Pseudo almost periodic solutions for Clifford-valued neutral-type fuzzy neural networks with multi-proportional delay and D operator 1

Abstract

In this paper, a class of Clifford-valued neutral fuzzy neural-type networks with proportional delay and D operator and whose self feedback coefficients are also Clifford numbers are considered. By using the Banach fixed point theorem and some differential inequality techniques, we directly study the existence and global asymptotic stability of pseudo almost periodic solutions by not decomposing the considered Clifford-valued systems into real-valued systems. Finally, two examples are given to illustrate our main results. Our results of this paper are new.

Keywords

Clifford-valued neural network fuzzy neural network proportional delay pseudo almost periodic solution global asymptotic stability.

1 Introduction

Clifford algebra was proposed by British mathematician William K. Clifford in 1878 [1]. It is a generalization of complex numbers and quaternions. At present, Clifford algebra has been widely used in many fields of natural science and engineering technology. For example, it has been successfully applied to neural computing, image and signal processing, computer and robot vision, control problems and other fields [2]. In particular, Clifford-valued neural network has been proved to be superior to real-valued neural network in many aspects [3]. Therefore, the study of Clifford-valued neural networks has attracted the interest of many researchers [4 –16]. However, because the multiplication of Clifford algebra does not satisfy the commutative law, the known results of Clifford-valued neural network dynamics obtained by direct method are still few [17 –22]. Therefore, it is of great theoretical and practical significance to further study the dynamics of Clifford valued neural networks by direct methods.

As we all know, the application field of neural network is becoming wider and wider [23 –26]. At the same time, the time delay is ubiquitous. Therefore, it is more reasonable to introduce time delay into the mathematical model describing the real process. In recent years, mathematical models with proportional time delay have important applications in many application fields, such as physics, biological systems, neural network systems and control science. Therefore, the dynamics of neural networks with proportional delay has been widely studied [27 –32]. On the other hand, periodic oscillation and almost periodic oscillation are one of the important dynamics of neural networks. In the past decades, many researchers have studied the existence and stability of periodic and almost periodic solutions of various types of neural networks, including neutral-type neural networks with D operator and fuzzy neural networks [33 –43].

Inspired by the above analysis, the main purpose of this paper is to study the existence and attraction of pseudo almost periodic solutions for a class of Clifford-valued fuzzy neural networks with proportional delay and D operator whose self feedback coefficients are also Clifford numbers. As far as we know, this is the first paper to study the existence and attraction of pseudo almost periodic solutions for Clifford-valued neural networks with proportional delay and D operator whose self feedback coefficients are also Clifford numbers.

The rest of the paper is organized as follows. In the second section, we review some definitions, introduce some lemmas and give a description of the model. In Section 3, we study the existence and uniqueness of pseudo almost periodic solutions for the network under consideration. In Section 4, we study the global asymptotic stability of the pseudo almost periodic solution. In Section 5, we give two numerical examples to illustrate the feasibility of our results. Finally, we draw a brief conclusion in Section 6.

2 Model description and preliminaries

The real Clifford algebra over $ℝ^{m}$ is defined as $\begin{matrix} A = {\sum_{A \in Ω} a^{A} e_{A}, a^{A} \in ℝ}, \end{matrix}$ where e_A = e_{h
₁}e_{h
₂} ⋯ e_{h
_v} = e_{h₁h₂⋯h_v} with A = h₁h₂ ⋯ h_v, 1 < h₁ < h₂ < ⋯ < h_v < m. Moreover, e_∅ = e₀ = 1 and e_h, h = 1, 2, ⋯ m are said to be Clifford generators and satisfy $e_{p}^{2} = 1, p = 1, 2, \dots, s$ , $e_{p}^{2} = - 1, p = s + 1, s + 2, \dots, m$ where s < m, and e_pe_q + e_qe_p = 0, p ≠ q, p, q = 1, 2, …, m. Let Ω = {∅ , 1, 2, ⋯ A, ⋯ , 12 ⋯ m}, then it is easy to see that $A = {\sum_{A} a^{A} e_{A}, a^{A} \in R}$ , where ∑_A is short for ∑_A∈Ω and $dim A = 2^{m}$ .

For $x = \sum_{A} x^{A} e_{A} \in A$ , we define $∥ x ∥_{A} = max_{A} {| x^{A} |}$ and for $x = (x_{1}, x_{2}, \dots, x_{n})^{T} \in A^{n}$ , we define $∥ x ∥_{A^{n}} = max_{1 \leq p \leq n} {∥ x_{p} ∥_{A}}$ , then $(A, ∥ \cdot ∥_{A})$ and $(A^{n}, ∥ \cdot ∥_{A^{n}})$ are Banach spaces. For x = ∑_Ax^Ae_A, we define x^c = ∑_A≠∅x^Ae_A and x^∅ = x - x^c. The derivative of x (t) = ∑_Ax^A (t) e_A is given by $\dot{x} (t) = \sum_{A} {\dot{x}}^{A} (t) e_{A}$ . For more knowledge about Clifford analysis, we refer to [44].

According to [8], for $x, y \in ℝ$ , we define $\begin{matrix} x ⋁ y = {\begin{matrix} y, if x \leq y \\ x, if x > y, \end{matrix} x ⋀ y = {\begin{matrix} x, if x \leq y \\ y, if x > y, \end{matrix} \end{matrix}$ and For x = ∑_A∈Ωx^Ae_A and $y = \sum_{A \in Ω} y^{A} e_{A} \in A$ , we define $\begin{matrix} x ⋁ y = \sum_{A \in Ω} (x^{A} ⋁ y^{A}) e_{A}, \\ x ⋀ y = \sum_{A \in Ω} (x^{A} ⋀ y^{A}) e_{A} . \end{matrix}$

In this paper, we are concerned with the following Clifford-valued neutral-type fuzzy neural network with proportional delay and D operator:

$\begin{matrix} [x_{i} (t) - p_{i} (t) x_{i} (η_{i} t)]^{'} \\ = - a_{i} (t) x_{i} (t) + \sum_{j = 1}^{n} b_{ij} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} c_{ij} (t) g_{j} (x_{j} (q_{ij} t)) + ⋁_{j = 1}^{n} α_{ij} (t) h_{j} (x_{j} (q_{ij} t)) \\ + ⋀_{j = 1}^{n} β_{ij} (t) l_{j} (x_{j} (q_{ij} t) + ⋁_{j = 1}^{n} s_{ij} v_{j} (t) \\ + ⋀_{j = 1}^{n} r_{ij} v_{j} (t) + I_{i} (t), \end{matrix}$ (2.1) where $x_{i} (t) \in A, i = 1, 2, \dots, n$ , corresponds to the state of the ith unit at time t, and $A$ is a Clifford algebra; $a_{i} (t) \in A$ represents the rate with which the ith unit will reset its potential to the resting state when disconnected from the network and external inputs at time t, $p_{ij} (t), b_{ij} (t), c_{ij} (t) \in A$ correspond to the connection weights of the neural network; $α_{ij} (t) \in A$ is the fuzzy feedback max template; $β_{ij} (t) \in A$ is the fuzzy feedback min template; $s_{ij} (t) \in A$ is the fuzzy feed-forward max template, $r_{ij} (t) \in A$ is the fuzzy feed-forward min template; $v_{j} (t) \in A$ represents the input of the jth neuron; $f_{j} (t), g_{j} (t), h_{j} (t), l_{j} (t) \in A$ are the activation functions; η_i, q_ij ∈ (0, 1) are proportional delay factors; $I_{i} (t) \in A$ is the external input at time t.

