In this section, we will study the existence and global exponential stability of weighted pseudo-almost periodic solutions of system (2).
Let
$$\begin{aligned} \mathbb{B}={}& \bigl\{ \varphi =\bigl(\varphi _{1}^{R}, \ldots , \varphi _{n} ^{R}, \varphi _{1}^{I}, \ldots , \varphi _{n}^{I}, \varphi _{1}^{J}, \ldots , \varphi _{n}^{J},\varphi _{1}^{K}, \ldots , \\ & \varphi _{n}^{K}\bigr)^{T}:= (\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})^{T} \in \mathit{PAP}\bigl(\mathbb{R},\mathbb{R}^{4n},\nu \bigr) \bigr\} \end{aligned}$$
with the norm \(\|\varphi \|_{\mathbb{B}}=\max_{p\in S} \{ \max_{l\in T} \{\sup_{t\in \mathbb{R}}|\varphi _{p} ^{l}(t)| \} \}\), then \(\mathbb{B}\) is a Banach space.
Let
$$\begin{aligned} \varphi ^{0}(t) ={}& \bigl(\bigl(\varphi ^{0} \bigr)_{1}^{R}(t),\ldots ,\bigl(\varphi ^{0} \bigr)_{n} ^{R}(t),\bigl(\varphi ^{0} \bigr)_{1}^{I}(t),\ldots , \bigl(\varphi ^{0}\bigr)_{n}^{I}(t), \\ &\bigl(\varphi ^{0}\bigr)_{1}^{J}(t), \ldots ,\bigl(\varphi ^{0}\bigr)_{n}^{J}(t), \bigl(\varphi ^{0}\bigr)_{1}^{K}(t),\ldots , \bigl(\varphi ^{0}\bigr)_{n}^{K}(t) \bigr)^{T}, \end{aligned} $$
where \((\varphi ^{0})_{p}^{l}(t)=\int _{-\infty }^{t}e^{-\int _{s}^{t}c _{p}(u)\,\mathrm{d}u}J_{p}^{l}(s)\,\mathrm{d}s\), \(p\in S\), \(l\in T\) and κ is a constant satisfying \(\kappa \geq \|\varphi ^{0}\|_{ \mathbb{B}}\).
Lemma 4
Fix
\(\nu \in \mathbb{W}_{\infty }^{\mathrm{Inv}}\). Suppose that assumptions
\((H_{1})\)
and
\((H_{2})\)
hold. For each
\(\varphi =(\varphi _{1}^{R}, \ldots , \varphi _{n}^{R}, \varphi _{1}^{I}, \ldots , \varphi _{n}^{I}, \varphi _{1}^{J}, \ldots , \varphi _{n}^{J}, \varphi _{1}^{K}, \ldots , \varphi _{n}^{K})^{T}\in \mathbb{B}\), define a nonlinear operator
Φ
as follows:
$$\begin{aligned}& \bigl(\varphi _{1}^{R}, \ldots , \varphi _{n}^{R}, \varphi _{1}^{I}, \ldots , \varphi _{n}^{I}, \varphi _{1}^{J}, \ldots , \varphi _{n}^{J}, \varphi _{1}^{K}, \ldots , \varphi _{n}^{K}\bigr)^{T} \\& \quad \rightarrow \bigl(\bigl(x^{\varphi }\bigr)_{1}^{R}, \ldots ,\bigl(x^{\varphi }\bigr)_{n} ^{R}, \bigl(x^{\varphi }\bigr)_{1}^{I},\ldots , \bigl(x^{\varphi }\bigr)_{n}^{I}, \bigl(x^{ \varphi }\bigr)_{1}^{J}, \ldots , \bigl(x^{\varphi }\bigr)_{n}^{J}, \bigl(x^{\varphi }\bigr)_{1} ^{K},\ldots , \bigl(x^{\varphi }\bigr)_{n}^{K} \bigr)^{T}, \end{aligned}$$
where
$$\begin{aligned} \bigl(x^{\varphi }\bigr)^{l}_{p}(t)= \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u) \,\mathrm{d}u}\varOmega _{p}^{l}(s)\,\mathrm{d}s, \qquad \varOmega _{p}^{l}(t)=F_{p}^{l}\bigl(t, \varphi (t)\bigr)+J_{p}^{l}(t), \quad p\in S, l\in T, \end{aligned}$$
then
Φ
maps
\(\mathbb{B}\)
into itself.
