$(\mu ,\nu )$-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks

Abstract

This article is concerned with a high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and mixed delays. Sufficient conditions are derived for the existence, the uniqueness and the exponential stability of $(\mu ,\nu )$-pseudo-almost automorphic solutions of the considered model. Banach fixed-point theorem is applied for the existence and the uniqueness results. Global exponential stability is derived via differential inequalities. Finally, two examples are provided to support the feasibility of the theoretical results.

Appendices

Appendix A Proof of Lemma 1

Proof

By definition, we can express $\phi$ as $\phi =f+g$ such that $f\in AA({\mathbb {R}},{\mathbb {R}})$ and $g \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).$

Clearly, $\phi (.-h)=f(.-h)+g(.-h).$

It is easily seen that $f(.-h)\in AA({\mathbb {R}},{\mathbb {R}}).$ For each $h\in {\mathbb {R}},$ letting $\mu _h=\mu (\{t+h,t\in A\})$ for $A \in B,$ it follows from (H4) that $\mu$ and $\mu _h$ are equivalent. For $h \ge 0,$ we have

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t )\\&\quad =\frac{1}{\nu ([-r,r])} \int _{-r-h}^{r-h} | g(t)| d\mu _{h} (t) \\&\quad =\frac{1}{\nu ([-r,r])} \left( \int _{-r-h}^{r+h} | g(t)| d\mu _{h} (t) - \int _{r-h}^{r+h} | g(t)| d\mu _{h}(t) \right) \\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r-h}^{r+h}| g(t)| d\mu _{h} (t). \end{aligned}$$

For $h \le 0,$ we have

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t )\\&\quad =\frac{1}{\nu ([-r,r])} \int _{-r-h}^{r-h} | g(t)| d\mu _{h}(t) \\&\quad =\frac{1}{\nu ([-r,r])} \left( \int _{-r+h}^{r-h} | g(t)| d\mu _{h} (t))-\int _{-r+h}^{-r-h} | g(t)| d\mu _{h} (t) \right) \\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r+h}^{r-h}| g(t)| d\mu _{h}(t) . \end{aligned}$$

So we obtain

$$\begin{aligned}&\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t)\\&\quad \le \frac{1}{\nu ([-r,r])} \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu _{h}(t) \\&\quad \le \frac{\nu ([-r-|h|,r+|h|])}{\nu ([-r,r])}\frac{1}{\nu ([-r-|h|,r+|h|])}\\&\qquad \times \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu _h(t)\\&\quad \le \frac{\nu ([-r-|h|,r+|h|])}{\nu ([-r,r])}\frac{\beta }{\nu ([-r-|h|,r+|h|])}\\&\qquad \times \int _{-r-\mid h \mid }^{r+ \mid h \mid } |g(t)| d\mu (t)\\ \end{aligned}$$

Since $\nu$ satisfies (H4) and $g \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ),$ then

$$\begin{aligned} \lim _{r\mapsto \infty }\frac{1}{\nu ([-r,r])} \int _{-r}^{r} | g(t-h)| d\mu (t)=0. \end{aligned}$$

The space $PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$ is translation invariant. $\square$

Appendix B Proof of Lemma 2

Proof

By definition, we can write $\phi = \phi _{1}+ \phi _{2} , \psi =\psi _{1}+\psi _{2}$ where $\phi _{1} , \psi _{1} \in AA({\mathbb {R}},{\mathbb {R}})$ and $\phi _{2} , \psi _{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$.

Clearly, $\phi \psi =\phi _{1}\psi _{1}+\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\phi _{2}\psi _{2},$$\phi _{1}\psi _{1} \in AA({\mathbb {R}},{\mathbb {R}})$ is hold.

For $[\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\phi _{2}\psi _{2}],$ we have

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{-r}^{r} |\phi _{1} \psi _{2} + \varphi _{2} \psi _{1} + \varphi _{2} \psi _{2} | d\mu (t)\\&\quad \le \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int _{-r}^{r} ( \parallel \phi _{1} \parallel _{\infty } | \psi _{2} | + |\phi _{2}| \Vert \psi _{1} \Vert _{\infty } \\&\qquad + \parallel \phi _{2} \parallel _{\infty } | \psi _{2}| ) d\mu (t)= 0, \end{aligned}$$

from which $[\phi _{1}\psi _{2}+\phi _{2}\psi _{1} +\varphi _{2}\psi _{2}] \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$.

Then, $\phi \times \psi \in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$. $\square$

Appendix C Proof of Lemma 3

Proof

We have $y = y_{1}+ y_{2},$ where $y_{1} \in AA({\mathbb {R}},{\mathbb {R}})$ and $y_{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).$ Let

$$\begin{aligned} {\hat{E}}(t)&= {} g(y(t-\tau ))\\&= g(y_{1}(t-\tau ))+ [g(y_{1}(t-\tau )+y_{2}(t-\tau ))\\ & \quad - g(y_{1}(t-\tau ))]\\&= {\hat{E}}_{1}(t) + {\hat{E}}_{2}(t). \end{aligned}$$

Since $y_{1}$ is almost automorphic, so for every sequence of real numbers $\left( s_{n}^{\prime }\right)$, we can extract a subsequence $\left( s_{n}\right)$ such that $\lim \nolimits _{n\rightarrow +\infty }y_{1}\left( t+s_{n}\right) =y_{1}^{1}\left( t\right) ,$$\lim \nolimits _{n\rightarrow +\infty }y_{1}^{1}\left( t-s_{n}\right) =y_{1}\left( t\right) ,$ for $t \in {\mathbb {R}}$. Clearly,

$$\begin{aligned}&\mid {\hat{E}}_{1}(t +s_{n})- g(y_{1}^{1}(t - \tau ))\mid \\&\quad \le \mid g(y_{1}(t - \tau +s_n))- g(y_{1}^{1}(t - \tau )) \mid \\&\quad \le m\mid y_{1}(t - \tau +s_n)- y_{1}^{1}(t - \tau ) \mid \rightarrow 0, n\rightarrow \infty \end{aligned}$$

Reasoning in a similar way, we can show easily that for $n > N,$$\{ y_{1}^{1}(t - \tau -s_{n} )\}$ converges to $y_{1}(t - \tau )$ uniformly on ${\mathbb {R}}.$

So $y_{1}(t - \tau )$ is almost automorphic . ${\hat{E}}_{1}(t)\in AA({\mathbb {R}},{\mathbb {R}})$.

Now, we show that ${\hat{E}}_{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | {\hat{E}}_{2}(t) | d\mu (t) \\&\quad =\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | g(y_{1}(t-\tau )\\&\qquad + y_{2}(t-\tau ))- g(y_{1}(t-\tau ))| d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{ m_j}{\nu ([-r,r])} \int _{[-r,r]} | y_{2}(t-\tau ))| d\mu (t)=0. \end{aligned}$$

Thus, ${\hat{E}}_{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$. So, $E(t) \in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$. $\square$

Appendix D Proof of Lemma 4

Proof

We know that

$$\begin{aligned} \phi _{ij}(t)= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) h_{i}(x_{i}(s)) {\mathrm{d}}s=\int \limits _{0}^{\infty } N_{ij}(s) h_{i}(x_{i}(t-s)) {\mathrm{d}}s.\\ \end{aligned}$$

First, we have

$$\begin{aligned} |\phi _{ij}(t) |\le & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) |h_{i}^{2}(x_{i}(s))| {\mathrm{d}}s\\\le & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) |h_{i}^{2}| {\mathrm{d}}s = |h_{i}^{2}| \end{aligned}$$

which proves that the integral $\int \nolimits _{-\infty }^{t} N_{ij}(t-s) h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s$ is absolutely convergent and the function $\phi _{ij}$ is bounded.

Here, we have to prove the continuity of the function $\phi _{ij}$.

Let $(w_{n})_{n}$ be a sequence of real numbers such that $\lim \nolimits _{n\longrightarrow \infty } w_{n} = 0$.