For convenience, we will adopt the following notations: $\begin{matrix} p_{i}^{+} = sup_{t \in ℝ} ∥ p_{i} (t) ∥_{A}, {\bar{a}}_{i} = sup_{t \in ℝ} ∥ a_{i} (t) ∥_{A}, \\ {\bar{a}}_{i}^{c} = sup_{t \in ℝ} ∥ a_{i}^{c} (t) ∥_{A}, {\hat{a}}_{i}^{\emptyset} = inf_{t \in ℝ} ∥ a_{i}^{\emptyset} (t) ∥_{A}, \\ {\bar{a}}_{i}^{\emptyset} = sup_{t \in ℝ} ∥ a_{i}^{\emptyset} (t) ∥_{A}, \\ b_{ij}^{+} = sup_{t \in ℝ} ∥ b_{ij} (t) ∥_{A}, c_{ij}^{+} = sup_{t \in ℝ} ∥ c_{ij} (t) ∥_{A}, \\ α_{ij}^{+} = sup_{t \in ℝ} ∥ α_{ij} (t) ∥_{A}, β_{ij}^{+} = sup_{t \in ℝ} ∥ β_{ij} (t) ∥_{A} . \end{matrix}$

The initial values of system (2.1) are given by

$x_{i} (s) = φ_{i} (s), s \in [ρ t_{0}, t_{0}], φ_{i} \in C ([ρ t_{0}, t_{0}], A),$ (2.2) where $ρ = min_{1 \leq i \leq n} {η_{i}, min_{1 \leq j \leq n} {q_{ij}}}, i = 1, 2, \dots, n .$

Let $BC (ℝ, A^{n})$ denote the set of continuous and bounded function from $ℝ$ to $A^{n}$ . Then $(BC (ℝ, A^{n}), ∥ \cdot ∥_{0})$ is a Banach space with the norm $\begin{matrix} ∥ f ∥_{0} = max_{1 \leq i \leq n} {sup_{t \in ℝ} ∥ f_{i} (t) ∥_{A}}, \end{matrix}$ where $f = (f_{1}, f_{2}, \dots, f_{n})^{T} \in BC (ℝ, A^{n})$ .

Definition 2.1. [10] A function $f \in BC (ℝ, A^{n})$ is said to be almost periodic, if for every ɛ > 0, it is possible to find a real number l = l (ɛ) such that, for any interval with length l (ɛ), there is a number τ = τ (ɛ) in this interval satisfying $\begin{matrix} ∥ f (t + τ) - f (t) ∥_{A^{n}} < ɛ, \end{matrix}$ for all $t \in ℝ$ . The collection of all such functions will be denoted by $AP (ℝ, A^{n})$ .

Let $\begin{matrix} {PAP}_{0} (ℝ, A^{n}) \\ = {f \in BC (ℝ, A^{n}) : lim_{T \to + \infty} \frac{1}{2 T} \int_{- T}^{T} {∥ f (t) ∥}_{A} dt = 0} . \end{matrix}$

Definition 2.2. [10] A function $f \in BC (ℝ, A^{n})$ is said to be pseudo almost periodic if it can be expressed as $\begin{matrix} f = g + h, \end{matrix}$ where $g \in AP (ℝ, A^{n})$ and $h \in {PAP}_{0} (ℝ, A^{n})$ . The collection of all such functions will be denoted by $PAP (ℝ, A^{n})$ .

Lemma 2.1. [10] If $λ \in ℝ, f, g \in PAP (ℝ, A^{n})$ , then $λ f \in PAP (ℝ, A^{n}), f + g, fg \in PAP (ℝ, A^{n})$ .

Similar to the proof of Lemma 2.1 in [45], one can easily show the following two lemmas:

Lemma 2.2. Let $φ \in PAP (ℝ, A^{n})$ and $β \in ℝ$ be a constant. Then, $φ (β t) \in PAP (ℝ, A^{n})$ .

Lemma 2.3. [10] Let $f \in C (A, A^{n})$ satisfy the Lipschitz condition and $x \in PAP (ℝ, A)$ , then $f (x (\cdot)) \in PAP (ℝ, A^{n})$ .

Similar to the proof of Lemma 2.2 in [46], one can easily show that

Lemma 2.4. Assume that $x, y \in PAP (ℝ, A)$ , then we have $x ⋁ y, x ⋀ y \in PAP (ℝ, A)$ .

Lemma 2.5. [10] If $a \in AP (ℝ, ℝ^{+})$ with $inf_{t \in ℝ} a (t) > 0$ , and $F (s) \in PAP (ℝ, A)$ , then the function $G : ℝ \to A$ define by $\begin{matrix} G (t) = \int_{- \infty}^{t} e^{- \int_{s}^{t} a (u) du} F (s) ds \end{matrix}$ is pseudo almost periodic.

Lemma 2.6. [10] The space $(PAP (ℝ, A^{n}), ∥ \cdot ∥_{0})$ is a Banach space.

Similar to the proof of Corollary 1 in [47], one can easily show that

Lemma 2.7. If $α_{ij}, β_{ij} \in C (ℝ, A), f \in C (ℝ, A), x, y \in A$ . Then we have $\begin{matrix} ∥ ⋀_{j = 1}^{n} α_{ij} (t) f_{j} (x) - ⋀_{j = 1}^{n} α_{ij} (t) f_{j} (y) ∥_{A} \\ \leq \sum_{j = 1}^{n} ∥ α_{ij} (t) ∥_{A} ∥ f_{j} (x) - f_{j} (y) ∥_{A}, \\ ∥ ⋁_{j = 1}^{n} β_{ij} (t) f_{j} (x) - ⋁_{j = 1}^{n} β_{ij} (t) f_{j} (y) ∥_{A} \\ \leq \sum_{j = 1}^{n} ∥ β_{ij} (t) ∥_{A} ∥ f_{j} (x) - f_{j} (y) ∥_{A} . \end{matrix}$

The assumptions used in this paper are as follows:

For i, j = 1, 2, ⋯ n, $a_{i}^{\emptyset} \in AP (ℝ, ℝ^{+})$ with ${\hat{a}}_{i}^{\emptyset} > 0$ , $a_{i}^{c}, p_{i}, b_{ij}, c_{ij}, α_{ij}, β_{ij}, s_{ij}, r_{ij} \in PAP (ℝ, A^{n})$ .

For $i, j = 1, 2, \dots, n, f_{j}, g_{j}, h_{j}, l_{j} \in C (A, A^{n})$ and there exist positive constants $L_{j}^{f}, L_{j}^{g}, L_{j}^{h}, L_{j}^{l}$ such that, for any $x, y \in A$ , $\begin{matrix} ∥ f_{j} (x) - f_{j} (y) ∥_{A} \leq L_{j}^{f} ∥ x - y ∥_{A}, \\ ∥ g_{j} (x) - g_{j} (y) ∥_{A} \leq L_{j}^{g} ∥ x - y ∥_{A}, \\ ∥ h_{j} (x) - h_{j} (y) ∥_{A} \leq L_{j}^{h} ∥ x - y ∥_{A}, \\ ∥ l_{j} (x) - l_{j} (y) ∥_{A} \leq L_{j}^{l} ∥ x - y ∥_{A}, \end{matrix}$ where f_j (0) = g_j (0) = h_j (0) = l_j (0) =0.

For i = 1, 2, ⋯ n, there are positive functions ${\tilde{a}}_{i} (t) \in BC (ℝ, ℝ)$ and constants K_i > 0 satisfying $\begin{matrix} e^{- \int_{s}^{t} a_{i}^{\emptyset} (u) du} \leq K_{i} e^{- \int_{s}^{t} {\tilde{a}}_{i} (u) du}, \end{matrix}$ for all $t, s \in ℝ$ , t - s ≥ 0, and ${\tilde{a}}_{i}^{-} = inf_{t \in ℝ} | {\tilde{a}}_{i} (t) | > 0$ .

There are positive constants ξ_i > 0, i = 1, 2, ⋯ n satisfying $\begin{matrix} u : = max_{1 \leq i \leq n} & {p_{i}^{+} + \frac{1}{{\tilde{a}}_{i}^{-}} K_{i} ({\bar{a}}_{i}^{\emptyset} p_{i}^{+} + {\bar{a}}_{i}^{c} + ξ_{i}^{- 1} \\ \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} + ξ_{i}^{- 1} \\ \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j})} < 1, \end{matrix}$ where ${\tilde{a}}_{i}^{-}$ is defined in (H₃).

There are positive constants ξ_i > 0, i = 1, 2, ⋯ n satisfying $\begin{matrix} ν : = max_{1 \leq i \leq n} & {- {\tilde{a}}_{i}^{-} + K_{i} [{\bar{a}}_{i}^{c} + {\bar{a}}_{i} p_{i}^{+} \frac{1}{1 - p_{i}^{+}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} \frac{1}{1 - p_{i}^{+}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \frac{1}{1 - p_{i}^{+}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} \frac{1}{1 - p_{i}^{+}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j} \frac{1}{1 - p_{i}^{+}}]} < 0 . \end{matrix}$

3 The existence of pseudo almost periodic solutions

In this section, we will use the contracting mapping principle to study the existence of pseudo almost periodic solutions.

Let $X = {φ \in BC (ℝ, A^{n}) : φ \in PAP (ℝ, A^{n})}$ . For φ ∈ X, we denote the norm of φ as $∥ φ ∥_{0} = sup_{t \in ℝ} ∥ φ (t) ∥_{A^{n}}$ , then X with this norm is a Banach space.

