Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays

Abstract: We consider the existence and stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. In order to overcome the incompleteness of the space composed of Weyl almost periodic functions, we first obtain the existence of a bounded continuous solution of the system under consideration by using the fixed point theorem, and then prove that the bounded solution is Weyl almost periodic by using a variant of Gronwall inequality. Then we study the global exponential stability of the Weyl almost periodic solution by using the inequality technique. Even when the system we consider degenerates into a real-valued one, our results are new. A numerical example is given to illustrate the feasibility of our results.


Introduction
Since shunting inhibitory cellular neural networks were proposed by Bouzerdoum and Pinter [1] as a new type of neural networks, they have received more and more attention and have been widely applied in optimisation, psychophysics, speech and other fields. At the same time, since time delays are ubiquitous, many research results have been obtained on the dynamics of shunting inhibitory cellular neural networks with time delays [2][3][4][5][6].
On the one hand, the quaternion is a generalization of real and complex numbers [7]. The skew field of quaternions is defined by H := {q = q R + iq I + jq J + kq K }, where q R , q I , q J , q K ∈ R and the elements i, j and k obey the Hamilton's multiplication rules: i j = − jk = k, jk = −k j = i, ki = −ik = j, i 2 = j 2 = k 2 = −1.
On the other hand, because periodic and almost periodic oscillations are important dynamics of neural networks, the periodic and almost periodic oscillations of neural networks have been studied a lot in the past few decades [25][26][27][28][29][30][31][32][33]. Weyl almost periodicity is a generalization of Bohr almost periodicity and Stepanov almost periodicity [34][35][36][37]. It is a more complex recurrent oscillation. Because the spaces composed of Bohr almost periodic functions and Stepanov almost periodic functions are Banach spaces, it brings some convenience to study the existence of almost periodic solutions in these two senses of differential equations. Therefore, many results have been obtained on the Bohr almost periodic oscillation and Stepanov almost periodic oscillation of neural networks. However, the space composed of Weyl almost periodic functions is incomplete [38]. Therefore, the results of Weyl almost periodic solutions of neural networks are still very rare. Therefore, it is a meaningful and challenging work to study the existence of Weyl almost periodic solutions of neural networks.
Motivated by the above, in this paper, we consider the following shunting inhibitory cellular neural networks with time-varying delays: C kl i j (t)g i j (x kl (t − τ kl (t)))x i j (t) + I i j (t), (1.1) where i j ∈ {11, 12, . . . , 1n, . . . , m1, m2, . . . , mn} := Λ, C i j denotes the cell at the (i, j) position of the lattice. The r-neighborhood N r (i, j) of C i j is given as and N s (i, j) is similarly specified; x i j (t) ∈ H denotes the activity of the cell of C i j , I i j (t) ∈ H is the external input to C i j , a i j (t) ∈ H is the coefficient of the leakage term, which represents the passive decay rate of the activity of the cell C i j , B kl i j (t) ≥ 0 and C kl i j (t) ≥ 0 represent the connection or coupling strength of postsynaptic of activity of the cell transmitted to the cell C i j , the activity functions f i j , g i j : H → H are continuous functions representing the output or firing rate of the cell C i j , τ kl (t) corresponds to the transmission delay and satisfies 0 ≤ τ kl (t) ≤ τ.
The purpose of this paper is to use the fixed point theorem and a variant of Gronwall inequality to establish the existence and global exponential stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks whose coefficients of the leakage terms are quaternions. This is the first paper to study the existence and global exponential stability of Weyl almost periodic solutions of system (1.1) by using the fixed point theorem and a variant of Gronwall inequality. Our result of this paper is new, and our method can be used to study other types of quaternion-valued neural networks.
For convenience, we introduce the following notations: The initial condition of system (1.1) is given by Throughout this paper, we assume that: The rest of this paper is arranged as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, we study the existence and global exponential stability of Weyl almost periodic solutions of (1.1). In Section 4, an example is given to verify the theoretical results. This paper ends with a brief conclusion in Section 5.

Preliminaries
Let (X, · X ) be a Banach space and BC(R, X) be the set of all bounded continuous functions from R to X. Definition 2.1. [38] A function f ∈ BC(R, X) is said to be almost periodic, if for every > 0, there exists a constant l = l( ) > 0 such that in every interval of length l( ) contains at least one σ such that Denote by AP(R, X) the set of all such functions.
For p ∈ [1, ∞), we denote by L p loc (R, X) the space of all functions from R into X which are locally p-integrable. For f ∈ L p loc (R, X), we define the following seminorm: Definition 2.2. [38] A function f ∈ L p loc (R, X) is said to be p-th Weyl almost periodic (W p -almost periodic for short), if for every > 0, there exists a constant l = l( ) > 0 such that in every interval of length l( ) contains at least one σ such that This σ is called on -translation number of f . The set of all such functions will be denoted by APW p (R, X).
Remark 2.1. By Definitions 2.1 and 2.2, it is easy to see that if f ∈ AP(R, X), then f ∈ APW p (R, X).
Similar to the proofs of the lemma on page 83 and the lemma on page 84 of [39], it is not difficult to prove the following two lemmas.
Lemma 2.1. If f ∈ APW p (R, X), then f is bounded and uniformly continuous on R with respect to the seminorn · W p .
Using the argumentation contained in the proof of Proposition 3.21 in [38], one can easily prove the following.

