Existence and Global Exponential Stability of Almost Automorphic Solution for Clifford-Valued High-Order Hopfield Neural Networks with Leakage Delays
In this paper, we study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by direct method. That is to say, we do not decompose the systems under consideration into real-valued systems, but we directly study Clifford-valued systems. Our methods and results are new. Finally, an example is given to illustrate our main results.
In recent years, high-order Hopfield neural networks have become the object of intensive analysis by many scholars because of their stronger approximation characteristics, faster convergence speed, larger storage capacity, and higher fault tolerance than low-order Hopfield neural networks. A lot of excellent research results about their dynamic characteristics have been obtained [1–14].
On the one hand, the Clifford algebra was proposed by the British mathematician William K. Clifford  in 1878 and is a generalization of the plural, quaternion, and Glassman algebra. Currently, Clifford algebra has been widely used in various fields such as neural computing, computer and robot vision, image and signal processing, and control problems. Studies have shown that Clifford-valued neural networks are superior to commonly used real-valued neural networks [16, 17], so they have become an active research field in recent years. However, because the multiplication of Clifford numbers does not satisfy the commutative law, it has brought great difficulties to the research of Clifford-valued neural networks. Therefore, the current results on the dynamics of Clifford-valued neural networks are still very rare. At present, only a few papers have been published on the dynamics of Clifford-valued neural networks [18–22]. It is worth mentioning that the results of these mentioned papers are obtained by decomposing Clifford-valued neural networks into real-valued networks. Therefore, it is meaningful to study the dynamics of Clifford-valued neural networks by direct method.
On the other hand, it is well known that periodic and almost periodic oscillations are important dynamic behaviors of neural networks. Almost automorphy is an extension of almost periodicity and plays an important role in better understanding of almost periodicity. Therefore, almost automorphic oscillation is more complex than almost periodic oscillation. Considering the interaction between neurons in a neural network is very complex, it is meaningful to study the almost automorphic oscillation of neural networks.
In addition, time delays are inevitable and may affect and change the dynamic behavior of dynamic systems [23, 24]. Therefore, neural networks with various delays have been extensively studied. In particular, since K. Gopalsamy  first studied the stability of neural networks with leakage delays, a lot of research has been done on neural networks with leakage delays [6, 11, 12, 26]. However, there is no research on the Clifford-valued neural network with leakage delays.
Inspired by the above analysis and discussion, the main purpose of this paper is to study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays by direct method; that is, we will study the considered Clifford-valued neural networks directly, instead of converting them to real-valued ones. As far as we know, this is the first paper to study the almost automorphic solutions of Clifford-valued high-order Hopfield neural networks with leakage delays. In addition, this is the first paper to study almost automorphic solutions of Clifford neural networks by direct method. So, our methods and results of this paper are new. Besides, our methods proposed in this paper can be used to study the problem of almost automorphic solutions for other types of Clifford-valued neural networks.
This paper is organized as follows. In Section 2, we introduce some basic definitions and lemmas and give a model description. In Section 3, we study the existence of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays. In Section 4, we investigate the global exponential stability of almost automorphic solutions of the neural networks. In Section 5, an example is given to demonstrate the proposed results. Finally, we draw a brief conclusion in Section 6.
2. Preliminaries and Model Description
The real Clifford algebra over is defined as where with , Moreover, and , are said to be Clifford generators and satisfy , , and , , . For convenience, we will denote the product of Clifford generators as . Let ; then it is easy to see that , where is short for and .
For , we define and for , we define .
The derivative of is given by . For more knowledge about Clifford algebra, we refer the reader to .
In this paper, we are concerned with the following Clifford-valued high-order Hopfield neural network with delays in the leakage term:where corresponds to the state of the th unit at time , represents the rate with which the th unit will reset its potential to the resting state when disconnected from the network and external inputs at time , denotes the strength of th unit on th unit at time , is the second-order synaptic weight of the neural networks, denote the activation functions, is the external input on the th at time , and denote the transmission delays.
Remark 1. When , the number of the generators of , equals , and , system (2) degenerates into real-valued, complex-valued, and quaternion-valued systems, respectively.
We will adopt the following notation:
The initial value of system (2) is given by where
Denote by the set of all uniformly continuous functions from to . Let denote the set of all bounded continuous functions from to . Then is a Banach space with the norm , where .
Definition 2. A function is said to be almost automorphic, if for every sequence of real numbers there exists a subsequence such that is well defined for each , and for each .
From the above definition, similar to the proofs of the corresponding results in , it is not difficult to prove the following two lemmas.
Lemma 3. If , then
Lemma 4. Let satisfy the Lipschitz condition and ; then
Lemma 5. If , , then .
Proof. Since and , is uniformly continuous for . Hence, for each , there exists a positive number such that implies Since , for every sequence of real numbers there exists a subsequence such that for every . Therefore, there exists a positive integer such that for and . Hence, we have Consequently, converges to for each . Similarly, we can obtain that converges to for each . Therefore, . The proof is complete.
Throughout this paper, we make the following assumptions:(H1)For , and , .(H2)For , and there exist positive constants such that Moreover, for (H3), where
3. The Existence of Almost Automorphic Solutions
In this section, we study the existence of almost automorphic solutions by the contracting mapping principle.
Let For any , we define the norm of as , where ; then is a Banach space.
Let , and take a positive number .
Set and then, for every , we have
Theorem 6. Assume that - hold. Then system (2) has at least one almost automorphic solution in .
Proof. Firstly, it is easy to check that if is a solution of the integral equationwhere , then is also a solution of system (2).
Secondly, we define an operator by where , We will prove that maps into itself. To this end, let Then by Lemmas 3–5, . We will prove that .
Let be a sequence of real numbers; since and , we can extract a subsequence of such that, for each , and Set and then we have By the Lebesgue dominated convergence theorem, we obtain that for each . Similarly, one can prove that for each . Hence, .
Noticing that similar to the above, we can prove that . Therefore, maps into itself.
Thirdly, we will prove that is a self-mapping from to . In fact, for each , we have and Thus, we obtain Fourthly, we will prove that is a contracting mapping. In fact, for any , we have that and Hence, by , is a contracting mapping principle. Therefore, there exists a unique fixed-point such that , which implies that system (2) has an almost automorphic solution in . The proof is complete.
4. Global Exponential Stability
In this section, we investigate the global exponential stability of almost automorphic solutions by reduction to absurdity.
Definition 7. Let be an almost automorphic solution of system (2) with the initial value and let be an arbitrary solution of system (2) with the initial value , respectively. If there exist positive constants and such that where Then the almost automorphic solution of system (2) is said to be globally exponentially stable.
Theorem 8. Assume that - hold. Then system (2) has a unique pseudo almost automorphic solution that is globally exponentially stable.
Proof. By Theorem 6, system (2) has a pseudo almost periodic solution. Let be an almost automorphic solution of (2) with the initial value and let be an arbitrary solution with the initial value . Set , , and we havewhere Let and be defined by and where When , we get Since are continuous on and as , there exist such that and for and for , Let ; then we have So we can take a positive constant . Obviously, we have which implies thatwhere Let From , we have . Hence, for , we can obtain By (31), we haveMultiplying (41) by and integrating on , we haveIt is easy to see that We claim thatTo prove (44), we show that, for any , the following inequality holds:If (45) is false, then there must be some such thatandBy (39), (42), (46), and (47), we haveFinding the derivative of (42), then by (39), (46), and (47), we have