The continuity of the function $x_{i}(.)$ implies that for all $\epsilon > 0$, $\exists N \in \mathbb {N}$ such that for all $n \ge N,$$\mid x_{i}(s+w_{n})- x_{i}(s)\mid \le \frac{\epsilon }{ D_{i}}.$ We have

$$\begin{aligned}&\mid \phi _{ij}(t+w_{n})- \phi _{ij}(t)\mid \\&\quad = \mid \int \limits _{-\infty }^{t+w_{n}} N_{ij}(t+w_{n}-s) h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s- \int \limits _{-\infty }^{t} N_{ij}(t-s)h_{i}^{2}(x_{i}(s)) {\mathrm{d}}s \mid \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s)\mid h_{i}^{2}(x_{i}(s+w_{n}))-h^{2}_{i} (x_{i}(s)) \mid {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) D_{i} \mid x_{i}(s+w_{n})- x_{i}(s) \mid {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) {\mathrm{d}}s D_{i}\frac{\epsilon }{ D_{i}} \le \epsilon ; \end{aligned}$$

it remains to be proven whether the function $\phi _{ij} \in PAA({\mathbb {R}}, {\mathbb {R}},\mu ,\nu ).$ Since $h_{i}^{2}(x_{i}(.))\in PAA({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$, we can write $h_{i}^{2}(x_{i}(s))= u_{i}(s)+v_{i}(s)$ such that $u_{i} (.)\in AA({\mathbb {R}},{\mathbb {R}})$ and $v_{i}(.) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$. Consequently,

$$\begin{aligned} \phi _{ij}(t)= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) [u_{i}(s)+v_{i}(s)] {\mathrm{d}}s \\= & {} \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}(s) {\mathrm{d}}s + \int \limits _{-\infty }^{t} N_{ij}(t-s) v_{i}(s) {\mathrm{d}}s \\= & {} \phi _{ij}^{1}(t)+ \phi _{ij}^{2}(t). \end{aligned}$$

Let $\left( s_{n}^{\prime }\right)$ be a sequence of real numbers; we can extract a subsequence $\left( s_{n}\right)$ of $\left( s_{n}^{\prime }\right)$ such that for all $t\in {\mathbb {R}}:$$\lim \nolimits _{n\rightarrow +\infty }u_{i}\left( t+s_{n}\right) =u_{i}^{1}\left( t\right) ,$ and, $\lim \nolimits _{n\rightarrow +\infty }u_{i}^{1}\left( t-s_{n}\right) =u_{i}\left( t\right) ,$

Let $\phi _{ij}^{1*}(t) = \int _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s.$ Hence,

$$\begin{aligned}&\mid \phi _{ij}^{1}(t+s_{n})- \phi _{ij}^{1*}(t)\mid \\&\quad =\, \mid \int \limits _{-\infty }^{t+s_{n}} N_{ij}(t+s_{n}-s) u_{i}(s) {\mathrm{d}}s - \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s \mid \\&\quad =\, \mid \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}(s+s_{n}) {\mathrm{d}}s - \int \limits _{-\infty }^{t} N_{ij}(t-s) u_{i}^{1}(s) {\mathrm{d}}s \mid \\&\quad \le \int \limits _{-\infty }^{t} N_{ij}(t-s) \mid u_{i}(s+s_{n}) - u_{i}^{1}(s) \mid {\mathrm{d}}s \\&\quad \le\, \epsilon \mid u_{i}(s+s_{n}) - u_{i}^{1}(s) \mid . \end{aligned}$$

We obtain immediately that $\lim \nolimits _{n\rightarrow +\infty } \phi _{ij}^{1}(t+s_{n}) = \phi _{ij}^{1*}(t),\; \forall \; t \in {\mathbb {R}}.$

Reasoning in a similar way, we can show easily that $\lim \nolimits _{n\rightarrow +\infty } \phi _{ij}^{1*}(t- s_{n}) = \phi _{ij}^{1}(t), \;\forall \; t \in {\mathbb {R}},$ which implies that for all , $1\le i\le n,$ we have $\phi _{ij}^{1} \in AA({\mathbb {R}},{\mathbb {R}}).$

Second, let us show that for all $1\le i\le n,$$1\le j\le p,$ the function $(\phi _{ij}^{2}(t)) \in \xi ({\mathbb {R}},{\mathbb {R}}^n,\mu ,\nu )$.

We obtain the following estimate

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]}| \phi _{ij}^{2}(t)| d\mu (t)\\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{0}^{\infty } N_{ij}(s) | v_{i}(t-s) | {\mathrm{d}}s\right) d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int \limits _{0}^{\infty } N_{ij}(s) \left( \int \limits _{[-r,r]} | v_{i}(t-s) | d\mu (t)\right) {\mathrm{d}}s \\&\quad = \int \limits _{0}^{\infty } N_{ij}(s)\left( \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int \limits _{[-r,r]} | v_{i}(t-s) | d\mu (t)\right) {\mathrm{d}}s \\&\quad =0 \end{aligned}$$

which implies that $\phi _{ij}^{2}(t) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$.

Consequently, $\phi _{ij}$ is $(\mu ,\nu )$-pseudo-almost automorphic function. $\square$

Appendix E Proof of Lemma 5

Proof

By using Lemma 1-4, for all $1 \le i \le n,$$1 \le j \le p,$ the functions $F_{i}(.)$ and $G_{j}(.)$ are $(\mu ,\nu )$-pseudo-almost automorphic.

Consequently, for all, $1 \le i \le n,$$1 \le j \le p$, we can write $F_{i}=F_{i}^{1}+F_{i}^{2}$ and $G_{j}=G_{j}^{1}+G_{j}^{2}$ such that $F_{i}^{1},G_{j}^{1} \in AA({\mathbb {R}},{\mathbb {R}}),$ and $F_{i}^{2},G_{j}^{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).$

First, we prove that the function $(\varGamma _i F_i)(t)= \int \nolimits _{-\infty }^{t} e^{-\int \limits _{s}^{t} a_{i}^{1}(u){\mathrm{d}}u} F_i(s){\mathrm{d}}s$ is $(\mu ,\nu )$-pseudo-almost automorphic function.

$$\begin{aligned} (\varGamma _i F_i)(t)= & {} \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u) {\mathrm{d}}u} F_{i}^{1}(s){\mathrm{d}}s + \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u) {\mathrm{d}}u } F_{i}^{2}(s){\mathrm{d}}s \\=\, & {} (\varGamma _{i}F_{i}^{1})(t) +(\varGamma _{i}F_{i}^{2})(t), \end{aligned}$$

Let $\left( s_{n}^{\prime }\right) _{n \in \mathbb {N}}$ be a sequence of real numbers. By hypothesis, we can extract a subsequence $(s_{n})_{n \in \mathbb {N}}$ of $\left( s_{n}^{\prime }\right) _{n \in \mathbb {N}}$ such that $\lim \nolimits _{n\rightarrow +\infty }a_{i}^{1} \left( t+s_{n}\right) =a_{i}^{1*}\left( t\right) , \;\; \lim \nolimits _{n\rightarrow +\infty }a_{i}^{1*}\left( t-s_{n}\right) =a_{i}^{1}\left( t\right) ,\;\forall \; t\in {\mathbb {R}}.$

$\lim \nolimits _{n\rightarrow +\infty }F_{i}^{1} \left( t+s_{n}\right) = F_{i}^{1*}\left( t\right) , \;\; \lim \nolimits _{n\rightarrow +\infty }F_{i}^{1*}\left( t-s_{n}\right) = F_{i}^{1}\left( t\right) ,\;\forall \; t\in {\mathbb {R}}.$

Pose $(\varGamma _{i}^{1}F_{i}^{1*})(t)=\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1*}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s$. We write

$$\begin{aligned}&(\varGamma _{i}F_{i}^{1})(t+s_{n})-(\varGamma _{i}^{1}F_{i}^{1*})(t)\\&\quad =\int \limits _{-\infty }^{t+s_{n}} e^{-\int \limits _{s}^{t+s_{n}}a_{i}^{1}(u) {\mathrm{d}}u } F_{i}^{1}(s){\mathrm{d}}s - \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t+s_{n}} e^{-\int \limits _{s-s_{n}}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1}(s){\mathrm{d}}s -\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1}(s+s_{n}){\mathrm{d}}s - \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1*}(s){\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(\rho +s_{n}) d\rho } F_{i}^{1*}(s){\mathrm{d}}s -\int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } F_{i}^{1*}(s){\mathrm{d}}s \\&\quad = \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u } (F_{i}^{1}(s+s_{n}) - F_{i}^{1*}(s)){\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t} \left( e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u } - e^{-\int \limits _{s}^{t}a^{1*}_{i}(u) {\mathrm{d}}u } \right) F_{i}^{1*}(s){\mathrm{d}}s. \\ \end{aligned}$$