Let

$\begin{matrix} φ_{0} (t) = & (\int_{- \infty}^{t} e^{\int_{s}^{t} - a_{1}^{\emptyset} (u) du} \\ (⋁_{j = 1}^{n} s_{1 j} v_{j} (s) + ⋀_{j = 1}^{n} r_{1 j} v_{j} (s) + I_{1} (s)) ds, \\ \int_{- \infty}^{t} e^{\int_{s}^{t} - a_{2}^{\emptyset} (u) du} \\ (⋁_{j = 1}^{n} s_{2 j} v_{j} (s) + ⋀_{j = 1}^{n} r_{2 j} v_{j} (s) + I_{2} (s)) ds, \dots, \\ \int_{- \infty}^{t} e^{\int_{s}^{t} - a_{n}^{\emptyset} (u) du} \\ (⋁_{j = 1}^{n} s_{nj} v_{j} (s) + ⋀_{j = 1}^{n} r_{nj} v_{j} (s) + I_{n} (s)) ds)^{T} \end{matrix}$

and take a positive constant R ≥ ∥ φ₀ ∥ ₀. Set $\begin{matrix} X_{0} = {φ \in X, ∥ φ - φ_{0} ∥_{0} \leq \frac{uR}{1 - u}}, \end{matrix}$ then, for every φ ∈ X₀, we have $\begin{matrix} ∥ φ ∥_{0} \leq ∥ φ - φ_{0} ∥_{0} + ∥ φ_{0} ∥_{0} \leq \frac{uR}{1 - u} + R = \frac{R}{1 - u} . \end{matrix}$

Theorem 3.1. Assume that (H₁)-(H₄) hold, then system (2.1) has a unique pseudo almost periodic solution in X₀.

Proof. By making the following transformation: $y_{i} = ξ_{i}^{- 1} x_{i} (t), Y_{i} (t) = y_{i} (t) - p_{i} (t) y_{i} (η_{i} t)$ , system (2.1) is transformed into

$\begin{matrix} Y_{i}^{'} (t) = & - a_{i} (t) Y_{i} (t) - a_{i} (t) p_{i} (t) y_{i} (η_{i} t) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij} (t) f_{j} (ξ_{j} y_{j} (t)) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij} (t) g_{j} (ξ_{j} y_{j} (q_{ij} t)) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} α_{ij} (t) h_{j} (ξ_{j} y_{j} (q_{ij} t)) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} β_{ij} (t) l_{j} (ξ_{j} y_{j} (q_{ij} t)) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} s_{ij} v_{j} (t) + ξ_{i}^{- 1} ⋀_{j = 1}^{n} r_{ij} v_{j} (t) \\ + ξ_{i}^{- 1} I_{i} (t), i = 1, 2, \dots, n . \end{matrix}$ (3.1) It is easy to see that if y = {y_i, i = 1, ⋯ , n} is a solution of system (3.1), then x (t) = {ξ_iy_i, i = 1, ⋯ , n} is a solution of system (2.1).

For i = 1, 2, ⋯ , n, we denote $\begin{matrix} Γ_{i} (t, y) = & - a_{i}^{c} (t) y_{i} (t) + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij} (t) f_{j} (ξ_{j} y_{j} (t)) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij} (t) g_{j} (ξ_{j} y_{j} (q_{ij} t)) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} α_{ij} (t) h_{j} (ξ_{j} y_{j} (q_{ij} t)) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} β_{ij} (t) l_{j} (ξ_{j} y_{j} (q_{ij} t)) . \end{matrix}$ It is easy to check that if y = {y_i, i = 1, ⋯ , n} is a solution of the following integral equation

$\begin{matrix} y_{i} (t) = & p_{i} (t) y_{i} (η_{i} t) + \int_{- \infty}^{t} e^{- \int_{s}^{t} a_{i}^{\emptyset} (u) du} \\ [- a_{i}^{\emptyset} (s) p_{i} (s) y_{i} (η_{i} s) + Γ_{i} (s, y) + ξ_{i}^{- 1} ⋁_{j = 1}^{n} s_{ij} v_{j} (s) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} r_{ij} v_{j} (s) + ξ_{i}^{- 1} I_{i} (s)] ds, \end{matrix}$ (3.2)

then x = {ξ_iy_i, i = 1, ⋯ , n} is a solution of system (2.1).

Define an operator $T : X_{0} \to BC (ℝ, A^{n})$ by $T φ = (T_{1} φ, T_{2} φ, \dots, T_{n} φ)^{T},$ where for φ ∈ X₀, i = 1, 2, ⋯ , n and $\begin{matrix} (T_{i} φ) (t) = & p_{i} (t) y_{i} (η_{i} t) + \int_{- \infty}^{t} e^{- \int_{s}^{t} a_{i}^{\emptyset} (u) du} \\ [- a_{i}^{\emptyset} (s) p_{i} (s) y_{i} (η_{i} s) + Γ_{i} (s, φ) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} s_{ij} v_{j} (s) + ξ_{i}^{- 1} ⋀_{j = 1}^{n} r_{ij} v_{j} (s) \\ + ξ_{i}^{- 1} I_{i} (s)] ds . \end{matrix}$

First of all, it follows from Lemmas 2-2 that Tφ ∈ X for every φ ∈ X. Now, we will prove that the mapping T is a self-mapping from X₀ to X₀. In fact, for each φ ∈ X₀, we have $\begin{matrix} ∥ T φ - φ_{0} ∥_{0} \\ \leq & max_{1 \leq i \leq n} {sup_{t \in ℝ} [∥ p_{i} (t) φ_{i} (η_{i} t) ∥_{A} + \int_{- \infty}^{t} e^{- \int_{s}^{t} a_{i}^{\emptyset} (u) du} \\ [∥ a_{i}^{\emptyset} (s) p_{i} (s) φ_{i} (η_{i} s) ∥_{A} + ∥ a_{i}^{c} (s) φ_{i} (s) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ b_{ij} (s) f_{j} (φ_{j} (s)) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ c_{ij} (s) g_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥_{A} \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} ∥ α_{ij} (s) h_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥_{A} \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} ∥ β_{ij} (s) l_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥_{A}] ds]} \\ \leq & max_{1 \leq i \leq n} {sup_{t \in ℝ} (∥ p_{i} (t) φ_{i} (η_{i} t) ∥_{A} + \int_{- \infty}^{t} e^{- \int_{s}^{t} {\tilde{a}}_{i} (u) du} \\ K_{i} [∥ a_{i}^{\emptyset} (s) p_{i} (s) φ_{i} (η_{i} s) ∥_{A} + ∥ a_{i}^{c} (s) φ_{i} (s) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ b_{ij} (s) f_{j} (ξ_{j} φ_{j} (s)) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} {∥ c_{ij} (s) g_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥}_{A} \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} ∥ α_{ij} (s) h_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥_{A} \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} ∥ β_{ij} (s) l_{j} (ξ_{j} φ_{j} (q_{ij} s)) ∥_{A}] ds)} \\ \leq & max_{1 \leq i \leq n} {p_{i}^{+} ∥ φ ∥_{0} + \int_{- \infty}^{t} e^{- {\tilde{a}}_{i}^{-} (t - s)} \\ K_{i} [{\bar{a}}_{i}^{\emptyset} p_{i}^{+} ∥ φ ∥_{0} + {\bar{a}}_{i}^{c} ∥ φ ∥_{0} + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} ∥ φ ∥_{0} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} ∥ φ ∥_{0} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} ∥ φ ∥_{0} + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ∥ φ ∥_{0}] ds} \\ \leq & max_{1 \leq i \leq n} {[p_{i}^{+} + \frac{1}{{\tilde{a}}_{i}^{-}} K_{i} ({\bar{a}}_{i}^{\emptyset} p_{i}^{+} + {\bar{a}}_{i}^{c} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j})] ∥ φ ∥_{0}} \\ \leq & \frac{uR}{1 - u} . \end{matrix}$ Hence, T is a self-mapping.

