping as follows: where
By homotopy invariance theorem (see [35]), topological degree theory (see [36]), (), and the proof, which is similar to Theorem 1 in [28], we can conclude that (13) has at least one solution.
That is, system (1) has at least an equilibrium point.
Part 2: Global Existence of the Solutions to System (1). Since is an -matrix, there exists a constant vector such that (see [15]); that is,
Suppose is a solution to system (1) and also satisfies the initial condition , , .
Let us choose a positive constant such that
From (15) we know that
Then, we will show that
If (17) is not true, there must be some positive integer and , such that
From , (15), and (18), we know
So which leads by contradiction to (18).
Hence, (17) holds. That is, the solutions to system (1) are bounded. So the solutions to system (1) are of global existence.
Part 3: Global Exponential Stability of System (1). From , we know that there exists constant , such that
So, from , we can choose a constant sufficiently small, such that
Let , and
Then, we will show that there exists such that
Define a Lyapunov functional by
Its Dini derivative reads
If , we have
Then, we will prove that
If (28) is not true, there exists and such that
From (29), we have
Because and , for all , , we can obtain
It is obvious that (31) is in contradiction to (30). Hence, (28) holds. That is,
So where .
If there exists another equilibrium of system (1), we have , , .
From above proof, the system (1) has a unique equilibrium point , which is globally exponentially robustly stable.
The proof of Theorem 2 is completed.