Weighted pseudo almost periodic solutions for Clifford-valued neutral-type neural networks with leakage delays on time scales

In this paper, we consider a class of Clifford-valued neutral-type neural networks with leakage delays on time scales. We do not decompose the networks under consideration into real-valued systems, but we directly study the Clifford-valued networks. We first establish the existence of weighted pseudo almost periodic solutions of this class of neural networks by the theory of calculus on time scales and the Banach fixed point theorem. Then, we study the global exponential stability of weighted pseudo almost periodic solutions of this class of neural networks by using inequality techniques and the proof by contradiction. Finally, we give an example to illustrate the feasibility of the obtained results.

as their special cases, and they have more advantages than real-valued networks (see [8]). Recently, more and more attention has been paid to the study of Clifford-valued neural networks [9][10][11][12][13]. However, because Clifford-valued neural networks are a kind of multidimensional neural networks, and the multiplication of Clifford numbers does not satisfy the commutativity law, it is more difficult to study the dynamics of Clifford-valued neural networks than that of real-valued and complex-valued neural networks. Nevertheless, it is well known that the dynamics of neural networks is of great significance to the design and implementation of neural networks. Recently, in [12][13][14], by decomposing Cliffordvalued systems into real-valued systems, the stability of the equilibrium, anti-periodicity, and S P -almost periodicity of three types of Clifford-valued neural networks have been studied, respectively. At present, it is rare to study the dynamics of Clifford-valued neural networks by direct method, that is, not decomposing the Clifford-valued neural networks into real-value systems [15,16]. In particular, until now, no paper has been published on the dynamics of Clifford-valued neural networks on time scales by direct method.
On the one hand, time delays are inevitable in real systems, and the existence of time delays may change the dynamic behavior of systems. The three types of time delays usually considered in neural network systems are transmission delay, leakage delay, and neutraltype delay. Therefore, neural network models with these types of delays are widely studied [17][18][19][20][21].
On the other hand, as we all know, the continuous-time systems and the discrete-time systems have the same importance in theory and practice, and the discrete-time systems are more convenient for computer processing than the continuous-time ones. So it is necessary to study the continuous-time systems and the discrete-time systems at the same time. The time scale calculus theory can unify the research of continuous analysis and discrete analysis, so the study of neural networks on time scales has important theoretical and practical significance.
In addition, in the design, implementation, and application of neural networks, the existence and stability of periodic solutions, almost periodic solutions, pseudo almost periodic solutions, and almost automorphic solutions of non-autonomous neural networks are of equal importance to the existence and stability of equilibrium solutions of autonomous neural networks. The weighted pseudo almost periodicity is the generalization of almost periodicity and pseudo almost periodicity, so it is of great theoretical significance and practical value to study the existence and stability of weighted pseudo almost periodic solutions of neural networks [22][23][24].
Based on the above discussion, one can see that studying Clifford-valued neural networks on time scales cannot only unify the research of real-valued neural networks, complex-valued neural networks, and quaternion-valued neural networks, but also unify the research of continuous-time neural networks and discrete-time neural networks. At the same time, the weighted pseudo almost periodic oscillation is one of the important dynamic behaviors of neural networks. Therefore, the main purpose of this paper is to study the existence and global exponential stability of weighted pseudo almost periodic solutions for a class of Clifford-valued neutral-type neural networks with leakage delays on time scales by direct method. The results of this paper are new. The method can be used to study the existence and stability of almost periodic function solutions of other types of neural networks on time scales. This paper is organized as follows: In Sect. 2, we introduce some definitions, preliminary lemmas and give a model description. In Sect. 3, we study the existence and global exponential stability of weighted pseudo almost periodic solutions for a class of Cliffordvalued neural networks on time scales. In Sect. 4, we investigate the global exponential stability of weighted pseudo almost periodic solutions of this class of neural networks. In Sect. 5, we give an example to demonstrate the feasibility of our results. This paper ends with a brief conclusion in Sect. 6.
and (A n , · A n ) are Banach spaces. For more information about Clifford algebra, we refer the reader to [25]. A time scale T is an arbitrary nonempty closed subset of the real set R. We use ρ(t) and ν(t) to denote the backward jump operator and the graininess function of T, respectively. Definition 2.1 ([26]) Assume that f : T → R is a function, and let t ∈ T k . Then we define f ∇ (t) to be the number (provided it exists) with the property that given any Let R ν = {f : T → R : 1ν(t)r(t) = 0, ∀t ∈ T k } and R + ν = {r ∈ R ν : 1ν(t)r(t) > 0, ∀t ∈ T}. Then, for p ∈ R ν , we define the nabla exponential function bŷ For p, q ∈ R ν , we define a circle plus addition by (p ⊕ ν q)(t) = p(t) + q(t)ν(t)p(t)q(t) for all t ∈ T k . For p ∈ R ν , we define a circle minus p by ν p = -p 1-νp . For the basic theory and notation of the time scale theory, we refer the reader to [26,27].
Proof If t is left-dense, then there exists {s n } ⊂ T such that lim n→∞ s n = t and s n < t.
Since T is an almost periodic time scale, we have s n + h ∈ T and s n + h < t + h for each h ∈ Π . Thus, we obtain that Now, we shall show that if t is left-scattered, then t + h is also left-dense. Otherwise, there exists h ∈ Π such that t + h is left-dense. Then there exists {s n + h} ⊂ T such that lim n→∞ s n + h = t + h and s n + h < t + h.
Hence, lim n→∞ s n = t and s n < t.
Since T is an almost periodic time scale, we have {s n } ⊂ T. That is, ρ(t) = t, which contradicts the fact that t is left-scattered. Hence, t + h is also left-scattered for every h ∈ Π . The proof is complete.