Then, we will prove that T is a contracting mapping. Indeed, for any φ, ψ ∈ X₀, we have that $\begin{matrix} ∥ T φ - T ψ ∥_{0} \\ \leq max_{1 \leq i \leq n} {sup_{t \in ℝ} [∥ p_{i} (t) (φ_{i} (η_{i} t) - ψ_{i} (η_{i} t)) ∥_{A} \\ + \int_{- \infty}^{t} e^{- \int_{s}^{t} {\tilde{a}}_{i} (u) du} K_{i} (∥ a_{i}^{\emptyset} (s) p_{i} (s) (φ_{i} (η_{i} s) \\ - ψ_{i} (η_{i} s)) ∥_{A} + ∥ a_{i}^{c} (s) (φ_{i} (s) - ψ_{i} (s)) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ b_{ij} (f_{j} (ξ_{j} φ_{j} (s)) - f_{j} (ξ_{j} ψ_{j} (s))) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ c_{ij} (g_{j} (ξ_{j} φ_{j} (q_{ij} s)) - g_{j} (ξ_{j} ψ_{j} (q_{ij} s))) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ α_{ij} (h_{j} (ξ_{j} φ_{j} (q_{ij} s)) - h_{j} (ξ_{j} ψ_{j} (q_{ij} s))) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ β_{ij} (l_{j} (ξ_{j} φ_{j} (q_{ij} s)) \\ - l_{j} (ξ_{j} ψ_{j} (q_{ij} s))) ∥_{A}) ds]} \\ \leq & max_{1 \leq i \leq n} [p_{i}^{+} ∥ φ - ψ ∥_{0} + \int_{- \infty}^{t} e^{- {\tilde{a}}_{i}^{-} (t - s)} K_{i} \\ ({\bar{a}}_{i}^{\emptyset} p_{i}^{+} ∥ φ - ψ ∥_{0} + {\bar{a}}_{i}^{c} ∥ φ - ψ ∥_{0} + ξ_{i}^{- 1} \sum_{j = 1}^{n} \\ b_{ij}^{+} L_{j}^{f} ξ_{j} ∥ φ - ψ ∥_{0} + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} ∥ φ - ψ ∥_{0} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} ∥ φ - ψ ∥_{0} + ξ_{i}^{- 1} \\ \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j} ∥ φ - ψ ∥_{0}) ds] \\ \leq & max_{1 \leq i \leq n} [p_{i}^{+} + \frac{1}{{\tilde{a}}_{i}^{-}} K_{i} ({\bar{a}}_{i}^{\emptyset} p_{i}^{+} + {\bar{a}}_{i}^{c} + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} \\ L_{j}^{f} ξ_{j} + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j}) ∥ φ - ψ ∥_{0}] \\ \leq & u ∥ φ - ψ ∥_{0}, \end{matrix}$ noticing that u < 1, T is a contracting mapping. Therefore, T has a unique fixed point in X₀, that is, system (2.1) has a unique pseudo almost periodic solution in X₀. The proof is complete.

4 Global asymptotic stability

In this section, we study the global asymptotic stability of pseudo almost periodic solutions of system (2.1).

Theorem 4.1. Suppose that (H₁)-(H₅) hold. Let x (t) be the pseudo almost periodic solution of system (2.1) with initial value φ and x^* (t) be an arbitrary solution of system (2.1) with initial value ψ. Then there exist positive constants σ and M such that $\begin{matrix} ∥ x (t) - x^{*} (t) ∥_{A^{n}} \leq M ∥ φ - ψ ∥_{ξ} e^{- σ ln \frac{1 + t}{1 + t_{0}}}, t > t_{0}, \end{matrix}$ where $∥ φ - ψ ∥_{ξ} = max_{1 \leq i \leq n} {sup_{s \in [ρ t_{0}, t_{0}]} ∥ [φ_{i} (t) - p_{i} (t) φ_{i} (η_{i} t)] - [ψ_{i} (t) - p_{i} (t) ψ_{i} (η_{i} t)] ∥_{A^{n}}}$ .

Proof. Using the following transformation

$\begin{matrix} y_{i} (t) = ξ_{i}^{- 1} x_{i} (t), y_{i}^{*} (t) = ξ_{i}^{- 1} x_{i}^{*} (t), z_{i} (t) \\ = y_{i} (t) - y_{i}^{*} (t), Z_{i} (t) = z_{i} (t) - p_{i} (t) z_{i} (η_{i} t), \end{matrix}$ (4.1) then system (2.1) can be written as

$\begin{matrix} Z_{i}^{'} (t) = & - a_{i} (t) Z_{i} (t) - a_{i} (t) p_{i} (t) z_{i} (η_{i} t) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij} (t) (f_{j} (ξ_{j} y_{j} (t)) - f_{j} (ξ_{j} y_{j}^{*} (t))) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij} (t) (g_{j} (ξ_{j} y_{j} (q_{ij} t)) - g_{j} (ξ_{j} y_{j}^{*} (q_{ij} t))) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} α_{ij} (t) (h_{j} (ξ_{j} y_{j} (q_{ij} t)) - h_{j} (ξ_{j} y_{j}^{*} (q_{ij} t))) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} β_{ij} (t) (l_{j} (ξ_{j} y_{j} (q_{ij} t)) - l_{j} (ξ_{j} y_{j}^{*} (q_{ij} t))), \\ i = 1, 2, \dots, n . \end{matrix}$ (4.2)

Without loss of generality, let $\begin{matrix} ∥ φ - ψ ∥_{A} > 0 . \end{matrix}$ In view of (H₅), we have that there exists a constant $σ \in (0, min_{1 \leq i \leq n} {\tilde{a}}_{i}^{-})$ such that $p_{i}^{+} e^{σ ln \frac{1}{η_{i}}} < 1$ and

$\begin{matrix} max_{1 \leq i \leq n} {σ - {\tilde{a}}_{i}^{-} + K_{i} [{\bar{a}}_{i}^{c} + {\bar{a}}_{i} p_{i}^{+} \frac{1}{1 - p_{i}^{+} e^{σ ln \frac{1}{η_{i}}}} e^{σ ln \frac{1}{η_{i}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1}{q_{ij}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1}{q_{ij}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1}{q_{ij}}}]} < 0, \\ i = 1, 2, \dots, n . \end{matrix}$ (4.3)

By (4.2), we have

$\begin{matrix} Z_{i}^{'} (s) + a_{i}^{\emptyset} (s) Z_{i} (s) \\ = - a_{i}^{c} (s) Z_{i} (s) - a_{i} (s) p_{i} (s) z_{i} (η_{i} s) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij} (s) (f_{j} (ξ_{j} y_{j} (s)) - f_{j} (ξ_{j} y_{j}^{*} (s))) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij} (s) (g_{j} (ξ_{j} y_{j} (q_{ij} s)) - g_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} α_{ij} (s) (h_{j} (ξ_{j} y_{j} (q_{ij} s)) - h_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} β_{ij} (s) (l_{j} (ξ_{j} y_{j} (q_{ij} s)) - l_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))), \\ i = 1, 2, \dots, n . \end{matrix}$ (4.4) Multiply both sides of formula (4.4) by $e^{\int_{t_{0}}^{s} a_{i}^{\emptyset} du}$ and integrating over the interval [t₀, t], we have

$\begin{matrix} Z_{i} (t) = & Z_{i} (t_{0}) e^{- \int_{t_{0}}^{t} a_{i}^{\emptyset} (u) du} + \int_{t_{0}}^{t} e^{- \int_{s}^{t} a_{i}^{\emptyset} (u) du} \\ [- a_{i}^{c} (s) Z_{i} (s) - a_{i} (s) p_{i} (s) z_{i} (η_{i} s) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij} (s) (f_{j} (ξ_{j} y_{j} (s)) - f_{j} (ξ_{j} y_{j}^{*} (s))) \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij} (s) (g_{j} (ξ_{j} y_{j} (q_{ij} s)) - g_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) \\ + ξ_{i}^{- 1} ⋁_{j = 1}^{n} α_{ij} (s) (h_{j} (ξ_{j} y_{j} (q_{ij} s)) - h_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) \\ + ξ_{i}^{- 1} ⋀_{j = 1}^{n} β_{ij} (s) (l_{j} (ξ_{j} y_{j} (q_{ij} s)) - l_{j} (ξ_{j} y_{j}^{*} (q_{ij} s)))] ds . \end{matrix}$ (4.5)

Take a constant M such that

$M > \sum_{i = 1}^{N} K_{i} + 1,$ (4.6) then, for any ɛ > 0, it is clear that

$\begin{matrix} ∥ Z (t) ∥_{A^{n}} < M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + t}{1 + t_{0}}}, \\ t \in (ρ t_{0}, t_{0}] . \end{matrix}$ (4.7) We claim that for any ɛ > 0,

$∥ Z (t) ∥_{A^{n}} < M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + t}{1 + t_{0}}}, t > t_{0} .$ (4.8) If (4.8) does not hold, then there must be some θ > t₀ such that

${\begin{matrix} ∥ Z (θ) ∥_{A^{n}} = M (∥ φ - ψ |_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} \\ ∥ Z (t) ∥_{A^{n}} < M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + t}{1 + t_{0}}}, t \in (ρ t_{0}, θ) . \end{matrix}$ (4.9)