Then, there exists $\theta \in ]0,1[$ such that

$$\begin{aligned}&|(\varGamma _{i} F_{i}^{1} )(t+s_{n})-(\varGamma _{i} ^{1}F_{i}^{1*} )(t) |\\&\quad \le |F_{i}^{1*}|_\infty \int \limits _{-\infty }^{t}\left( e^{-\int \limits _{s}^{t}a_{i}^{1}\left( u +s_{n}\right) {\mathrm{d}}u }-e^{-\int \limits _{s}^{t}a_{i}^{1*}\left( u \right) {\mathrm{d}}u }\right) {\mathrm{d}}s\\&\qquad +\int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a_{i}^{1}\left( u +s_{n}\right) {\mathrm{d}}u }\left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| {\mathrm{d}}s \\&\quad \le \int \limits _{-\infty }^{t}\left( e^{- \left[ \int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u + \theta \left( \int \limits _{s}^{t}a_{i}^{1*}( u) {\mathrm{d}}u-\int \limits _{s}^{t}a_{i}^{1}(u+s_{n}) {\mathrm{d}}u\right) \right] } \right. \\&\left. \qquad \times \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u \right) {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad +\int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u }\left| F_{i}^{1} ( s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\le \int \limits _{-\infty }^{t}\left( e^{-\left[ \int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u + \theta \left( \int \limits _{s}^{t}a_{i}^1(u) {\mathrm{d}}u-\int \limits _{s}^{t}a_{i}^{1}(u +s_{n}) {\mathrm{d}}u\right) \right] }\right. \\&\left. \quad \times \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u \right) {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad +\int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)}\left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| {\mathrm{d}}s\\&\quad \le \int \limits _{-\infty }^{t}\left\{ e^{-a_{i*}^{1}(t-s)} e^{- \theta \left( \int \limits _{s}^{t}\left| a_{i}^1*( u )-a_{i}^{1}(u +s_{n})\right| {\mathrm{d}}u \right) }\right. \\&\left. \qquad \times \int _s^t \left| a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})\right| {\mathrm{d}}u \right\} {\mathrm{d}}s |F_{i}^{1*}|_\infty \\&\qquad + \int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} (s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\quad \le |F_{i}^{1*}|_\infty \int \limits _{-\infty }^{t} \left\{ e^{-a_{i*}^{1}(t-s)} \int _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u +s_{n})|{\mathrm{d}}u \right\} {\mathrm{d}}s \\&\qquad + \int \limits _{-\infty }^{t}e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} (s+s_{n}) -F_{i}^{1*}(s) \right| {\mathrm{d}}s\\&\quad = \int \limits _{-\infty }^{t}\varPhi _{i}(t,s){\mathrm{d}}s+\int \limits _{-\infty }^{t}\Psi _{i}(t,s){\mathrm{d}}s, \end{aligned}$$

such that $\varPhi _{i}(t,s)=e^{-a_{i*}^{1}(t-s)}|F_{i}^{1*}|_\infty \int \nolimits _s^t |a_{i}^{1*}(u)-a_{i}^{1}(u+s_{n})|{\mathrm{d}}u,$ and $\Psi _{i}(t,s)=e^{-a_{i*}^{1}(t-s)} \left| F_{i}^{1} \left( s+s_{n}\right) -F_{i}^{1*}\left( s\right) \right| .$

By the Lebesgue dominated convergence theorem, we have for all $t\in {\mathbb {R}},$$\lim \nolimits _{n\rightarrow +\infty }(\varGamma _{i} F_{i}^{1} )(t+s_{n})=(\varGamma _{i} ^{1}F_{i}^{1*} )(t).$

The same approach proves that, $\forall \; t\in {\mathbb {R}}$$\lim \nolimits _{n\rightarrow +\infty }(\varGamma _{i}^{1}F_{i}^{1*})(t-s_{n})=(\varGamma _{i} F_{i}^{1})(t),$ so $(\varGamma _{i}F_{i}^{1}) \in AA({\mathbb {R}},{\mathbb {R}}).$

Reasoning in a similar way, we can show that $(\varGamma _{j}G_{j}^{1}) \in AA({\mathbb {R}},{\mathbb {R}}).$ Now, we focus on $(\varGamma _{i}F_{i}^{2}) \in \xi ({\mathbb {R}},{\mathbb {R}}^n,\mu ,\nu ).$

$$\begin{aligned}&\lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | (\varGamma _{i}F_{i}^{2})(t) | d\mu (t)\\&\quad = \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]}\left| \int \limits _{-\infty }^{t} e^{-\int \limits _{s}^{t}a_{i}^{1} (u) {\mathrm{d}}u} F_{i}^{2}(s) {\mathrm{d}}s \right| d\mu (t) \\&\quad \le \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \int \limits _{-\infty }^{t} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s d\mu (t) \\&\quad \le E_{1} + E_{2}, \text{ such } \text{ that }\\&\quad E_{1} = \lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int _{-r}^{t} | e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s) | {\mathrm{d}}s \right) d\mu (t), \\&\quad E_{2} = \lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s\right) d \mu (t).\\ \end{aligned}$$

Let $m= t-s,$ then by Fubini's theorem, we obtain

$$\begin{aligned} E_{1}= & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int _{-r}^{t} e^{-(t-s)a_{i*}^{1}}| F_{i}^{2}(s) | {\mathrm{d}}s \right) d\mu (t) \\\le & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}}| F_{i}^{2}(t-m) |{\mathrm{d}}m \right) d\mu (t) \\\le & {} \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} | F_{i}^{2}(t-m) |d \mu (t) \right) {\mathrm{d}}m \\= & {} \int \limits _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int \limits _{-r-m}^{r-m} | F_{i}^{2}(t) | d\mu _m(t) \right) {\mathrm{d}}m \\\le & {} \int _{0}^{+ \infty } e^{-m a_{i*}^{1}} \left( \lim _{r\longrightarrow \infty } \frac{\nu ([-r-m,r+m] ) }{\nu ([-r,r]}\frac{\beta }{\nu ([-r-m,r+m])} \right. \\&\times \int \limits _{[-r-m,r+m] } | F_{i}^{2}(t) | d\mu (t) ) {\mathrm{d}}m. \end{aligned}$$

Since $\nu$ satisfies (H4) and $F_{i}^{2} \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$, then $E_1=0.$

On the other hand, let, $| F_{i}^{2} |_{\infty }= \sup \nolimits _{t \in {\mathbb {R}}} | F_{i}^{2} (t)|< \infty$ then

$$\begin{aligned} E_{2}= & {} \lim _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} | e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s)| {\mathrm{d}}s\right) d\mu (t) \\= & {} \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int _{[-r,r]} \left( \int \limits _{- \infty }^{-r} e^{-(t-s)a_{i*}^{1}} |F_{i}^{2}(s)| {\mathrm{d}}s \right) d\mu (t) \\\le & {} \lim _{r\longrightarrow \infty }\frac{1}{\nu ([-r,r])} \int \limits _{- \infty }^{- r} e^{s a_{i*}^{1}} | F_{i}^{2}(s)| {\mathrm{d}}s\int \limits _{-r}^{r} e^{- t a_{i*}^{1}} d\mu (t) \\= & {} \lim _{r\longrightarrow \infty } \frac{|F_{i}^{2}|_{\infty }}{ a_{i*}^{1}} e^{- 2r a_{i*}^{1}}.\\&\text{ So }\\&\lim \limits _{r\longrightarrow \infty } \frac{1}{\nu ([-r,r])} \int _{-r}^{r} \left| \int \limits _{-\infty }^{t} e^{-(t-s)a_{i*}^{1}} F_{i}^{2}(s) {\mathrm{d}}s \right| d\mu (t) =0. \end{aligned}$$

Consequently, the function $(\varGamma _{i}F_{i}^{2})\in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu )$.

