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.


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][4][5][6][7][8][9][10][11][12][13], there have been extensive results about the dynamical behaviors of SICNNs with time delays [4][5][6][7][8][9][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][12][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, quaternionvalued 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][18][19][20][21][22][23][24].For example, in [21,22], the global μ-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][26][27][28][29][30][31][32][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 pq ∈ 11, 12, … , 1n, … , m1, m2, … , mn ≔ ℬ; C pq denotes the cell at the p, q position of the lattice.The r neighborhood N r p, q of C pq is given by x pq ∈ ℚ is the activity of the cell C pq ; T pq ℚ → ℚ is the external input to C pq , a pq t > 0 represents the passive decay rate of the cell activity, C kl pq t ≥ 0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell C pq , and the activity function f ℚ → ℚ is a continuous function representing the output or firing rate of the cell C pq ; K pq t corresponds to the transmission delay kernel; and t h , h ∈ ℕ is a sequence of real numbers such that 0 < t 1 < t 2 < ⋯ < t h → ∞ as h → ∞, there exists a μ ∈ ℕ such that t h+μ = t h + ω/2, I pq h+μ x pq t h+μ = −I pqh −x pq t h , h ∈ ℕ.Without loss of generality, we also assume that 0, ω/2 ∩ t h h ∈ ℕ = t 1 , t 2 , … , t μ .For convenience, we denote g = sup t∈ℝ g t and g = inf t∈ℝ g t , where g 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 x pq = x R pq + ix I pq + jx J pq + kx K pq ∈ ℚ, assume that the activity function f ℚ → ℚ of (1) can be expressed as where f ν ∈ C ℝ 4 , ℝ , ν ∈ R, I, J, K ≔ E; the impulsive operators I pqh ℚ → ℚ can be expressed as where where f ν and I ν pqh u and ν ∈ E are mentioned in H1.
We will adopt the following notations: The initial value of (1) is given by 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.

Preliminaries
The quaternion was invented in 1843 by Hamilton [34].The skew field of the quaternion is denoted by where q 0 , q 1 , q 2 , and q 3 are real numbers and the elements i,j, and k 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: 1) can be decomposed as the following realvalued system: with the initial values where φ ν pq −∞, 0 → ℝ is continuous and bounded.
and pq ∈ ℬ; (i) x t satisfies (16) for t ≥ 0; (ii) x t is continuous everywhere except for some t h and left continuous at t = t h , and the right limit x t + h exists for h ∈ ℕ.
Definition 2. A solution x of system ( 16) is said to be the ω/2-antiperiodic solution of ( 16), if mn T be a solution of (16) If there exist constants λ > 0 and M > 0 such that then the solution x of ( 16) is said to be globally exponentially stable, where is an ω/2-antiperiodic solution to (16), then u = x 11 , … , x mn T , where x pq = x R pq + ix I pq + jx J pq + kx K pq , pq ∈ E must be an ω/2-antiperiodic solution to (1).Thus, the problem of finding an ω/2-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).

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 X and Y be Banach spaces, and let L Dom L ⊂ X → Y be linear and N X → Y be continuous.Assume that L is one-to-one and K ≔ L −1 N is compact.Furthermore, assume that there exists a bounded and open subset Ω ⊂ X with 0 ∈ Ω such that equation Lu = λNu has no solutions in ∂Ω ∩ Dom L for any λ ∈ 0, 1 .Then the problem Lu = Nu has at least one solution in Ω.
Proof 1.One has Then X is a Banach space with the norm x X = max pq∈B max ν∈E sup t∈ℝ x ν pq t , and Y is also a Banach space with the norm where Dom L = x x ∈ X, x ∈ X and a continuous operator It is easy to see that

28
where , by applying the Arzela-Ascoli theorem, we know that K is compact.Corresponding to the operator equation Lx = λNx, λ ∈ 0, 1 , we have Repeating the above procession, for ν = I, J, K, we can obtain that Integrating both sides of ( 29) over the interval 0, ω , we can obtain Repeating the above procession, for ν = I, J, K, we have Since for any t 1 , t 2 ∈ 0, ω , and pq ∈ ℬ, we have x ν pq t dt Thus, from (39) and (40) we have has no solutions in ∂Ω ∩ Dom L for any λ ∈ 0, 1 .Thus, by Lemma 1 we conclude that Lx = Nx has at least one ω/2-antiperiodic solution in X, that is, ( 16) has at least one ω/2-antiperiodic solution.By Remark 1, we see that (1) has at least one ω/2-antiperiodic solution.

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 satisfy There exists a positive constant λ > 0 satisfying where
Proof 2. By Theorem 1, ( 16) has an ω/2-periodic solution.Let x t be the ω/2-periodic solution with the initial value φ t and y t be an arbitrary solution with the initial value ψ t .Set z ν pq t = x ν pq t − y ν pq t , pq ∈ and ν ∈ E. By ( 16), for t > 0 and t ≠ t h , we have For t = t h , h ∈ ℕ, from A2 we can have the following: that is where pq ∈ ℬ and ν ∈ E Construct the Lyapunov function V t as follows: where V ν t = ∑ pq∈B z ν pq t t e λt , pq ∈ ℬ, and ν ∈ E.
Calculating the upper right derivative of V R t along the solutions of ( 16), for t ≠ t h , we obtain

51
Repeat the same calculation and we can get It follows from A3, (51), and (52) that for t ≠ t h , By (49), we also have Hence, V t ≤ V 0 for all t ≥ 0.
On the other hand, we have

Complexity
Let M = 1, and we can easily obtain the following: Therefore, the ω/2-antiperiodic solution of ( 16) is globally exponentially stable.According to Remark 1, we know that the ω/2-antiperiodic solution of ( 16) is globally exponentially stable.The proof is complete.

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 x pq t = x R pq t + ix I pq t + jx J pq t + kx K pq t ∈ ℚ, K pq u = e −u , p, q = 1, 2, and the coefficients are as follows: Thus conditions H1-H4 and A1-A3 hold.Therefore, according to Theorems 1 and 2, (57) has at least one π-antiperiodic solution, which is globally exponentially stable (see Figures 1  and 2).

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.

Figure 1 :
Figure 1: The states of four of x 11 , x 12 , x 21 , x .

Figure 2 :
Figure 2: Curves of x in 3-dimensional space for stable case.
x pq t = −a pq t x pq t − 〠 Δx pq t h = x pq t + h − x pq t − h = I pqh x pq t h , t = t h , pq t f x kl t − u du + T pq t , t ≥ 0, t ≠ t h ,