By (4.1), we have

$\begin{matrix} e^{σ ln \frac{1 + v}{1 + t_{0}}} ∥ z_{i} (v) ∥_{A} \\ \leq e^{σ ln \frac{1 + v}{1 + t_{0}}} ∥ z_{i} (v) - p_{i} (t) z_{i} (η_{i} v) ∥_{A} \\ + e^{σ ln \frac{1 + v}{1 + t_{0}}} ∥ p_{i} (t) z_{i} (η_{i} v) ∥_{A} \\ \leq e^{σ ln \frac{1 + v}{1 + t_{0}}} ∥ Z_{i} (v) ∥_{A} + p_{i}^{+} e^{σ ln \frac{1 + v}{1 + η_{i} v}} e^{σ ln \frac{1 + η_{i} v}{1 + t_{0}}} ∥ z_{i} (η_{i} v) ∥_{A} \\ \leq M (∥ φ - ψ ∥_{ξ} + ɛ) + p_{i}^{+} e^{σ ln \frac{1}{η_{i}}} sup_{s \in [ρ t_{0}, t]} \\ e^{σ ln \frac{1 + s}{1 + t_{0}}} ∥ z_{i} (s) ∥_{A}, \end{matrix}$ (4.10)

for all v ∈ [ρt₀, t₀] , i = 1, 2, …, n. From (4.10), we have

$\begin{matrix} e^{σ ln \frac{1 + t}{1 + t_{0}}} ∥ z_{i} (t) ∥_{A} & \leq sup_{s \in [ρ_{i} t_{0}, t]} e^{σ ln \frac{1 + s}{1 + t_{0}}} ∥ z_{i} (s) ∥_{A} \\ \leq \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{i}^{+} e^{σ ln \frac{1}{η_{i}}}}, \\ t \in [ρ t_{0}, θ), i = 1, 2, \dots, n . \end{matrix}$ (4.11) From (4.3), (4.5), (4.6), (4.9) and (4.11) it follows that for i = 1, 2, …, n,

$\begin{matrix} ∥ Z_{i} (θ) ∥_{A} \leq (∥ φ - ψ ∥_{ξ} + ɛ) K_{i} e^{- \int_{t_{0}}^{θ} {\tilde{a}}_{i} (u) du} \\ + \int_{t_{0}}^{θ} K_{i} e^{- \int_{s}^{θ} {\tilde{a}}_{i} (u) du} [∥ a_{i}^{c} (s) Z_{i} (s) ∥_{A} \\ + ∥ a_{i} (s) p_{i} (s) z_{i} (η_{i} s) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ b_{ij} (s) (f_{j} (ξ_{j} y_{j} (s)) - f_{j} (ξ_{j} y_{j}^{*} (s))) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ c_{ij} (s) (g_{j} (ξ_{j} y_{j} (q_{ij} s)) - g_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) ∥_{A} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ α_{ij} (s) (h_{j} (ξ_{j} y_{j} (q_{ij} s)) - h_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) ∥_{R} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} ∥ β_{ij} (s) (l_{j} (ξ_{j} y_{j} (q_{ij} s)) - l_{j} (ξ_{j} y_{j}^{*} (q_{ij} s))) ∥_{A}] ds \\ \leq & (∥ φ - ψ ∥_{ξ} + ɛ) K_{i} e^{- \int_{t_{0}}^{θ} {\tilde{a}}_{i} (u) du} \\ + \int_{t_{0}}^{θ} K_{i} e^{- \int_{s}^{θ} {\tilde{a}}_{i} (u) du} [{\bar{a}}_{i}^{c} M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + s}{1 + t_{0}}} \\ + {\bar{a}}_{i} p_{i}^{+} \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{i}^{+} e^{σ ln \frac{1}{η_{i}}}} e^{- σ ln \frac{1 + η_{i} s}{1 + t_{0}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{- σ ln \frac{1 + s}{1 + t_{0}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{- σ ln \frac{1 + q_{ij} s}{1 + t_{0}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{- σ ln \frac{1 + q_{ij} s}{1 + t_{0}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j} \frac{M (∥ φ - ψ ∥_{ξ} + ɛ)}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{- σ ln \frac{1 + q_{ij} s}{1 + t_{0}}}] ds \\ \leq & M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} {\frac{K_{i}}{M} e^{- \int_{t_{0}}^{θ} ({\tilde{a}}_{i}^{-} - σ) du} \\ + \int_{t_{0}}^{θ} e^{- \int_{s}^{θ} ({\tilde{a}}_{i}^{-} - σ) du} K_{i} [{\bar{a}}_{i}^{c} + {\bar{a}}_{i} p_{i}^{+} \frac{1}{1 - p_{i}^{+} e^{σ ln \frac{1}{η_{i}}}} e^{σ ln \frac{1 + θ}{1 + η_{i} s}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} b_{ij}^{+} L_{j}^{f} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} c_{ij}^{+} L_{j}^{g} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1 + θ}{1 + q_{ij} s}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} α_{ij}^{+} L_{j}^{h} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1 + θ}{1 + q_{ij} s}} \\ + ξ_{i}^{- 1} \sum_{j = 1}^{n} β_{ij}^{+} L_{j}^{l} ξ_{j} \frac{1}{1 - p_{j}^{+} e^{σ ln \frac{1}{η_{j}}}} e^{σ ln \frac{1 + θ}{1 + q_{ij} s}}] ds} \\ \leq & M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} \\ [\frac{K_{i}}{M} e^{- \int_{t_{0}}^{θ} ({\tilde{a}}_{i}^{-} - σ) du} + \int_{t_{0}}^{θ} e^{- \int_{s}^{θ} ({\tilde{a}}_{i}^{-} - σ) du} ({\tilde{a}}_{i}^{-} - σ) ds] \\ \leq & M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} [1 - (1 - \frac{K_{i}}{M}) e^{- \int_{t_{0}}^{θ} ({\tilde{a}}_{i}^{-} - σ) du}] \\ < & M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} . \end{matrix}$

Consequently, we obtain $∥ Z (t) ∥_{A^{n}} < M (∥ φ - ψ ∥_{ξ} + ɛ) e^{- σ ln \frac{1 + θ}{1 + t_{0}}} .$ This contradicts (4.9), hence, (4.8) holds. Letting ɛ → 0⁺, we get

$∥ Z (t) ∥_{A^{n}} \leq M (∥ φ - ψ ∥_{ξ}) e^{- σ ln \frac{1 + t}{1 + t_{0}}}, t > t_{0} .$ (4.12)

Therefore, the pseudo almost periodic solution of system (2.1) is globally asymptotically stable. The proof is completed.

5 Two examples

In this section, we present two examples to show the feasibility of our main results of this paper.

Example 5.1. In system (2.1), let n = m = 2, s = 1, $η_{i} = \frac{1}{2}$ , $q_{ij} = \frac{1}{3}$ , and for i, j = 1, 2, take $\begin{matrix} x_{i} (t) = & x_{i}^{0} (t) e_{0} + x_{i}^{1} (t) e_{1} + x_{i}^{2} (t) e_{2} + x_{i}^{12} (t) e_{12} \in A, \\ f_{j} (x_{j}) = & \frac{1}{35} e_{0} sin x_{j}^{0} + \frac{1}{32} e_{1} sin x_{j}^{1} \\ + \frac{1}{26} e_{2} sin x_{j}^{2} + \frac{1}{37} e_{12} tan x_{j}^{12}, \\ g_{j} (x_{j}) = & \frac{1}{24} e_{0} tan x_{j}^{0} + \frac{1}{40} e_{1} sin x_{j}^{2} \end{matrix}$ $\begin{matrix} + \frac{1}{45} e_{2} sin x_{j}^{1} + \frac{1}{39} e_{12} sin x_{j}^{12}, \\ h_{j} (x_{j}) = & \frac{1}{50} e_{0} sin x_{j}^{0} + \frac{1}{45} e_{1} tan x_{j}^{1} \\ + \frac{1}{56} e_{2} sin x_{j}^{12} + \frac{1}{60} e_{12} sin x_{j}^{2}, \\ l_{j} (x_{j}) = & \frac{1}{62} e_{0} sin x_{j}^{0} + \frac{1}{64} e_{1} sin x_{j}^{12} \\ + \frac{1}{70} e_{2} tan x_{j}^{2} + \frac{1}{66} e_{12} sin x_{j}^{1}, \end{matrix}$ $\begin{matrix} a_{1} (t) = & (2 + 0.1 sin \sqrt{2} t) e_{0} + (1.8 + 0.1 sin 3 t) e_{1} \\ + (1.7 + 0.2 sin \sqrt{3} t) e_{2} \\ + (1.8 + 0.2 sin \sqrt{3} t) e_{12}, \\ a_{2} (t) = & (2.1 + 0.1 cos t) e_{0} + (1.7 + 0.2 sin 2 t) e_{1} \\ + (1.8 + 0.3 sin \sqrt{5} t) e_{2} \\ + (1.9 + 0.2 sin 4 t) e_{12}, \\ p_{1} (t) = & (0.0007 + 0.0002 sin \sqrt{2} t) e_{0} \\ + (0.0005 + 0.0003 sin t) e_{1} \\ + (0.0003 + 0.0002 sin \sqrt{3} t) e_{2} \\ + (0.0006 + 0.0001 cos t) e_{12}, \\ p_{2} (t) = & (0.0008 + 0.0001 cos \sqrt{2} t) e_{0} \\ + (0.0006 + 0.0004 sin \sqrt{2} t) e_{1} \\ + (0.0004 + 0.0001 sin t) e_{2} \\ + (0.0006 + 0.0004 sin t) e_{12}, \end{matrix}$