The same approach proves that, $\forall \; t\in {\mathbb {R}}$, $(\varGamma _{j}G_{j}^{2}) \in \xi ({\mathbb {R}},{\mathbb {R}},\mu ,\nu ).$

Therefore, $\varGamma _{Z}$ maps $PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu )$ into itself. $\square$

Appendix F Proof of Theorem 3

Proof

We have

$$\begin{aligned} \Vert Z_{0}\Vert= & {} \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{1}_{i}(u){\mathrm{d}}u}I_{i}(s){\mathrm{d}}s \right| \right\} ;\right. \\&\left. \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{2}_{j}(u){\mathrm{d}}u}J_{j}(s){\mathrm{d}}s\right| \right\} \right\} \\\le & {} \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{1}_{i}(u){\mathrm{d}}u} \left| I_{i}(s)\right| {\mathrm{d}}s\right\} ,\right. \\&\left. \sup _{t \in {\mathbb {R}}} \max _{1 \le j\le p}\left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t}a^{2}_{j}(u){\mathrm{d}}u} \left| J_{j}(s)\right| {\mathrm{d}}s \right\} \right\} \\\le & {} \max \left\{ \max \limits _{ 1 \le i\le n}\left\{ \frac{I_{i}^{*}}{a_{i*}^{1}}\right\} ;\max \limits _{ 1 \le j\le p}\left\{ \frac{J_{j}^{*}}{a_{j*}^{2}}\right\} \right\} =\varpi , \end{aligned}$$

after,

$$\begin{aligned} \Vert Z\Vert \le \Vert Z-Z_{0}\Vert _{\infty }+\Vert Z_{0}\Vert _{\infty }\le & {} \frac{\lambda }{1-\lambda }\varpi +\varpi \\= & {} \frac{\varpi }{1-\lambda }.\\ \end{aligned}$$

Set $S^{*}=\left\{ Z \in PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu ) ; \Vert Z-Z_{0}\Vert _{\infty }\le \frac{\lambda }{1-\lambda }\varpi \right\} .$ Clearly, $S^{*}$ is a closed convex subset of $PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu ).$ For any $Z \in S^{*}$ by using the estimate just obtained, we obtain

$$\begin{aligned}&\Vert \varGamma _{Z}-Z_{0}\Vert \\&\quad = \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \left| \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t} a_{i}^{1}(u) {\mathrm{d}}u} \left( \sum \limits _{j=1}^{p}b^{1}_{ji}(s) f_{j}^{1}(\psi _{j}(s))\right. \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}c^{1}_{jil}(s) g_{j}^{1}(\psi _{j}(s-\tau ))g^{1}_{l}(\psi _{l}(s-\tau ))\\&\qquad +\sum \limits _{j=1}^{p}d^{1}_{ji}(s) \int \limits _{-\infty }^{s} K_{ji}(s-m) h_{j}^{1}(\psi _{j}(m)){\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(s)\int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\psi _{j}(m)){\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\psi _{l}(m)){\mathrm{d}}m ){\mathrm{d}}s)\right| \right\} ,\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j \le p}\left\{ \left| \int _{-\infty }^{t}e^{-\int _{s}^{t} a_{j}^{2}(u){\mathrm{d}}u}\left( \sum \limits _{i=1}^{n}b^{2}_{ij}(s) f_{i}^{2}(\varphi _{i}(s))\right. \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s)g_{i}^{2}(\varphi _{i}(s-\gamma )g^{2}_{l}(\varphi _{l}(s-\gamma ))\\&\qquad +\sum \limits _{i=1}^{n} d^{2}_{ij}(s)\int \limits _{-\infty }^{s}N_{ij}(s-m) h_{i}^{2}(\varphi _{i}(m)){\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(s) \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) k^{2}_{i}(\varphi _{i}(m)){\mathrm{d}}m \\&\left. \left. \left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) k^{2}_{l}(\varphi _{l}(m)){\mathrm{d}}m){\mathrm{d}}s\right) \right| \right\} \right\} \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \int \limits _{-\infty }^{t}e^{-\int \limits _{s}^{t} a_{i}^{1}(u) {\mathrm{d}}u} \left( \sum \limits _{j=1}^{p}\left| b^{1}_{ji}(s)\right| \left| f_{j}^{1}(\psi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}\left| c^{1}_{jil}(s)\right| \left| g_{j}^{1}(\psi _{j}(s-\tau ) g^{1}_{l}(\psi _{l}(s-\tau ))\right| \\&\qquad +\sum \limits _{j=1}^{p}\left| d^{1}_{ji}(s)\right| \int \limits _{-\infty }^{s} K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}\left| r^{1}_{jil}(s)\right| \int \limits _{-\infty }^{s} P_{jil}(s-m)\left| k^{1}_{j}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s\right) \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j \le p}\left\{ \int _{-\infty }^{t}e^{-\int _{s}^{t} a_{j}^{2}(u){\mathrm{d}}u} \left( \sum \limits _{i=1}^{n}\left| b^{2}_{ij}(s)\right| \left| f_{i}^{2}(\varphi _{i}(s))\right| \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n} \left| c^{2}_{ijl}(s)\right| \left| g_{i}^{2}(\varphi _{i}(s-\gamma ) g^{2}_{l}(\varphi _{l}(s-\gamma ))\right| \\&\qquad +\sum \limits _{i=1}^{n}\left| d^{2}_{ij}(s)\right| \int \limits _{-\infty }^{s}N_{ij}(s-m) \left| h_{i}^{2}(\varphi _{i}(m)\right| {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}\left| r^{2}_{ijl}(s)\right| \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i} (\varphi _{i}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s)\right\} \right\} \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1 \le i\le n} \left\{ \int _{-\infty }^{t}e^{-a^{1}_{i*}(t-s)} \left( \sum \limits _{j=1}^{p}b^{1*}_{ji}\left| f_{j}^{1}(\psi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1*}_{jil}\left| g_{j}^{1}(\psi _{j}(s-\tau )g^{1}_{l}(\psi _{l}(s-\tau ))\right| \\&\qquad +\sum \limits _{j=1}^{p}d^{1*}_{ji} \int \limits _{-\infty }^{s} K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1*}_{jil}\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\psi _{j}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\psi _{l}(m))\right| ){\mathrm{d}}s\right) \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{1\le j\le p}\left\{ \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left( \sum \limits _{i=1}^{n}b^{2*}_{ij}\left| f_{i}^{2}(\varphi _{i}(s))\right| \right. \right. \\&\qquad + \sum \limits _{i=1}^{n}\sum \limits _{i=1}^{n}c^{2*}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )g^{2}_{l}(\varphi _{l}(s-\gamma )) \right| \\&\qquad +\sum \limits _{i=1}^{n} d^{2*}_{ij}\int \limits _{-\infty }^{s}N_{ij} (s-m) \left| h_{i}^{2}(\varphi _{i}(m)\right| {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2*}_{ijl} \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m)\left| k^{2}_{i}(\varphi _{i}(m))\right| {\mathrm{d}}m\\&\left. \left. \qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m){\mathrm{d}}s)\right\} \right\} \\ \end{aligned}$$

$$\begin{aligned}&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}}\max _{1 \le i\le n} \left\{ \int _{-\infty }^{t}e^{-a^{1}_{i*}(t-s)} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}\left( b^{1*}_{ji}l_{j}+c^{1*}_{jil}m_{j}e_{l}+d^{1*}_{ij}d_{j} \right. \right. \right. \\&\left. \left. \qquad +\,r^{1*}_{jil}\alpha _{j}w_{l}\right) {\mathrm{d}}s\right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{1 \le j\le p} \left\{ \int _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}\left( b^{2*}_{ij}L_{i}+c^{2*}_{ijl}M_{i}E_{l}\right. \right. \\&\left. \left. \left. \qquad +\,d^{2*}_{ij}D_{i}+r^{2*}_{ijl}\beta _{i}W_{l}\right) {\mathrm{d}}s\right\} \right\} \Vert Z\Vert \\&\quad = \max \left\{ \max _{1\le i\le n}\left\{ \frac{1}{a^{1}_{i*}} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}(b^{1*}_{ji}l_{j} +c^{1*}_{jil}m_{j}e_{l}+d^{1*}_{ji}d_{j}+r^{1*}_{jil}\alpha _{j}w_{l}) \right\} ;\right. \\ &\left.\qquad \max_{ 1 \leq j\leq p} \left\{\frac{1}{a^{2}_{j*}} \sum\limits_{i=1}^{n}\sum\limits_{l=1}^{n}( b^{2*}_{ij}L_{i} +c^{2*}_{ijl}M_{i}E_{l}+d^{2*}_{ij}D_{i}+r^{2*}_{ijl}\beta_{i}W_{l})\right\}\right\}\| Z\|\\ &\quad= \lambda \| Z\|\leq \frac{\lambda}{1-\lambda} \varpi, \end{aligned}$$

then, $\varGamma _{Z}\in S^{*}.$ The mapping $\varGamma$ is a self-mapping from $S^*$ to $S^*.$