Main results
Let BUC(R, H m×n ) be a collection of bounded and uniformly continuous functions from R to H m×n , then, the space BUC(R, H m×n ) with the norm We will show that φ 0 is well defined under assumption (H 1 ). In fact, by I i j ∈ APW p (R, H) and Lemma 2.1, there exists a constant M > 0 such that I i j W p ≤ M for all i j ∈ Λ. According to the Hölder inequality, one has Then, for every φ ∈ Ω, one has and, for p = 2, then system (1.1) has a unique W p -almost periodic solution in Ω.
Define an operator T : Ω → H m×n by Now, we will prove that T φ is well defined. Actually, by (H 1 )-(H 3 ) and (3.1), for i j ∈ Λ, one deduces that That is, T φ is well defined. We will divide the rest of the proof into four steps. S tep 1, we will prove that T φ ∈ BUC(R, H m×n ), for every φ ∈ Ω. In fact, by (3.3), we see that T φ is bounded on R. So, we only need to show that T φ is uniformly continuous on R. Based on the Hölder inequality for 0 ≤ h ≤ 1 and q ≥ 1 with 1 p + 1 q = 1, one has S tep 2, we will prove that T is a self-mapping from Ω to Ω. Actually, for arbitrary φ ∈ Ω, from (H 2 )-(H 3 ), we have which implies that T φ ∈ Ω. Consequently, T is a self-mapping from Ω to Ω. S tep 3 , we will prove T is a contraction mapping. As a matter of fact, in view of (H 1 )-(H 2 ), for any φ, ν ∈ Ω, we can get

From this and (H 3 ), one has
Noticing that κ < 1, T is a contraction mapping. Consequently, system (1.1) has a unique solution x in Ω. S tep 4, we will prove that the unique solution x ∈ Ω is W p -almost periodic.
x is bounded and uniformly continuous. Hence, for every > 0, there exists a δ ∈ (0, ) such that for any t 1 , t 2 ∈ R with |t 1 − t 2 | < δ and i j ∈ Λ, we have Also, for this δ, in view of (H 1 ) and Lemma 2.2, we see that there exists a common δ-translation number σ such that and where i j, kl ∈ Λ. Consequently, from (3.4) and (3.9), we get Since x is a solution of system (1.1), by (3.2), for i j ∈ Λ, we have When p > 2, it follows from Hölder's inequality 2 p , p−2 p , Hölder's inequality 1 2 , 1 2 and (H 2 ) that for i j ∈ Λ. Similarly, we have and for i j ∈ Λ. In a similar way, one can get that   and, together with a change of variables, Fubini's theorem, Hölder's inequality, (3.6) and (3.12), we derive that By (H 4 ), we have γ < η. Thus, it follows from Lemma 2.3 that Hence, x ∈ APW p (R, H m×n ). When p = 2, similar to the proof of the case of p > 2, one can obtain By (H 4 ), we haveγ <η. Thus, it follows from Lemma 2.3 that which means that x ∈ APW 2 (R, H m×n ). The proof is complete.
Then the solution x of system (1.1) is said to be globally exponentially stable.
Theorem 3.2. Assume that (H 1 )-(H 3 ) hold, then system (1.1) has a unique W p -almost periodic solution that is globally exponentially stable.
Proof. Let x(t) be the W p -almost periodic solution with the initial value ϕ(t) and y(t) be an arbitrary solution with the initial value ψ(t). Taking − g i j (y kl (t − τ kl (t)))y i j (t)), i j ∈ Λ. (3.28) For i j ∈ Λ, we define the following functions: From (H 3 ), we get Hence, we can choose a positive constant λ such that 0 < λ < min{ς, a m } and Π i j (λ) > 0. Thus, one has From (3.28), we havė − g i j (y kl (t − τ kl (t)))y i j (t) ds, i j ∈ Λ.
Hence, for any > 0, it is easy to see that We claim that Otherwise, there exists t * > 0 such that and C kl i j (t) g i j (x kl (t − τ kl (t)))(x i j (t) − y i j (t)) H + (g i j (x kl (t − τ kl (t))) − g i j (y kl (t − τ kl (t))))y i j (t) H ds which contradicts the Eq (3.33). Hence, (3.32) holds. Letting → 0 + , from (3.32), we have Therefore, the W p -almost periodic solution of system (1.1) is globally exponentially stable. This completes the proof.

Conclusions
In this paper, the existence and global exponential stability of Weyl almost periodic solutions for a class of quaternion-valued neural networks with time-varying delays are established. Even when the system we consider is a real-valued system, our results are brand-new. In addition, the method in this paper can be used to study the existence of Weyl almost periodic solutions for other types of neural networks.