$\begin{matrix} (\begin{matrix} b_{11} (t) & b_{12} (t) \\ b_{21} (t) & b_{22} (t) \end{matrix}) = (\begin{matrix} 0.003 e_{0} sin t + 0.001 e_{1} cos \sqrt{5} t & 0.004 e_{2} sin \sqrt{2} t + 0.007 e_{12} sin \sqrt{3} t \\ 0.002 e_{1} sin \sqrt{3} t + 0.004 e_{2} sin t & 0.005 e_{0} cos t + 0.008 e_{12} sin \sqrt{2} t \end{matrix}), \\ (\begin{matrix} c_{11} (t) & c_{12} (t) \\ c_{21} (t) & c_{22} (t) \end{matrix}) = (\begin{matrix} 0.001 e_{0} cos t + 0.003 e_{1} sin \sqrt{3} t & 0.002 e_{2} sin t + 0.004 e_{12} sin \sqrt{2} t \\ 0.004 e_{1} sin \sqrt{2} t + 0.007 e_{2} sin \sqrt{3} t & 0.004 e_{0} sin \sqrt{3} t + 0.005 e_{12} cos t \end{matrix}), \\ (\begin{matrix} α_{11} (t) & α_{12} (t) \\ α_{21} (t) & α_{22} (t) \end{matrix}) = (\begin{matrix} 0.002 e_{0} cos t + 0.001 e_{1} sin \sqrt{3} t & 0.008 e_{2} cos \sqrt{5} t + 0.007 e_{12} sin t \\ 0.009 e_{1} sin \sqrt{5} t + 0.004 e_{2} sin \sqrt{3} t & 0.001 e_{0} sin t + 0.003 e_{12} cos \sqrt{5} t \end{matrix}), \\ (\begin{matrix} β_{11} (t) & β_{12} (t) \\ β_{21} (t) & β_{22} (t) \end{matrix}) = (\begin{matrix} 0.004 e_{0} sin \sqrt{3} t + 0.005 e_{1} cos \sqrt{5} t & 0.003 e_{2} cos t + 0.007 e_{12} sin t \\ 0.001 e_{1} sin \sqrt{5} t + 0.009 e_{2} cos t & 0.006 e_{0} cos 3 t + 0.004 e_{12} sin 2 t \end{matrix}), \end{matrix}$

$\begin{matrix} (\begin{matrix} s_{11} (t) & s_{12} (t) \\ s_{21} (t) & s_{22} (t) \end{matrix}) = (\begin{matrix} 0.002 e_{0} sin \sqrt{7} t + 0.005 e_{1} cos t & 0.004 e_{2} cos t + 0.008 e_{12} sin \sqrt{5} t \\ 0.003 e_{1} sin \sqrt{3} t + 0.004 e_{2} sin t & 0.005 e_{0} sin t + 0.007 e_{12} cos t \end{matrix}), \\ (\begin{matrix} r_{11} (t) & r_{12} (t) \\ r_{21} (t) & r_{22} (t) \end{matrix}) = (\begin{matrix} 0.001 e_{0} sin \sqrt{3} t + 0.004 e_{1} cos \sqrt{7} t & 0.004 e_{2} cos \sqrt{7} t + 0.008 e_{12} cos t \\ 0.002 e_{1} sin \sqrt{7} t + 0.003 e_{2} sin t & 0.005 e_{0} cos t + 0.007 e_{12} sin \sqrt{3} t \end{matrix}), \\ I_{1} (t) = 0.2 e_{0} (sin \sqrt{3} t + \frac{2}{1 + t^{2}}) + 0.24 e_{1} sin 2 t \\ + 0.26 e_{2} cos \sqrt{2} t + 0.3 e_{12} cos \sqrt{4} t, \\ I_{2} (t) = 0.56 e_{0} (sin \sqrt{3} t + \frac{3}{1 + t^{3}}) + 0.46 e_{1} sin 3 t \\ + 0.58 e_{2} cos \sqrt{4} t + 0.6 e_{12} cos \sqrt{3} t, \\ v_{1} (t) = v_{2} (t) = 0.001 e_{0} sin t + 0.009 e_{1} cos \sqrt{3} t \\ + 0.007 e_{2} sin \sqrt{2} t + 0.004 e_{12} sin \sqrt{7} t, \\ (\begin{matrix} {\tilde{a}}_{1} (t) \\ {\tilde{a}}_{2} (t) \end{matrix}) = (\begin{matrix} 2.2 + 0.1 sin \sqrt{3} t \\ 3 - 1 sin \sqrt{5} t \end{matrix}) . \end{matrix}$ Let K_i = 0.6, ξ_i = ξ_j = 1, i, j = 1, 2, by calculation, for j=1, 2, we have $L_{j}^{f} = \frac{1}{26}, L_{j}^{g} = \frac{1}{24}, L_{j}^{h} = \frac{1}{45}, L_{j}^{l} = \frac{1}{62}, {\tilde{a}}_{1}^{-} = 2.1, {\tilde{a}}_{2}^{-} = 2, p_{1}^{+} = 0.0009, p_{2}^{+} = 0.0009, {\bar{a}}_{1} = 2.1, {\bar{a}}_{2} = 2.2, {\bar{a}}_{1}^{\emptyset} = 2.1, {\bar{a}}_{2}^{\emptyset} = 2.2, {\bar{a}}_{1}^{c} = 2, {\bar{a}}_{2}^{c} = 2.1, b_{11}^{+} = 0.003, b_{12}^{+} = 0.007, b_{21}^{+} = 0.004, b_{22}^{+} = 0.008, c_{11}^{+} = 0.003, c_{12}^{+} = 0.004, c_{21}^{+} = 0.007, c_{22}^{+} = 0.005, α_{11}^{+} = 0.002, α_{12}^{+} = 0.008, α_{21}^{+} = 0.009, α_{22}^{+} = 0.003, β_{11}^{+} = 0.005, β_{12}^{+} = 0.007, β_{21}^{+} = 0.009, β_{22}^{+} = 0.006$ . So (H₁) and (H₂) are satisfied. Besides, we can get u₁ ≈ 0.601794, u₂ ≈ 0.6319332, ν₁ ≈ -0.8971, ν₁ ≈ -0.7367996, then $\begin{matrix} u = max {u_{1}, u_{2}} \approx 0.6319332 < 1, \\ ν = max {ν_{1}, ν_{2}} \approx - 0.8971 < 0 . \end{matrix}$ Therefore, all of the conditions of Theorem 4.1 are satisfied. Hence, system (2.1) has a unique pseudo almost periodic solutions that is globally asymptotically stable (see Figs. 1-4).

Fig. 1

States $x_{1}^{0} (t)$ , $x_{1}^{1} (t)$ , $x_{1}^{2} (t)$ and $x_{1}^{12} (t)$ of (2.1) with different initial values.

Fig. 2

States $x_{2}^{0} (t)$ , $x_{2}^{1} (t)$ , $x_{2}^{2} (t)$ and $x_{2}^{12} (t)$ of (2.1) with different initial values.

Fig. 3

Global asymptotic stability of states $x_{1}^{0} (t)$ , $x_{1}^{1} (t)$ , $x_{1}^{2} (t)$ and $x_{1}^{12} (t)$ of (2.1) with different initial values.

