Research Article | Open Access
Antiperiodic Solutions for Quaternion-Valued Shunting Inhibitory Cellular Neural Networks with Distributed Delays and Impulses
This paper is concerned with quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with distributed delays and impulses. By using a new continuation theorem of the coincidence degree theory, the existence of antiperiodic solutions for QVSICNNs is obtained. By constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of antiperiodic solutions for QVSICNNs. Finally, an example is given to show the feasibility of obtained results.
The shunting inhibitory cellular neural networks (SICNNs) [1, 2] have found many applications in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Since all of these applications heavily rely on the dynamics of SICNNs and time delays are unavoidable in a realistic system [3–13], there have been extensive results about the dynamical behaviors of SICNNs with time delays [4–10]. Besides, a wide variety of evolutionary processes which exist universally in nature and many signal transmission processes in neural networks are often subject to abrupt changes at certain moments due to instantaneous perturbations which lead to impulsive effects. Also, the existence of impulses is frequently a source of instability, bifurcation, and chaos for neural networks [11–13]. Therefore, many researchers have investigated various dynamical behaviors of SICNNs with time delays and impulses [12, 13].
On the other hand, quaternion-valued neural networks, which can be seen as a generic extension of complex-valued neural networks (CVNNs) or real-valued neural networks (RVNNs), are much more complicated than CVNNs for their quaternion-valued states, quaternion-valued connection weights, and quaternion-valued activation functions. In the past decades, QVNNs have found many practical applications in aerospace and satellite tracking, processing of polarized waves, image processing, 3D geometrical affine transformation, spatial rotation [14, 15], color night vision , and so on. Due to so many practical applications, it is necessary to study the dynamics of QVNNs. At present, only a few of the dynamical behaviors of QVNNs have been studied [17–24]. For example, in [21, 22], the global -stability criteria for QVNNs were studied, respectively; in , based on Mawhin’s continuation theorem of coincidence degree theory, the existence of periodic solutions for QVNNs was established; in , the multiplicity of periodic solutions for QVNNs was discussed by employing Brouwer’s and Leray-Schauder’s fixed point theorems. However, to the best of our knowledge, the antiperiodic oscillation of QVVNs with time-varying delays and impulses has not been reported. Since the existence and stability of antiperiodic solutions are an important topic in nonlinear differential equations and the signal transmission process of neural networks can often be described as an antiperiodic process, the antiperiodic oscillation of neural networks have been considered by many authors, see [13, 25–33]. So, it is necessary to study the antiperiodic solutions for QVNNs.
Motivated by the above, in this paper, we are concerned with the following QVSICNN with distributed delays and impulsive effects: where ; denotes the cell at the position of the lattice. The neighborhood of is given by
is the activity of the cell ; is the external input to , represents the passive decay rate of the cell activity, is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell , and the activity function is a continuous function representing the output or firing rate of the cell ; corresponds to the transmission delay kernel; and is a sequence of real numbers such that as , there exists a such that . Without loss of generality, we also assume that . For convenience, we denote and , where is a bounded function.
The main purpose of this paper is to establish the existence of antiperiodic solutions to (1) by using a new continuation theorem of coincidence degree theory and by constructing a suitable Lyapunov function to obtain the global exponential stability of the antiperiodic solution. The results of this paper are completely new and supplement the previously known results.
Throughout this paper, we assume Hypotheses 1 to 4. (H1)Let , assume that the activity function of (1) can be expressed as where , ; the impulsive operators can be expressed as where , , and , and the external inputs can be expressed as where , and .(H2)Functions , where , and there exists a constant such that and functions , where and , satisfying (H3)For , (H4)For , , there exist positive constants , , and such that and where and are mentioned in H1.
We will adopt the following notations:
The initial value of (1) is given by where
The remaining part of this paper is organized as follows. In Section 3, some definitions are given. In Section 4, we obtain sufficient conditions for the existence of antiperiodic solutions of (1). In Section 5, the global exponential stability of the antiperiodic solution is studied. In Section 6, we give an example to illustrate the feasibility of the obtained results.
The quaternion was invented in 1843 by Hamilton . The skew field of the quaternion is denoted by where are real numbers and the elements , and obey Hamilton’s multiplication rules:
In order to avoid the difficulty resulting from the noncommutativity of the quaternion multiplication, by Hamilton’s rules and H1, we decompose (1) into the following systems: where , , and , , , and . That is, (1) can be decomposed as the following real-valued system: with the initial values where is continuous and bounded.
Definition 1. A piecewise continuous function is said to be a solution of (16), if (i) for , where , , , and ;(i) satisfies (16) for ;(ii) is continuous everywhere except for some and left continuous at , and the right limit exists for .
Definition 3. Let be a solution of (16) with the initial value and be an arbitrary solution of (16) with the initial value . If there exist constants and such that then the solution of (16) is said to be globally exponentially stable, where
Remark 1. If is an -antiperiodic solution to (16), then , where must be an -antiperiodic solution to (1). Thus, the problem of finding an -antiperiodic solution for (1) reduces to finding one for the system of (16). For considering the stability of the solution of (1), we just need to consider the stability of the solutions of (16).
3. The Existence of Antiperiodic Solutions
In this section, based on a new continuation theorem of coincidence degree theory, we shall study the existence of antiperiodic solutions of (1).
Lemma 1.  Let and be Banach spaces, and let be linear and be continuous. Assume that is one-to-one and is compact. Furthermore, assume that there exists a bounded and open subset with such that equation has no solutions in for any . Then the problem has at least one solution in .
Theorem 1. Assume that H1–H4 holds. Furthermore, suppose that A1.If one hasthen (1) has at least one -antiperiodic solution.
Proof 1. One has
Then is a Banach space with the norm , and is also a Banach space with the norm , where is any norm of .
Define a linear operator by where and a continuous operator by
It is easy to see that
Hence, is reversible. Denote by the inverse of , one has where for all . Let , by applying the Arzela-Ascoli theorem, we know that is compact. Corresponding to the operator equation , we have
Suppose that is a solution of (29) for a certain , set , then we have
Repeating the above procession, for , we can obtain that
Integrating both sides of (29) over the interval , we can obtain
Repeating the above procession, for , we have
Since for any , and , we have then for any , , and , we obtain
Let such that , , , and ; by the arbitrariness of , we can obtain