Abstract
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.
1. Introduction
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 [16], 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 [23], based on Mawhin’s continuation theorem of coincidence degree theory, the existence of periodic solutions for QVNNs was established; in [24], 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.
2. Preliminaries
The quaternion was invented in 1843 by Hamilton [34]. 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 2. A solution of system (16) is said to be the -antiperiodic solution of (16), if
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. [35] 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
Set and
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
Hence,
Repeating the above procession, for , we have
Since for any , and , we have then for any , , and , we obtain
Dividing by on the both sides of (36) and (37), respectively, we have
Let such that , , , and ; by the arbitrariness of , we can obtain and
Thus, from (39) and (40) we have hence that is
Obviously, is independent of . Take then with such that equation has no solutions in for any . Thus, by Lemma 1 we conclude that has at least one -antiperiodic solution in , that is, (16) has at least one -antiperiodic solution. By Remark 1, we see that (1) has at least one -antiperiodic solution.
4. The Global Exponential Stability of Antiperiodic Solution
In this section, by constructing a suitable Lyapunov function we derive sufficient conditions ensuring the global exponential stability of antiperiodic solutions for (1).
Theorem 2. Let H1–H4 and A1 hold. Furthermore, assume that A2.The impulsive operators satisfyA3.There exists a positive constant satisfyingwhere then, the -antiperiodic solution of (1) is globally exponentially stable.
Proof 2. By Theorem 1, (16) has an -periodic solution. Let be the -periodic solution with the initial value and be an arbitrary solution with the initial value . Set , and . By (16), for and , we have
For , from A2 we can have the following: that is where and
Construct the Lyapunov function as follows: where , , and .
Calculating the upper right derivative of along the solutions of (16), for , we obtain
Repeat the same calculation and we can get
It follows from A3, (51), and (52) that for ,
By (49), we also have
Hence, for all .
On the other hand, we have
Let , and we can easily obtain the following:
Therefore, the -antiperiodic solution of (16) is globally exponentially stable. According to Remark 1, we know that the -antiperiodic solution of (16) is globally exponentially stable. The proof is complete.
5. An Illustrative Example
In this section, we give an example to show the feasibility and effectiveness of the results obtained in this paper.
Example 1. Consider the following QVSICNN:
where , , and the coefficients are as follows:
and the impulsive functions .
By a simple calculation, we can get
Taking , we obtain
Thus conditions H1–H4 and A1–A3 hold. Therefore, according to Theorems 1 and 2, (57) has at least one -antiperiodic solution, which is globally exponentially stable (see Figures 1 and 2).
6. Conclusion
In this paper, we investigated the existence and global exponential stability of antiperiodic solutions for a class of QVSICNNs with impulsive effects. We introduce a new method different from all other antiperiodic solutions of neural networks in a previous work. By using a new continuation theorem of the coincidence degree theory and constructing a suitable Lyapunov function, we obtain the existence and global exponential stability results for an antiperiodic solution. However, in this paper, we only investigate the antiperiodic solution problem of QVSICNNs with impulsive effects. In future work, periodic solution, almost periodic solution, and pseudo almost periodic solution in the quaternion field can be considered.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of the People’s Republic of China under Grant nos. 11861072 and 11361072.