In view of (H1), for any $Z_{1},Z_{2} \in S^{*},\;$ such that $Z_{1}=(\varphi _1,...,\varphi _n,\psi _1,...,\psi _p)^{T}$ and $Z_{2}=(\phi _1,...,\phi _n,\chi _1,...,\chi _p)^{T},$ we have

$$\begin{aligned}&\Vert \varGamma _{Z_{1}}-\varGamma _{Z_{2}}\Vert \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{1\le i\le n}\left\{ \left| \int \limits _{-\infty }^{t}e^{-a^{1}_{i*} (t-s)}\left[ \left( \sum \limits _{j=1}^{p} b^{1}_{ji}(s) \left( f_{j}^{1}(\psi _{j}(s))-f_{j}^{1}(\chi _{j}(s))\right) \right. \right. \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau ) -g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau ))\right. \\&\qquad +\sum \limits _{j=1}^{p}d^{1}_{ji}(s) \int \limits _{-\infty }^{s}K_{ji}(s-m) \left( (h_{j}^{1}(\psi _{j}(m))-h_{j}^{1}(\chi _{j}(m))\right) {\mathrm{d}}m\\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(s) \left( \int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\psi _{j}(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\psi _{l}(m)){\mathrm{d}}m\right. \\&\left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) k^{1}_{j}(\chi _{j}(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) k^{1}_{l}(\chi _{l}(m)){\mathrm{d}}m \right) \right] {\mathrm{d}}s\right| \right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \left| \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left[ \sum \limits _{i=1}^{n} b^{2}_{ij}(s) \left( f_{i}^{2}(\varphi _{i}(s))-f_{i}^{2}(\phi _{i}(s))\right) \right. \right. \right. \\&\qquad +\sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma ) g_{l}^{2}(\varphi _{l}(s-\gamma ))-g_{i}^{2} (\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma ) \right) \\&\qquad +\sum \limits _{i=1}^{n} d^{2}_{ij}(s)\int \limits _{-\infty }^{s} N_{ij}(s-m)\left( h_{i}^{2}(\varphi _{i}(m))-h_{i}^{2}(\phi _{i}(m)) \right) {\mathrm{d}}m\\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(s) \left( \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m)k^{2}_{i}(\varphi _{i}(m)){\mathrm{d}}m\int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) k^{2}_{l}(\varphi _{i}(m)){\mathrm{d}}m \right. \\&\left. \left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) k^{2}_{i}(\phi _{i}(m)){\mathrm{d}}m\int \limits _{-\infty }^{s}\tilde{Q}_{ijl} (s-m) k^{2}_{l}(\phi _{i}(m)){\mathrm{d}}m\right) \right] {\mathrm{d}}s\right| \right\} \right\} . \end{aligned}$$

Let

$$\begin{aligned} \Lambda (s,\psi ,\chi )= & {} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau )\right. \\&\left. -\,g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau )\right) \\ \left| \Lambda (s,\psi ,\chi )\right|= & {} \left| \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil}(s) \left( g_{j}^{1}(\psi _{j}(s-\tau )g_{l}^{1}(\psi _{l}(s-\tau )\right. \right. \\&\left. -\,g_{j}^{1}(\chi _{j}(s-\tau )g_{l}^{1}(\chi _{l}(s-\tau ))\right| \\\le & {} \sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1*}_{jil} \left| g_{j}^{1}(\psi _{j}(s-\tau )\right| \left| g_{l}^{1}(\psi _{l}(s-\tau )-g_{l}^{1}(\chi _{l}(s-\tau ) \right| \\&+ \left| g_{j}^{1}(\psi _{j}(s-\tau )-g_{j}^{1}(\chi _{j}(s-\tau ) \right| \left| g_{l}^{1}(\chi _{l}(s-\tau )\right| , \end{aligned}$$

and

$$\begin{aligned} \varOmega (s,\varphi ,\phi )= & {} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma ) g_{l}^{2}(\varphi _{l}(s-\gamma )\right. \\&\left. -\,g_{i}^{2}(\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma )\right) \\ \left| \varOmega (s,\varphi ,\phi )\right|= & {} \left| \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl}(s) \left( g_{i}^{2}(\varphi _{i}(s-\gamma )g_{l}^{2}(\varphi _{l}(s-\gamma )\right. \right. \\&\left. \left. -\,g_{i}^{2}(\phi _{i}(s-\gamma )g_{l}^{2}(\phi _{l}(s-\gamma )\right) \right| \\\le & {} \sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2*}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )\right| \\&\times \left| g_{l}^{2}(\varphi _{l}(s-\gamma )-g_{l}^{2}(\phi _{l} (s-\gamma ) \right| \\&+ \left| g_{i}^{2}(\varphi _{i}(s-\gamma )-g_{i}^{2} (\phi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\phi _{l}(s-\gamma )\right| .\\ \end{aligned}$$

Therefore,

$$\begin{aligned}&\Vert \varGamma _{Z_{1}}-\varGamma _{Z_{2}}\Vert \\&\quad \le \max \left\{ \sup _{t \in {\mathbb {R}}} \max _{ 1 \le i\le n}\left\{ \int \limits _{-\infty }^{t}e^{-a^{1}_{i*} (t-s)}\left[ \sum \limits _{j=1}^{p}\left| b^{1}_{ji}(s)\right| \left| f_{j}^{1}(\psi _{j}(s))-f_{j}^{1}(\chi _{j}(s))\right| \right. \right. \right. \\&\qquad +\sum \limits _{j=1}^{p}\sum \limits _{l=1}^{p}c^{1}_{jil} \left| g_{j}^{1}(\psi _{j}(s-\tau )\right| \left| g_{l}^{1}(\psi _{l}(s-\tau ) -g_{l}^{1}(\chi _{l}(s-\tau ) \right| \\&\qquad + \left| g_{j}^{1}(\psi _{j}(s-\tau )-g_{j}^{1}(\chi _{j}(s-\tau ) \right| \left| g_{l}^{1}(\chi _{l}(s-\tau )\right| \\&\qquad +\sum \limits _{j=1}^{p}\left| d^{1}_{ji}(s) \right| \int \limits _{-\infty }^{s}K_{ji}(s-m) \left| h_{j}^{1}(\psi _{j}(m))-h_{j}^{1}(\chi _{j}(m))\right| {\mathrm{d}}m \\&\qquad + \sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p} \left| r^{1}_{jil}(s)\right| \left( \int \limits _{-\infty }^{s} P_{jil}(s-m)\left| k^{1}_{j}(\psi _{j}(m)) \right| {\mathrm{d}}m \right. \\&\qquad \times \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m\\&\qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m \\&\qquad +\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m)\left| k^{1}_{l}(\psi _{l}(m))\right| {\mathrm{d}}m \\&\left. \left. \left. \qquad -\int \limits _{-\infty }^{s} P_{jil}(s-m) \left| k^{1}_{j}(\chi _{j}(m))\right| {\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(s-m) \left| k^{1}_{l}(\chi _{l}(m))\right| {\mathrm{d}}m\right) \right] {\mathrm{d}}s\right\} ;\\&\qquad \sup _{t \in {\mathbb {R}}} \max _{ 1 \le j\le p} \left\{ \int \limits _{-\infty }^{t}e^{-a^{2}_{j*}(t-s)} \left[ \sum \limits _{i=1}^{n}\left| b^{2}_{ij}(s)\right| \left| f_{i}^{2}(\varphi _{i}(s))-f_{i}(\phi _{i}(s))\right| \right. \right. \\&\qquad +\sum \limits _{i=1}^{n}\sum \limits _{l=1}^{n}c^{2}_{ijl} \left| g_{i}^{2}(\varphi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\varphi _{l}(s-\gamma )-g_{l}^{2} (\phi _{l}(s-\gamma ) \right| \\&\qquad + \left| g_{i}^{2}(\varphi _{i}(s-\gamma )-g_{i}^{2} (\phi _{i}(s-\gamma )\right| \left| g_{l}^{2}(\phi _{l}(s-\gamma )\right| \\&\qquad +\sum \limits _{i=1}^{n}\left| d^{2}_{ij}(s)\right| \int \limits _{-\infty }^{s} N_{ij}(s-m)\left| h_{i}^{2}(\varphi _{i}(m))-h_{i}^{2}(\phi _{i}(m))\right| {\mathrm{d}}m \\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n} \left| r^{2}_{ijl}(s)\right| \left( \int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\varphi _{i}(m))\right| {\mathrm{d}}m\right. \\&\qquad \times \int \limits _{-\infty }^{s}\tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\qquad +\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\varphi _{l}(m))\right| {\mathrm{d}}m\\&\left. \left. \left. \left. \qquad -\int \limits _{-\infty }^{s}\tilde{P}_{ijl}(s-m) \left| k^{2}_{i}(\phi _{i}(m))\right| {\mathrm{d}}m\int \limits _{-\infty }^{s} \tilde{Q}_{ijl}(s-m) \left| k^{2}_{l}(\phi _{l}(m))\right| {\mathrm{d}}m\right) \right] {\mathrm{d}}s \right\} \right\} \\&\quad \le \max \left\{ \max _{1 \le i \le n}\frac{1}{a_{i*}^{1}}\sum _{j=1}^{p}\sum _{l=1}^{p}( b_{ji}^{1*}l_{j} +c_{jil}^{1*}(e_{j}m_{l}+m_{j}e_{l})+ d_{ji}^{1*}d_{j}\right. \\&\qquad + r_{jil}^{1*}(\alpha _jw_l +w_j \alpha _l ); \max _{1 \le j \le p}\frac{1}{a_{j*}^{2}}\sum _{i=1}^{n}\sum _{l=1}^{n}(b_{ij}^{2*}L_{i} +c_{ijl}^{2*}(E_{i}M_{l}+M_{i}E_{l})\\&\left. \qquad + d_{ij}^{2*}D_{i} +r_{ijl}^{2*}( \beta _i W_l+W_i\beta _l)\right\} \Vert \ Z_{1}-Z_{2}\Vert \le \bar{\lambda }\Vert \ Z_{1}-Z_{2}\Vert , \end{aligned}$$