Fig. 4

Global asymptotic stability of states $x_{2}^{0} (t)$ , $x_{2}^{1} (t)$ , $x_{2}^{2} (t)$ and $x_{2}^{12} (t)$ of (2.1) with different initial values.

Example 5.2. In system (2.1), let n = 2, m = 3, s = 1, $η_{i} = \frac{1}{4}$ , $q_{ij} = \frac{1}{7}$ , and for i, j = 1, 2, take $\begin{matrix} x_{i} (t) = & x_{i}^{0} (t) e_{0} + x_{i}^{1} (t) e_{1} + x_{i}^{2} (t) e_{2} + x_{i}^{3} (t) e_{3} \\ + x_{i}^{12} (t) e_{12} + x_{i}^{13} (t) e_{13} \\ + x_{i}^{23} (t) e_{23} + x_{i}^{123} (t) e_{123} \in A, \end{matrix}$

$\begin{matrix} f_{j} (x_{j}) = & \frac{1}{176} e_{0} tan x_{j}^{0} + \frac{1}{167} e_{1} sin (x_{j}^{1} + x_{j}^{2}) \\ + \frac{1}{189} e_{2} sin x_{j}^{3} + \frac{1}{197} e_{3} sin x_{j}^{2} \\ + \frac{1}{178} e_{12} sin x_{j}^{12} + \frac{1}{156} e_{13} tan x_{j}^{23} \\ + \frac{1}{149} e_{23} tan x_{j}^{13} + \frac{1}{186} e_{123} sin x_{j}^{123}, \\ g_{j} (x_{j}) = & \frac{1}{188} e_{0} sin x_{j}^{1} + \frac{1}{197} e_{1} sin x_{j}^{2} \\ + \frac{1}{186} e_{2} tan (x_{j}^{0} + x_{j}^{3}) + \frac{1}{177} e_{3} sin x_{j}^{12} \\ + \frac{1}{167} e_{12} sin x_{j}^{3} + \frac{1}{156} e_{13} tan x_{j}^{13} \\ + \frac{1}{189} e_{23} sin x_{j}^{123} + \frac{1}{157} e_{123} sin x_{j}^{23}, \\ h_{j} (x_{j}) = & \frac{1}{100} e_{0} sin x_{j}^{0} + \frac{1}{123} e_{1} tan x_{j}^{2} \\ + \frac{1}{198} e_{2} sin (x_{j}^{0} + x_{j}^{1}) + \frac{1}{108} e_{3} tan x_{j}^{12} \\ + \frac{1}{197} e_{12} sin x_{j}^{23} + \frac{1}{179} e_{13} sin x_{j}^{13} \\ + \frac{1}{167} e_{23} sin x_{j}^{23} + \frac{1}{175} e_{123} tan x_{j}^{123}, \\ ł_{j} (x_{j}) = & \frac{1}{180} e_{0} tan (x_{j}^{0} + x_{j}^{2}) + \frac{1}{178} e_{1} sin x_{j}^{3} \\ + \frac{1}{198} e_{2} sin x_{j}^{1} + \frac{1}{120} e_{3} sin x_{j}^{23} \\ + \frac{1}{190} e_{12} sin x_{j}^{2} + \frac{1}{174} e_{13} sin x_{j}^{12} \\ + \frac{1}{169} e_{23} tan x_{j}^{123} + \frac{1}{175} e_{123} sin x_{j}^{13}, \\ p_{1} (t) = & (0.0023 + 0.003 cos \sqrt{3} t) e_{0} \\ + (0.0012 + 0.001 sin t) e_{1} \\ + (0.0012 + 0.0016 sin \sqrt{2} t) e_{2} \\ + (0.0018 + 0.0023 cos t) e_{3} \\ + (0.0017 + 0.0016 cos \sqrt{5} t) e_{12} \\ + (0.0014 + 0.0023 sin \sqrt{2} t) e_{13} \\ + (0.0019 + 0.0021 sin 2 t) e_{23} \\ + (0.0012 + 0.0031 cos \sqrt{5} t) e_{123}, \\ p_{2} (t) = & (0.0043 + 0.0031 sin 2 t) e_{0} \\ + (0.0031 + 0.0021 sin t) e_{1} \\ + (0.0034 + 0.0016 sin \sqrt{11} t) e_{2} \\ + (0.0018 + 0.0019 cos \sqrt{2} t) e_{3} \\ + (0.0019 + 0.0023 cos 2 t) e_{12} \\ + (0.0024 + 0.0037 sin 5 t) e_{13} \\ + (0.0023 + 0.0024 sin 2 t) e_{23} \\ + (0.0029 + 0.0016 sin \sqrt{13} t) e_{123}, \\ a_{1} (t) = & (2.3 + 0.1 sin \sqrt{2} t) e_{0} \\ + (0.7 + 0.02 cos \sqrt{3} t) e_{1} \\ + (0.8 + 0.23 sin \sqrt{11} t) e_{2} \\ + (0.76 + 0.01 cos \sqrt{5} t) e_{3} \\ + (0.9 + 0.2 sin t) e_{12} \\ + (0.8 + 0.1 sin 2 t) e_{13} \\ + (0.58 + 0.02 sin \sqrt{7} t) e_{23} \\ + (1.4 + 0.03 cos \sqrt{5} t) e_{123}, \\ a_{2} (t) = & (2.5 + 0.1 cos \sqrt{5} t) e_{0} \\ + (1.6 + 0.1 sin \sqrt{5} t) e_{1} \\ + (0.9 + 0.4 sin t) e_{2} \\ + (0.74 + 0.4 cos 2 t) e_{3} \\ + (0.85 + 0.2 sin 5 t) e_{12} \\ + (0.74 + 0.01 cos t) e_{13} \\ + (0.3 + 0.01 sin \sqrt{11} t) e_{23} \\ + (0.89 + 0.1 cos 2 t) e_{123}, \\ I_{1} (t) = & 0.06 e_{0} sin \sqrt{5} t + 0.016 e_{1} cos \sqrt{5} t \\ + 0.013 e_{2} sin t + 0.01 e_{3} cos \sqrt{7} t \\ + 0.061 e_{12} sin \sqrt{7} t + 0.014 e_{13} cos \sqrt{11} t \\ + 0.017 e_{23} sin \sqrt{6} t + 0.056 e_{123} cos \sqrt{3} t, \\ I_{2} (t) = & 0.023 e_{0} cos \sqrt{5} t + 0.026 e_{1} sin \sqrt{7} t \\ + 0.041 e_{2} cos t + 0.04 e_{3} sin \sqrt{11} t \\ + 0.06 e_{12} cos \sqrt{5} t + 0.025 e_{13} cos \sqrt{13} t \\ + 0.018 e_{23} sin 2 t + 0.06 e_{123} cos \sqrt{3} t, \\ v_{1} (t) = & v_{2} (t) = 0.016 e_{0} sin 2 t + 0.23 e_{1} cos 3 t \\ + 0.27 e_{2} sin \sqrt{3} t + 0.32 e_{3} sin \sqrt{3} t \\ + 0.18 e_{12} cos t + 0.19 e_{13} sin \sqrt{5} t \\ + 0.24 e_{23} cos 7 t + 0.34 e_{123} sin \sqrt{3} t, \\ b_{11} (t) = & 0.18 e_{0} cos t + 0.17 e_{1} sin \sqrt{2} t \\ + 0.15 e_{2} cos \sqrt{3} t + 0.13 e_{3} sin t, \\ b_{12} (t) = & 0.16 e_{12} sin 5 t + 0.23 e_{13} cos \sqrt{2} t \\ + 0.14 e_{23} sin \sqrt{3} t + 0.17 e_{123} cos t, \\ b_{21} (t) = & 0.19 e_{2} sin t + 0.26 e_{3} cos t \\ + 0.23 e_{12} sin \sqrt{5} t + 0.22 e_{13} sin \sqrt{5} t, \\ b_{22} (t) = & 0.23 e_{0} cos \sqrt{2} t + 0.21 e_{1} sin t \\ + 0.16 e_{23} cos \sqrt{3} t + 0.18 e_{123} sin \sqrt{3} t, \\ c_{11} (t) = & 0.34 e_{0} sin t + 0.21 e_{1} cos \sqrt{3} t \\ + 0.19 e_{2} cos \sqrt{2} t + 0.18 e_{3} sin t, \\ c_{12} (t) = & 0.21 e_{12} sin 2 t + 0.54 e_{13} sin t \\ + 0.18 e_{23} cos \sqrt{3} t + 0.12 e_{123} cos 3 t, \\ c_{21} (t) = & 0.19 e_{2} cos \sqrt{5} t + 0.27 e_{3} sin t \\ + 0.18 e_{12} cos \sqrt{3} t + 0.22 e_{13} cos t, \\ c_{22} (t) = & 0.24 e_{0} sin t + 0.28 e_{1} cos t \\ + 0.19 e_{23} sin 2 t + 0.17 e_{123} cos \sqrt{5} t, \\ α_{11} (t) = & 0.21 e_{0} cos \sqrt{11} t + 0.56 e_{1} sin \sqrt{3} t \\ + 0.27 e_{2} sin t + 0.23 e_{3} cos t, \\ α_{12} (t) = & 0.25 e_{12} cos 2 t + 0.43 e_{13} cos 3 t \\ + 0.39 e_{23} sin \sqrt{2} t + 0.27 e_{123} cos t, \\ α_{21} (t) = & 0.34 e_{2} sin \sqrt{13} t + 0.21 e_{3} cos t \\ + 0.19 e_{12} sin \sqrt{5} t + 0.16 e_{13} sin t, \\ α_{22} (t) = & 0.3 e_{0} cos t + 0.24 e_{1} sin 5 t + 0.19 e_{23} cos t \\ + 0.14 e_{123} sin \sqrt{3} t, \\ β_{11} (t) = & 0.17 e_{0} sin \sqrt{13} t + 0.24 e_{1} cos \sqrt{5} t \\ + 0.16 e_{2} sin \sqrt{3} t + 0.18 e_{3} cos t, \\ β_{12} (t) = & 0.25 e_{12} sin t + 0.19 e_{13} sin 2 t \\ + 0.13 e_{23} cos \sqrt{5} t + 0.47 e_{123} cos t, \\ β_{21} (t) = & 0.16 e_{2} cos t + 0.31 e_{3} sin \sqrt{5} t \\ + 0.37 e_{12} cos 3 t + 0.16 e_{13} sin t, \\ β_{22} (t) = & 0.19 e_{0} sin t + 0.24 e_{1} sin 3 t \\ + 0.18 e_{23} sin \sqrt{5} t + 0.17 e_{123} cos \sqrt{3} t, \\ s_{11} (t) = & 0.37 e_{0} sin t + 0.46 e_{1} sin \sqrt{3} t \\ + 0.35 e_{2} cos \sqrt{5} t + 0.47 e_{3} sin \sqrt{5} t, \\ s_{12} (t) = & 0.42 e_{12} cos t + 0.36 e_{13} cos \sqrt{5} t \\ + 0.46 e_{23} sin 2 t + 0.56 e_{123} cos \sqrt{3} t, \\ s_{21} (t) = & 0.48 e_{2} sin \sqrt{5} t + 0.57 e_{3} cos \sqrt{3} t \\ + 0.49 e_{12} sin \sqrt{5} t + 0.74 e_{13} cos \sqrt{5} t, \\ s_{22} (t) = & 0.59 e_{0} cos \sqrt{11} t + 0.64 e_{1} sin t \\ + 0.38 e_{23} cos t + 0.47 e_{123} sin \sqrt{5} t, \\ γ_{11} (t) = & 0.49 e_{0} cos t + 0.54 e_{1} sin \sqrt{3} t \\ + 0.64 e_{2} cos \sqrt{3} t + 0.37 e_{3} sin \sqrt{2} t, \\ γ_{12} (t) = & 0.34 e_{12} sin t + 0.47 e_{13} sin \sqrt{6} t \\ + 0.32 e_{23} cos 5 t + 0.29 e_{123} sin \sqrt{5} t, \\ γ_{21} (t) = & 0.37 e_{2} cos \sqrt{6} t + 0.48 e_{3} sin \sqrt{5} t \\ + 0.49 e_{12} cos t + 0.34 e_{13} sin t, \\ γ_{22} (t) = & 0.24 e_{0} cos \sqrt{13} t + 0.29 e_{1} cos \sqrt{5} t \\ + 0.37 e_{23} sin 2 t + 0.54 e_{123} sin t, \\ {\tilde{a}}_{1} (t) = & 4.1 + 0.1 sin \sqrt{5} t, \\ {\tilde{a}}_{2} (t) = 4.7 + 0.4 sin \sqrt{11} t . \end{matrix}$ Let K_i = 1, $ξ_{i} = \frac{1}{5}$ , ξ_j = 1, i, j = 1, 2, by calculate, for j=1, 2, we have $L_{j}^{f} = \frac{1}{149}$ , $L_{j}^{g} = \frac{1}{156}$ , $L_{j}^{h} = \frac{1}{100}$ , $L_{j}^{l} = \frac{1}{120}$ , ${\tilde{a}}_{1}^{-} = 4$ , ${\tilde{a}}_{2}^{-} = 4.3$ , $p_{1}^{+} = 0.0057$ , $p_{2}^{+} = 0.0074$ , ${\bar{a}}_{1}^{\emptyset} = 2.4$ , ${\bar{a}}_{2}^{\emptyset} = 2.6$ , ${\bar{a}}_{1}^{c} = 1.72$ , ${\bar{a}}_{2}^{c} = 2$ , $b_{11}^{+} = 0.18$ , $b_{12}^{+} = 0.23$ , $b_{21}^{+} = 0.26$ , $b_{22}^{+} = 0.23$ , $c_{11}^{+} = 0.34$ , $c_{12}^{+} = 0.54$ , $c_{21}^{+} = 0.27$ , $c_{22}^{+} = 0.28$ , $α_{11}^{+} = 0.56$ , $α_{12}^{+} = 0.43$ , $α_{21}^{+} = 0.34$ , $α_{22}^{+} = 0.3$ , $β_{11}^{+} = 0.24$ , $β_{12}^{+} = 0.47$ , $β_{21}^{+} = 0.37$ , $β_{22}^{+} = 0.24$ . So (H₁) and (H₂) are satisfied. Besides, we can get u₁ ≈ 0.4403286, u₂ ≈ 0.46526539, ν₁ ≈ -2.2515 and ν₁ ≈ -2.262, then $\begin{matrix} u = max {u_{1}, u_{2}} \approx 0.46526539 < 1, \\ ν = max {ν_{1}, ν_{2}} \approx - 2.262 < 0 . \end{matrix}$ Therefore, all of the conditions of Theorem 4.1 are satisfied. Hence, system (2.1) has a unique pseudo almost periodic solution that is globally asymptotic stable (see Figs. 5-10).