Lemma 2.2 Let T be an almost periodic time scale and h ∈ Π , then for each t ∈ T,
(1) Proof If t is left-dense, then, by Lemma 2.1, it is obvious that (1) holds.
If t is left-scattered, then, by Lemma 2.1, for each h ∈ Π , t + h is also left-scattered. Hence, for each h ∈ Π , ρ(t + h) < t + h. That is, Since T is an almost periodic time scale and h ∈ Π , we have that ρ(t + h)-h ∈ T. According to (2) and the definition of the backward jump operator, for each h ∈ Π , we have On the other hand, since t is left-scattered, ρ(t) < t. Hence, Since h ∈ Π , ρ(t) + h ∈ T. By (4) and the definition of the backward jump operator, we obtain that By (3) and (5), we obtain that the first equality of (1) holds. Similarly, one can prove that the second equality of (1) also holds. The proof is complete.
In the following, we always assume that T is an almost periodic time scale.
In the sequel, we denote by BC(T, A n ) the set of all bounded continuous functions from T to A n and by C 1 Similar to Definition 3.9 in [28], we give the following definition.
is a relatively dense set in R for all ε > 0; that is, for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains at least one τ (ε) ∈ E{ε, f } such that We denote by AP(T, A n ) the set of all almost periodic functions defined on T. Similar to the proofs of the corresponding results in [28], one can easily prove the following.
Let U be the set of all functions ς : T → (0, +∞) that are locally ∇-integrable over T such that ς > 0 almost everywhere. From now on, for ς ∈ U and r ∈ T with r > 0, we denote The space of weights U ∞ is defined by Similar to Definition 3.1 in [30], we introduce the following definition.
We denote by PAP(T, A n , ς) the set of all weighted pseudo almost periodic functions from T to A n . Denote Proof Since x ∈ PAP(T, A, ς), by Definition 2.4, we have x = x 1 + x 0 , where x 1 ∈ AP(T, A) and x 0 ∈ PAP 0 (T, A, ς). Then we have By Lemma 2.6, we get x 1 ∈ AP(T, A). Now, we will show that x 0 ∈ PAP 0 (T, A, ς). In fact, Similar to the proofs of the corresponding results in [30], it is not difficult to prove the following three lemmas.
In this paper, we consider the following Clifford-valued neutral-type neural networks with leakage delays on time scale T: where i ∈ {1, 2, . . . , n} =: I; n corresponds to the number of units in the neural network; x i (t) ∈ A is the state vector of the ith unit at time t; a i (t) > 0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; b ij (t), c ij (t) ∈ A are the delay connection weight and the neutral delay connection weight from neuron j to neuron i at time t, respectively; δ i (t) ≥ 0 denotes the leakage delay at time t and satisfies tδ i (t) ∈ T for t ∈ T, η ij (t), τ ij (t) ≥ 0 correspond to the transmission delays at time t and satisfy tη ij (t) and tτ ij (t) ∈ T for t ∈ T; u i (t) ∈ A is the external input at time t; f j , g j : A → A are the activation functions of signal transmission. For convenience, we introduce the following notation: The initial values of system (8) are as follows: where ϕ i : [t 0ϑ, t 0 ] T → A is a bounded and ∇-differentiable function. Throughout this paper, we assume that the following conditions hold:

The existence of weighted pseudo almost periodic solutions
then E is a Banach space.
Proof Firstly, it is easy to see that if x = (x 1 , x 2 , . . . , x n ) T ∈ BC(T, A n ) is a solution of the following system then x is also a solution of system (8).
Secondly, we define a mapping Φ : E → BC(T, A n ) by We will prove that Φ maps E into itself. To this end, let By Lemmas 2.8 and 2.9, we find that We will prove that F 1 i ∈ AP(T, A) and F 0 i ∈ PAP 0 (T, A, ς) for i ∈ I. In fact, since a i ∈ AP(T, R), Γ 1 i ∈ AP(T, A), by Lemma 2.7, for every ε > 0, there exists ξ ∈ Π such that Hence, from the definition of F 1 i and Lemma 2.3, we have , ς), i ∈ I and by Lebesgue's dominated convergence theorem, we have which implies that F 0 i ∈ PAP 0 (T, A, ς) for all i ∈ I. Therefore, Φ i ϕ ∈ PAP(T, A, ς), i ∈ I. Moreover, noting that for i ∈ I, Thirdly, we will prove that Φ is a self-mapping defined on E 0 . In fact, for each ϕ ∈ E 0 , we have Finally, we shall prove that Φ is a contraction mapping. In fact, for any ϕ = (ϕ 1 , ϕ 2 , . . . , ϕ n ) T , ψ = (ψ 1 , ψ 2 , . . . , ψ n ) T ∈ E 0 , we have

By (A 3 ), we have
Hence, Φ is a contraction mapping. Therefore, system (8) has a unique weighted pseudo almost periodic in the set E 0 = {ϕ ∈ E 0 : ϕ E ≤ Υ }. This completes the proof of Theorem 3.1.

The stability of weighted pseudo almost periodic solutions
In this section, we study the global exponential stability of the unique weighted pseudo almost periodic solution of system (8) by using the proof by contradiction and the technique of inequalities.

Definition 4.1
A solution x with the initial value ϕ of system (8) is called globally exponentially stable if there exist positive constants λ with ν λ ∈ R + ν andC > 1 such that for each solution y with the initial value ψ satisfies

An example
In this section, we give an example to illustrate the feasibility of our results obtained in this paper.
Example 5.1 In system (8), let m = 3, n = 2, the weight (t) = e -|t| , and take the coefficients as follows: f j (x) = g j (x) = 1 120 e 0 sin x 2 + x 0 + 3 400 cos x 1 + If T = R, then we take and if T = Z, then we take  Remark 5.1 There are no existing results that can be applied to Example 5.1.

Conclusion
In this paper, we have considered a class of Clifford-valued neutral-type neural networks with leakage delays on time scales. By using the Banach fixed point theorem, the theory of calculus on time scales, inequality techniques, and the proof by contradiction, we have established the existence and exponential stability of weighted pseudo almost periodic solutions for this class of neural networks. An example has been given to show the feasibility of our results. This is the first paper to study the existence and global exponential stability of weighted pseudo almost periodic solutions for Clifford-valued neutral-type neural networks with leakage delays on time scales. The study of Clifford-valued neural networks on time scales cannot only unify the research of real-valued neural networks, complex-valued neural networks, and quaternion-valued neural networks, but also unify the research of continuous-time neural networks and discrete-time neural networks. Our method of this paper can be applied to study other types of the Clifford-valued neural networks on times scales, such as BAM neural networks on time scales, shunting inhibitory cellular neural networks on time scales, high-order Hopfield neural networks, and so on.