which prove that $\varGamma$ is a contraction mapping. Then, by the Banach fixed-point theorem, $\varGamma$ has a unique fixed point which corresponds to the solution of (2) in $S^{*} \subset PAA({\mathbb {R}},{\mathbb {R}}^{n+p},\mu ,\nu )$. $\square$

Appendix G Proof of Theorem 4

Proof

Suppose that $Z(t)=(x_{1}(t),\ldots ,x_{n}(t),y_{1}(t),\ldots ,y_{p}(t))^{T},$ is an arbitrary solution of system (2) with initial value $\phi (t)=( \varphi _{1}(t) ,\ldots ,\varphi _{n}(t),\psi _{1}(t) ,\ldots ,\psi _{p}(t))^{T}.$

It follows from Theorem 3 that system (2) has one and only one ($\mu ,\nu )$-pseudo-almost automorphic solution $Z^{*}(t)=(x_{1}^{*}(t),\ldots ,x_{n}^{*}(t),y_{1}^{*}(t),\ldots ,y_{p}^{*}(t))^{T} \in S^{*},$ with initial value $\phi ^{*}(t)=( \varphi _{1}^{*}(t) ,\ldots ,\varphi _{n}^{*}(t),\psi _{1}^{*}(t) ,\ldots ,\psi _{p}^{*}(t))^{T}.$

Let $u_{i}(t)=x_{i}(t)-x_{i}^{*}(t),$ and $v_{j}(t)=y_{j}(t)-y_{j}^{*}(t),\;i=1 \ldots n, j=1 \ldots p.$ We can write that

$$\begin{aligned}&u'_{i}(t)+a_{i}^{1}(t) u_{i}(t) \nonumber \\&\quad = \sum _{j=1}^{p} b_{ji}^{1}(t) \left[ f_{j}^{1} (v_{j}(t)+y_{j}^{*}(t))-f_{j}^{1} (y_{j}^{*}(t))\right] \nonumber \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1}(t) \left[ g_{j}^{1}(v_{j}(t-\tau )+ y_{j}^{*}(t-\tau )) \right. \nonumber \\&\left. \qquad \times g_{l}^{1}(v_{l}(t-\tau )+ y_{l}^{*}(t-\tau )) -g_{j}^{1}(y_{j}^{*}(t-\tau ) g_{l}^{1}(y_{l}^{*}(t-\tau )) \right] \nonumber \\&\qquad +\sum _{j=1}^{n} d_{ji}^{1}(t)\int _{-\infty }^{t} K_{ji}(t-m) \nonumber \\&\qquad \times \left[ h_{j}^{1}(v_{j}(m)+y_{j}^{*}(m))-h_{j}^{1} (y_{j}^{*}(m))\right] {\mathrm{d}}m \nonumber \\&\qquad +\sum \limits _{j=1}^{p} \sum \limits _{l=1}^{p}r^{1}_{jil}(t) \left[ \int \limits _{-\infty }^{t} P_{jil}(t-m) k^{1}_{j}(v_{j}(m)+y_{j}^*(m)){\mathrm{d}}m \right. \nonumber \\&\qquad \times \int \limits _{-\infty }^{t} Q_{jil}(t-m) k^{1}_{l}(v_{l} (m)+y_{l}^*(m)){\mathrm{d}}m \nonumber \\&\left. \qquad -\int \limits _{-\infty }^{t} P_{jil}(t-m) k^{1}_{j}(y_{j}^*(m)){\mathrm{d}}m \int \limits _{-\infty }^{t} Q_{jil}(t-m) k^{1}_{l}(y_{l}^*(m)){\mathrm{d}}m \right] , \end{aligned}$$

(4)

$$\begin{aligned}&v'_{j}(t)+a_{j}^{2}(t) v_{j}(t)\nonumber \\&\quad = \sum _{i=1}^{p} b_{ij}^{2}(t) \left[ f_{i}^{2} (u_{i}(t)+x_{i}^{*}(t))-f_{i}^{2} (x_{i}^{*}(t))\right] \nonumber \\&\qquad + \sum _{i=1}^{n} \sum _{l=1}^{n} c_{ijl}^{2}(t) \left[ g_{i}^{2}(u_{i}(t-\gamma )+x_{i}^{*}(t-\gamma )) \right. \nonumber \\&\left. \qquad \times g_{l}^{2}(u_{l}(t-\gamma )+ x_{l}^{*}(t-\gamma ))- g_{i}^{2}(x_{i}^{*}(t-\gamma ) g_{l}^{2}(x_{l}^{*}(t-\gamma ))\right] \nonumber \\&\qquad +\sum _{i=1}^{n} d_{ij}^{2}(t)\int _{-\infty }^{t} N_{ij}(t-m)\nonumber \\&\qquad \times \left[ h_{i}^{2}(u_{i}(m)+x_{i}^{*}(m))-h_{i}^{2} (x_{i}^{*}(m))\right] {\mathrm{d}}m\nonumber \\&\qquad +\sum \limits _{i=1}^{n} \sum \limits _{l=1}^{n}r^{2}_{ijl}(t) \left[ \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(t-m) k^{2}_{i}(u_{i}(m)+x_{i}^*(m)){\mathrm{d}}m \right. \nonumber \\&\qquad \times \int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(t-m) k^{2}_{l}(u_{l}(m)-x_{l}^*(m)){\mathrm{d}}m \nonumber \\&\left. \qquad -\int \limits _{-\infty }^{t} \tilde{P}_{ijl}(t-m) k^{2}_{i} (x_{i}^*(m)){\mathrm{d}}m\int \limits _{-\infty }^{t} \tilde{Q}_{ijl} (t-m) k^{2}_{l}(x_{l}^*(m)){\mathrm{d}}m\right] \end{aligned}$$

(5)

