Abstract
Some sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of weighted pseudo-almost periodic solutions to a class of neutral type high-order Hopfield neural networks with distributed delays by employing fixed point theorem and differential inequality techniques. The results of this paper are new and they complement previously known results. Moreover, an example is given to show the effectiveness of the proposed method and results.
1. Introduction
Since high-order Hopfield neural networks (HHNNs) have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order Hopfield neural networks, the study of high-order Hopfield neural networks has recently gained a lot of attention and there have been extensive results on the problem of the existence and stability of equilibrium points, periodic solutions, and almost periodic solutions of high-order Hopfield neural networks in the literature. We refer the reader to [1–9] and the references cited therein. Also, since it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions [10], many authors investigated the dynamical behaviors of neutral type neural networks with delays [11–25]. Moreover, it is well known that, compared with periodic effects, almost periodic effects are more frequent, and many phenomena exhibit great regularity with being pseudo-almost periodic which allow complex repetitive phenomena to be represented as an almost periodic process plus an ergodic component. However, to the best of our knowledge, few authors have considered the exponential convergence on the pseudo-almost periodic solution for neutral type neural networks with delays. Motivated by the above, in this paper, we consider the following high-order Hopfield neural networks with neutral distributed delays: where corresponds to the number of units in a neural network, corresponds to the state vector of the th unit at the time , represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, and are the first- and second-order connection weights of the neural network, is the kernel, denote the external inputs at time , and and are the activation functions of signal transmission.
The initial conditions of (1) are the form where denotes a differential real-value bounded function defined on and satisfies that is bounded on .
To the best of our knowledge, there is no paper published on the global exponential stability and existence of weighted pseudo-almost periodic solution to system (1). Our main purpose of this paper is, for fixed will be defined in Section 2), to study the existence, uniqueness, and globally exponential stability of weighted pseudo-almost periodic solution by employing fixed point theorem and differential inequality techniques.
This paper is organized as follows. In Section 2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence of weighted pseudo-almost periodic solutions of (1). In Section 4, we prove that the weighted pseudo-almost periodic solution is globally exponentially stable. In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections.
2. Assumptions and Preliminaries
Throughout this paper we assume that(H1), are almost periodic functions, and , ;(H2)there exist positive constants , , , and such that , , , and , for all , and , ;(H3)for , the delay kernels are continuous and integrable with (H4)for fixed , is a weighted pseudo-almost periodic function.
Definition 1 (see [26, 27]). Let be continuous in ; is said to be almost periodic on if, for any , the set is relatively dense; that is, for all , it is possible to find a real number ; for any interval with length , there exists a number in this interval such that , for all .
For , we define .
Definition 2 (see [26, 27]). Let and let be an continuous matrix defined on . The linear system is said to admit an exponential dichotomy on if there exist positive constants , ; projection and the fundamental solution matrix of (4) satisfy
Lemma 3 (see [26, 27]). Let be an almost periodic function on ; then is bounded on and is uniformly continuous in .
The collection of all almost periodic functions which go from to will be denoted by . equipped with the sup-norm is a Banach space.
Let denote the collection of functions (weights) , which are locally integrable over such that almost everywhere. If and for , we set and Let and .
Definition 4 (see [28]). Fix . A continuous function is called weighted pseudo-almost periodic if it can be written as , with and , where the space is defined by The collection of all weighted pseudo-almost periodic functions will be denoted by .
Remark 5. If , then ; if , , then .
Lemma 6 (see [29]). Fix . Suppose that, for any , Then is translation-invariant.
Denote .
Remark 7. Fix . Let Then is a Banach space with the norm defined by , where , .
3. Existence of Weighted Pseudo-Almost Periodic Solution
To obtain the existence of weighted pseudo-almost periodic solution to system (1), we need the following lemmas.
Lemma 8 (see [26, 27]). If the linear system (4) admits an exponential dichotomy, then the almost periodic system has a unique almost periodic solution , and
Lemma 9 (see [26, 27]). Let be an almost periodic function on , and Then the linear system admits an exponential dichotomy on .
Lemma 10. Fix . If satisfies the Lipschitz condition, is continuous and integrable, and satisfying (where is a positive constant) and , then belongs to .
Proof. Since , there exist and such that ; then First, we prove that . Since satisfies the Lipschitz condition, there exists a positive constant , such that for all . For any , since , it is possible to find a real number ; for any interval with length , there exists a number in this interval such that for all ; then which implies that . Next, we prove that . Consider Consider the following function: obviously, is bounded, and, by using Lemma 6, we have . Consequently, by the Lebesgue dominated convergence theorem, we get which implies that . The proof is complete.