Fig. 5

States $x_{1}^{0} (t)$ , $x_{1}^{1} (t)$ , $x_{1}^{2} (t)$ , $x_{1}^{3}$ , $x_{1}^{12}$ , $x_{1}^{13} (t)$ , $x_{1}^{23} (t)$ , $x_{1}^{123} (t)$ of (2.1) with different initial values.

Fig. 6

States $x_{2}^{0} (t)$ , $x_{2}^{1} (t)$ , $x_{2}^{2} (t)$ , $x_{2}^{3}$ , $x_{2}^{12}$ , $x_{2}^{13} (t)$ , $x_{2}^{23} (t)$ , $x_{2}^{123} (t)$ of (2.1) with different initial values.

Fig. 7

Global asymptotic stability of states $x_{1}^{0} (t)$ , $x_{1}^{1} (t)$ , $x_{1}^{2} (t)$ and $x_{1}^{3} (t)$ of (2.1) with different initial values.

Fig. 8

Global asymptotic stability of states $x_{1}^{12} (t)$ , $x_{1}^{13} (t)$ , $x_{1}^{23} (t)$ and $x_{1}^{123} (t)$ of (2.1) with different initial values.

Fig. 9

Global asymptotic stability of states $x_{2}^{0} (t)$ , $x_{2}^{1} (t)$ , $x_{2}^{2} (t)$ and $x_{2}^{3} (t)$ of (2.1) with different initial values.

Fig. 10

Global asymptotic stability of states $x_{2}^{12} (t)$ , $x_{2}^{13} (t)$ , $x_{2}^{23} (t)$ and $x_{2}^{123} (t)$ of (2.1) with different initial values.

Remark 5.1. The results of Examples 5.1 and 5.2 cannot be deduced from [6–11 , 28] and any other known results.

6 Conclusions

In this paper, we obtain the existence and global asymptotic stability of pseudo almost periodic solutions for a class of Clifford-valued neutral-type fuzzy neural networks with proportional delay and D operator and whose self feedback coefficients are also Clifford numbers by direct method. Our results are new. Our method can be used to study the existence of almost automorphic solutions for Clifford-valued neural networks with proportional delay and D operator.

Data Availability

All the data are available within this article.

Conflicts of Interest

The authors declare no conflicts of interest.

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