Let $F_{i},G_{j}$ be defined by

$$\begin{aligned} F_{i}(w)= & {} a_{i*}^{1}-w- \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}-\sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j} + e_{j} m_{l}) e^{w \tau }\right. \\&-d_{ji}^{1*} d_{j} \int _{0}^{+\infty } K_{ji}(m) e^{w m} {\mathrm{d}}m- \sum \limits _{l=1}^{p}r^{1*}_{jil}\left( \alpha _j w_l\int \limits _{-\infty }^{t} P_{jil}(m)e^{wm}{\mathrm{d}}m\right. \\&\left. + w_j\alpha _l \int \limits _{-\infty }^{t} Q_{jil}(m) e^{wm}{\mathrm{d}}m \right) ,\\ \end{aligned}$$

and

$$\begin{aligned} G_{j}(w)= & {} a_{j*}^{2}-w- \sum \limits _{i=1}^{n} \left( b_{ij}^{2*} L_{i}-\sum \limits _{l=1}^{n}c_{ijl}^{2*} (E_{i} M_{l}+ E_{l} M_{i}) e^{w\gamma }\right. \\&-d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{w m} {\mathrm{d}}m- \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{wm}{\mathrm{d}}m \right. \\&\left. +\,W_j\beta _l\int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{wm}{\mathrm{d}}m \right) , \end{aligned}$$

for $i=1,\ldots ,n,$$j=1,\ldots ,p,$$w\in [0,+\infty [,$ by (H2) and $\bar{\lambda } < 1$, we obtain that

$$\begin{aligned} \left\{ \begin{array}{lll} F_{i}(0)&{}=&{}a_{i*}^{1}- \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}+ \sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j}+ e_{j} m_{l})\right. \\ &{}&{}+d_{ji}^{1*} d_{j}+ \sum \limits _{l=1}^{p}r^{1*}_{jil}(\alpha _j w_l+w_j\alpha _l))> 0, \\ G_{j}(0)&{}=&{}a_{j*}^{2}- \sum \limits _{i=1}^{n}\left( b_{ij}^{2*} L_{i}+\sum \limits _{l=1}^{n} c_{ijl}^{2*}(E_{i} M_{l}+ E_{l} M_{i})\right. \\ &{}&{}+d_{ij}^{2*} D_{i}+\sum \limits _{l=1}^{n}r^{2*}_{ijl}(\beta _i W_l+W_j\beta _l) )> 0. \end{array}\right. \end{aligned}$$

Both, $F_{i}(.)$ and $G_{j}(.)$ are continuous on $[0,\infty [$ such that $F_{i}(w) \longrightarrow - \infty$ when $w\longmapsto +\infty$, there exist $\varepsilon _{i}^{*} > 0$ such that $F_{i}(\varepsilon _{i}^{*}) = 0$ and $F_{i}(\varepsilon _{i}) >0$ for $\varepsilon _{i} \in (0,\varepsilon _{i}^{*})$, also $G_{j}(w) \longrightarrow - \infty$ when $w \longmapsto +\infty$, $\exists \zeta _{j}^{*} > 0$ such that $G_{j}(\zeta _{j}^{*}) = 0$ and $G_{j}(\zeta _{j}) > 0$ for $\zeta _{j} \in (0,\zeta _{j}^{*})$.

By choosing $\eta = \min \{ \varepsilon _{1}^{*},\ldots ,\varepsilon _{n}^{*},\zeta _{1}^{*},\ldots , \zeta _{p}^{*}\}$, we obtain

$$\begin{aligned} \left\{ \begin{array}{cc} F_{i}(\eta ) \ge 0 ,\; i=1,\ldots ,n,\\ G_{j}(\eta ) \ge 0 ,\, j=1,\ldots ,p. \end{array} \right. \end{aligned}$$

So, we can choose a positive constant $\lambda$ such that $0<\lambda < \min \{ \eta , a_{1*}^{1},\ldots ,a_{n*}^{1},a_{1*}^{2},\ldots ,a_{p*}^{2},\lambda _{0} \}$ such that $F_{i}(\lambda )$ and $G_{j}(\lambda )$ are nonnegative ,which implies that, for $i=1,\ldots ,n,$$j=1,\ldots ,p,$

$$\begin{aligned} \left\{ \begin{array}{lll} &{}\frac{1}{a_{i*}^{1}-\lambda } \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j} +\sum \limits _{l=1}^{p}c_{jil}^{1*}(e_{l} m_{j}+ e_{j} m_{l}) e^{\lambda \tau }\right. \\ &{}+d_{ji}^{1*} d_{j}\int _{0}^{+\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\ &{}+ \sum \limits _{l=1}^{p}r^{1*}_{jil}\left( \alpha _j w_l \int \nolimits _{-\infty }^{t} P_{jil}(m)e^{wm}{\mathrm{d}}m \right. \\ &{}\left. \left. + w_j\alpha _l\int \nolimits _{-\infty }^{t} Q_{jil}(m) e^{wm}{\mathrm{d}}m \right) \right)< 1, \\ &{}\frac{1}{a_{j*}^{2}-\lambda } \sum \limits _{i=1}^{n}\left( b_{ij}^{2*} L_{i} +\sum \limits _{l=1}^{n}c_{ijl}^{2*}(E_{i} M_{l}+ E_{l} M_{i})e^{\lambda \gamma } \right. \\ &{}+ d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \\ &{}+ \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \nolimits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{wm}{\mathrm{d}}m \right. \\ &{}+W_j\beta _l \int \nolimits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{wm}{\mathrm{d}}m) < 1. \end{array}\right. \end{aligned}$$

Let

$$\begin{aligned} \left\{ \begin{array}{lll} N= \max \nolimits _{1 \le i\le n} \left( \frac{a_{i*}^{1}}{\sum \nolimits _{j=1}^{p}(b_{ji}^{1*} l_{j}+\sum \nolimits _{l=1}^{p}c_{jil}^{1*} (e_{l} m_{j}+ e_{j} m_{l}) +d_{ji}^{1*} d_{j}+\sum \nolimits _{l=1}^{p}r_{jil}^{1*} (\alpha _{j}w_{l} + w_{j} \alpha _{l}))}\right) ,\\ \bar{N}=\max \nolimits _{1 \le j\le p} \left( \frac{a_{j*}^{2}}{\sum \nolimits _{i=1}^{n}(b_{ij}^{2*} L_{i}+\sum \nolimits _{l=1}^{n}c_{ijl}^{2*} (E_{i} M_{l}+ E_{l} M_{i})+d_{ij}^{2*} D_{i}+\sum \nolimits _{l=1}^{n}r_{ijl}^{2*} ( \beta _{l}W_{i}+ W_{l} \beta _{i}))}\right) . \end{array}\right. \end{aligned}$$

Clearly $N> 1$, $\bar{N}> 1,$ and we take $M=\max \{N ,\bar{N}\}> 1.$

$$\begin{aligned} \left\{ \begin{array}{l} \left(\frac{1}{M} - \frac{1}{a_{i*}^{1}-\lambda } \sum \limits _{j=1}^{p}\left( b_{ji}^{1*} l_{j}+\sum \limits _{l=1}^{p}c_{jil}^{1*} (e_{l} m_{j}+ e_{j} m_{l}) e^{\lambda \tau }\right. \right. \\ + d_{ji}^{1*} d_{j} \int _{0}^{+\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m +\sum \limits _{l=1}^{p}r^{1*}_{jil}(\alpha _j w_l \int \limits _{-\infty }^{t} P_{jil}(m)e^{\lambda m}{\mathrm{d}}m\\ \left. \left. \left. + w_j\alpha _l\int \limits _{-\infty }^{t} Q_{jil}(m) e^{\lambda m}{\mathrm{d}}m \right) \right) \right) \le 0\\ \left( \frac{1}{M} - \frac{1}{a_{j*}^{2}-\lambda } \sum \limits _{i=1}^{n} \left( b_{ij}^{2*} L_{i} +\sum \limits _{l=1}^{n}c_{ijl}^{*2} (E_{i} M_{l}+ E_{l} M_{i}) e^{v \gamma }\right. \right. \\ +d_{ij}^{2*} D_{i} \int _{0}^{+\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m+ \sum \limits _{l=1}^{n}r^{2*}_{ijl}\left( \beta _i W_l \int \limits _{-\infty }^{t} \tilde{P}_{ijl}(m)e^{\lambda m}{\mathrm{d}}m \right. \\ \left. \left. \left. +W_j\beta _l\int \limits _{-\infty }^{t} \tilde{Q}_{ijl}(m) e^{\lambda m}{\mathrm{d}}m\right) \right) \right) \le 0 \end{array}\right. \end{aligned}$$

where $0<\lambda < \min \{\eta , a_{1*}^{1},\ldots ,a_{n*}^{1},a_{1*}^{2},\ldots ,a_{p*}^{2}, \lambda _{0} \}$.