Lemma 11. Fix . Suppose that assumptions (H1)–(H3) hold. For each , define a nonlinear operator as follows: where, for , and then maps into itself.
Proof. Let ; by Lemma 10 and in view of (H1)–(H4), we have ; that is, can be rewritten as , where and . Hence, Consider the following almost periodic system: Since , from Lemmas 8 and 9, system (22) has an almost periodic solution which can be expressed as follows: that is, . Let in order to prove that , we will prove . Notice that where . Since the function , then the functions defined by are bounded and satisfy . Consequently, by the Lebesgue dominated convergence theorem, we have that is, . Now, we can get that , and obviously, ; that is, maps into itself.
For the sake of convenience, we introduce the following notations:
Theorem 12. Suppose that (H1)–(H4) and hold; then there exists a unique continuously differentiable weighted pseudo-almost periodic solution of system (1) in the region .
Proof. , from Lemma 11, maps into itself. By the definition of the norm of Banach space , we have Hence, , we obtain Next, we show that maps the closed set into itself. In fact, for any , we obtain by (H2)-(H3) that Furthermore, we have Thus, it follows from (33) and (34) that which implies that . So, the mapping is a self-mapping from to . Finally, we prove that is a contraction mapping of the . In fact, in view of , for any , we obtain Thus, Notice that ; it means that the mapping is a contraction mapping. By Banach fixed point theorem, there exists a unique fixed point such that , which implies that system (1) has a unique weighted pseudo-almost periodic solution. This completes the proof.
Since and are Banach spaces, we can get the following corollary.
Corollary 13. If (H1)–(H3) and hold, furthermore, assume that are almost periodic functions; then there exists a unique continuously differentiable almost periodic solution of system (1) in the region where .
Corollary 14. If (H1)–(H3) and hold, furthermore, assume that are pseudo-almost periodic functions; then there exists a unique continuously differentiable pseudo-almost periodic solution of system (1) in the region where .
4. Global Exponential Stability of Weighted Pseudo-Almost Periodic Solution
Definition 15. Fix . The weighted pseudo-almost periodic solution of system (1) with initial value is said to be globally exponentially stable. If there exist constants and such that for every solution of system (1) with any initial value satisfies where
Theorem 16. Fix . If conditions (H1)–(H5) hold, then system (1) has a unique continuously differentiable weighted pseudo-almost periodic solution which is globally exponentially stable.
Proof. It follows from Theorem 12 that system (1) has a unique weighted pseudo-almost periodic solution
with initial value . Let be an arbitrary solution of system (1) with initial value . Let , ; then
where . Let and be defined by
where . By , we obtain that
Since are continuous on and as , there exist such that , and for , for . By choosing , we have
So, we can choose a positive constant such that and , which implies that
where .
Multiplying both sides of (44) by and integrating it over , we have
Let
By , we have . Thus,
where as in (48). We claim that
To prove (53), we first show that, for any , the following inequality holds:
If (54) is not true, then there must be some and some , such that
By (48)–(51), (56), and , we get
Direct differentiation of (49) gives
where . Thus, we have by (48) and (58) and (H2)-(H3) that
In view of (57) and (59), we obtain
which contradicts the equality (55), and so (54) holds. Letting , (53) holds. Hence, the weighted pseudo-almost periodic solution of system (1) is globally exponentially stable.
Corollary 17. If conditions (H1)–(H3) and hold, furthermore, assume that are almost periodic functions; then system (1) has a unique continuously differentiable almost periodic solution which is globally exponentially stable.
Corollary 18. If conditions (H1)–(H3) and hold, furthermore, assume that are pseudo-almost periodic functions; then system (1) has a unique continuously differentiable pseudo-almost periodic solution which is globally exponentially stable.
5. An Example
In this section, we give one example to illustrate our result. Consider the weight and let
Then system (1) has exactly one continuously differentiable weighted pseudo-almost periodic solution, which is globally exponentially stable.
Proof. By calculating, hence we have It is obvious that (H1)–(H5) are satisfied. By Theorems 12 and 16, system (1) has exactly one continuously differentiable weighted pseudo-almost periodic solution, which is globally exponentially stable (see Figures 1, 2, and 3).
6. Conclusion
In this paper, we employ fixed point theorem and differential inequality techniques to study the existence, uniqueness, and global exponential stability of weighted pseudo-almost periodic solutions to a class of neutral type high-order Hopfield neural networks with infinitely distributed delays. Our results of this paper are new and complement previously known results, and our methods used in this paper can be used to investigate other types of neural networks such as neutral type BAM neural networks, neutral type Cohen-Grossberg neural networks, and neutral type high-order Hopfield neural networks with delays in the leakage term and so on.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of China under Grant 11361072.