Besides, $\;\forall \; t \in (-\infty ,0]$

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\\ \parallel v\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel , \end{array}\right. \end{aligned}$$

(6)

We claim that, $\text{ for }\; t>0,$

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\\ \parallel v\parallel &{}\le &{}Me^{-\lambda t}\parallel \phi -\phi ^* \parallel ,\; \end{array}\right. \end{aligned}$$

(7)

If (7) is false, then there must be some $t_{1} > 0$ some $i \in \{ 1,\ldots ,n\},$$j \in \{ 1,\ldots ,p\},$ for any $p > 1$ and some k such that

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u(t_1) \parallel &{}=&{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}},\\ \parallel v(t_1) \parallel &{}=&{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}}, \end{array}\right. \end{aligned}$$

(8)

and

$$\begin{aligned} \left\{ \begin{array}{lll} \parallel u(t) \parallel &{}\le &{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t}, \; \forall \; t \in (-\infty , t_{1}]\\ \parallel v(t) \parallel &{}\le &{} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t}, \; \forall \; t \in (-\infty , t_{1}]. \end{array}\right. \end{aligned}$$

(9)

Now,we have the following:

$$\begin{aligned}&\mid u_{i}(t_{1})\mid \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}\left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} \parallel v_{j}(s)\parallel \right. \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) \parallel v_{j}(s-\tau )\parallel \\&\qquad + \sum _{j=1}^{p} d_{ji}^{1*} \int _{-\infty }^{s} K_{ji}(s-m) d_{j} \parallel v_{j}(m)\parallel {\mathrm{d}}m \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l} \int \limits _{-\infty }^{s} P_{jil}(s-m) \parallel v_{j}(m)\parallel {\mathrm{d}}m \right. \\&\left. \qquad + w_{j}\alpha _{l} \int \limits _{-\infty }^{s} Q_{jil}(s-m) \parallel v_{j}(m)\parallel {\mathrm{d}}m\right) {\mathrm{d}}s \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}\\&\qquad \times \left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} p M \parallel \phi -\phi ^*\parallel e^{-\lambda s} \right. \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-\tau )} \\&\qquad +\sum _{j=1}^{p} d_{ji}^{1*}d_{j} \int _{0}^{\infty } K_{ji}(m) p M \parallel \phi -\phi ^*\parallel e^{- \lambda (s-m)} {\mathrm{d}}m \\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l} \int \limits _{0}^{\infty } P_{jil}(m) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-m)} {\mathrm{d}}m\right. \\&\left. \qquad +\, w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) p M \parallel \phi -\phi ^*\parallel e^{-\lambda (s-m)}) {\mathrm{d}}m\right) {\mathrm{d}}s \\&\quad \le \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{i*}^{1}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{i*}^{1}}p M \parallel \phi -\phi ^*\parallel e^{-\lambda s}\\&\qquad \times \left( \sum _{j=1}^{p} b_{ji}^{1*} l_{j} + \sum _{j=1}^{p}\sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) e^{\lambda \tau }\right. \\&\qquad + \sum _{j=1}^{p} d_{ji}^{1*}d_{j} \int _{0}^{\infty } K_{ji}(m) e^{ \lambda m} {\mathrm{d}}m\\&\qquad + \sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right. \\&\left. \qquad + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m \right) {\mathrm{d}}s \\&\quad \le p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }[e^{(\lambda - a_{i*}^{1})t_{1}}\left( \frac{1}{M}-\frac{1}{a_{i*}^{1}-\lambda }\right. \\&\qquad \times \sum _{j=1}^{p}\left( b_{ji}^{1*}l_{j} + \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j})e^{\lambda \tau }\right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m \right. \\&\left. \qquad + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m)\right) \\&\qquad +\left( \frac{1}{a_{i*}^{1}-\lambda }\left( \sum _{j=1}^{p}( b_{ji}^{1*} l_{j} + \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j})e^{\lambda \tau }\right. \right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\left. \qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m+ w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right) \right) \\&\quad \le p M\parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} } \left( \frac{1 }{a_{i*}^{1} - \lambda } \sum _{j=1}^{p}\left( b_{ji}^{1*} l_{j} \sum _{l=1}^{p} c_{jil}^{1*} (m_{j}e_{l}+m_{l}e_{j}) e^{\lambda \tau }\right. \right. \\&\qquad + d_{ji}^{1*} d_{j} \int _{0}^{\infty } K_{ji}(m) e^{\lambda m} {\mathrm{d}}m \\&\left. \left. \qquad +\sum _{j=1}^{p}\sum _{l=1}^{p} r_{jil}^{1*} \left( \alpha _{j}w_{l}\int \limits _{0}^{\infty } P_{jil}(m) e^{\lambda m} {\mathrm{d}}m + w_{j}\alpha _{j} \int \limits _{0}^{\infty } Q_{jil}(m) e^{\lambda m} {\mathrm{d}}m\right) \right) \right. \\&\quad < p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1}}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \mid v_{j}(t_{1})\mid\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}} \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} \parallel u_{i}(s)\parallel \right. \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) \parallel u_{i}(s-\gamma )\parallel \\&+ \sum _{i=1}^{n} d_{ij}^{2*} \int _{-\infty }^{s} N_{ij}(s-m) D_{i} \parallel u_{i}(m)\parallel {\mathrm{d}}m \\&+\sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) \parallel u_{i}(s)\parallel {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) \parallel u_{i}(s)\parallel {\mathrm{d}}m\right) {\mathrm{d}}s\\\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}}\left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} p M \right. \\&\times \parallel \phi -\phi ^* \parallel e^{-\lambda s} + \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) q M \\&\times \parallel \phi -\phi ^*\parallel e^{-\lambda (s-\gamma )}\\&+ \sum _{i=1}^{n} d_{ij}^{2*} \int _{0}^{\infty } N_{ij}(m) D_{i} p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m\\&+\sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m \right) {\mathrm{d}}s\\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) p M \parallel \phi -\phi ^* \parallel e^{- \lambda (s-m)} {\mathrm{d}}m \right) {\mathrm{d}}s\\\le & {} \parallel \phi -\phi ^* \parallel e^{-t_{1} a_{j*}^{2}} + \int _{0}^{t_{1}} e^{-(t_{1}-s) a_{j*}^{2}}p M \parallel \phi -\phi ^* \parallel e^{\lambda s}\\&\times \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i} +\sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma } \right. \\&+\sum _{i=1}^{n} d_{ij}^{2*} \int _{0}^{\infty } N_{ij}(m) D_{i} e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m)e^{-\lambda m} {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right) {\mathrm{d}}s\\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }\left[ \frac{e^{(\lambda - a_{j*}^{2}) t_{1} }}{p M} + \frac{1}{a_{j*}^{2} - \lambda }\right. \\&\times \left( \sum _{i=1}^{n} b_{ij}^{2*} L_{i}+ \sum _{i=1}^{n}\sum _{l=1}^{n} c_{ijl}^{2*}(M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma }\right. \\&+ \sum _{i=1}^{n} d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m\right) \left( 1- e^{(\lambda - a_{j*}^{2}) t_{1} }\right) \right] \\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }[e^{(\lambda - a_{j*}^{2})t_{1}}\left( \frac{1}{M}-\frac{1}{a_{j*}^{2}-\lambda }\right. \\&\times \sum _{i=1}^{n}\left( b_{ij}^{2*}L_{i} + \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i})e^{\lambda \gamma }+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \right. \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m\right) \right) \\&+\left( \frac{1}{a_{j*}^{2}-\lambda }\left( \sum _{i=1}^{n}\left( b_{ij}^{1*} L_{i} + \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i})e^{\lambda \gamma }\right. \right. \right. \\&+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m\\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. \left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right) \right) \\\le & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} } \left( \frac{1 }{a_{j*}^{2} - \lambda } \sum _{i=1}^{n}( b_{ij}^{2*} L_{i}\right. \\&+ \sum _{l=1}^{n} c_{ijl}^{2*} (M_{i}E_{l}+M_{l}E_{i}) e^{\lambda \gamma }\\&+ d_{ij}^{2*} D_{i} \int _{0}^{\infty } N_{ij}(m) e^{\lambda m} {\mathrm{d}}m \\&+ \sum _{i=1}^{n}\sum _{l=1}^{n} r_{ijl}^{2*} \left( \beta _{i}W_{l}\int \limits _{0}^{\infty } \tilde{P}_{ijl}(m) e^{- \lambda m} {\mathrm{d}}m \right. \\&\left. + W_{j}\beta _{j} \int \limits _{0}^{\infty } \tilde{Q}_{ijl}(m) e^{-\lambda m} {\mathrm{d}}m \right) \\< & {} p M \parallel \phi -\phi ^* \parallel e^{-\lambda t_{1} }. \end{aligned}$$

which contradicts 8, then 7 holds.

Letting $p \longrightarrow 1,$ then 7 holds.

Hence, the $(\mu ,\nu )$-pseudo-almost automorphic solution $Z(t)=(x(t),y(t))^T$ of system (2) is globally exponentially stable. $\square$