Proof
Let \(\varphi \in \mathbb{B}\). By \((H_{2})\) and Lemma 3, we have \(f_{q}^{l}[t,\varphi ]\in \mathit{PAP}(\mathbb{R},\mathbb{R},\nu )\) and by \((H_{1})\) and Lemma 3, we have \(g_{q}^{l}[t,\tau ,\varphi ]\in \mathit{PAP}(\mathbb{R},\mathbb{R},\nu )\). Hence, from Lemma 1, we obtain \(\varOmega _{p}^{l}\in \mathit{PAP}(\mathbb{R}, \mathbb{R},\nu )\) for all \(p\in S\), \(l\in T\). Consequently, \(\varOmega _{p} ^{l}\) can be written as \(\varOmega _{p}^{l}=\varOmega _{p1}^{l}+\varOmega _{p2} ^{l}\), where \(\varOmega _{p1}^{l}\in \mathit{AP}(\mathbb{R},\mathbb{R})\), \(\varOmega _{p2}^{l}\in \mathit{PAP}_{0}(\mathbb{R},\mathbb{R},\nu )\). Hence,
$$\begin{aligned} \bigl(x^{\varphi }\bigr)^{l}_{p}(t) &= \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u) \,\mathrm{d}u}\varOmega _{p1}^{l}(s)\,\mathrm{d}s + \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\varOmega _{p2}^{l}(s)\,\mathrm{d}s \\ &=:\varTheta _{p1}^{l}(t)+\varTheta _{p2}^{l}(t), \quad p\in S, l\in T. \end{aligned}$$
First, we will prove that \(\varTheta _{p1}^{l}\in \mathit{AP}(\mathbb{R}, \mathbb{R})\) for all \(p\in S\), \(l\in T\). For every \(\epsilon > 0\), since \(\varOmega _{p1}^{l}, c_{p}\in \mathit{AP}(\mathbb{R},\mathbb{R})\), it is possible to find a real number \(l=l(\epsilon )>0\), for each interval with length \(l(\epsilon )\), there exists a number \(\tau =\tau (\epsilon )\) in this interval such that \(|\varOmega _{p1}^{l}(t+\tau )-\varOmega _{p1}^{l}(t)|<\epsilon \) and \(|c_{p}(t+\tau )-c_{p}(t)|<\epsilon \), then
$$\begin{aligned}& \bigl\vert \varTheta _{p1}^{l}(t+ \tau )-\varTheta _{p1}^{l}(t) \bigr\vert \\& \quad = \biggl\vert \int _{-\infty }^{t+\tau }e^{-\int _{s}^{t+\tau }c_{p}(u) \,\mathrm{d}u}\varOmega _{p1}^{l}(s)\,\mathrm{d}s - \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\varOmega _{p1}^{l}(s)\,\mathrm{d}s \biggr\vert \\& \quad = \biggl\vert \int _{-\infty }^{t}e^{-\int _{s+\tau }^{t+\tau }c_{p}(u) \,\mathrm{d}u}\varOmega _{p1}^{l}(s+\tau )\,\mathrm{d}s - \int _{-\infty }^{t}e ^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\varOmega _{p1}^{l}(s)\,\mathrm{d}s \biggr\vert \\& \quad = \biggl\vert \int _{-\infty }^{t}e^{-\int _{s+\tau }^{t+\tau }c_{p}(u) \,\mathrm{d}u}\varOmega _{p1}^{l}(s+\tau )\,\mathrm{d}s - \int _{-\infty }^{t}e ^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\varOmega _{p1}^{l}(s+\tau )\,\mathrm{d}s \\& \qquad {}+ \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\varOmega _{p1} ^{l}(s+\tau )\,\mathrm{d}s - \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u) \,\mathrm{d}u}\varOmega _{p1}^{l}(s)\,\mathrm{d}s \biggr\vert \\& \quad \leq \int _{-\infty }^{t} \bigl\vert e^{-\int _{s+\tau }^{t+\tau }c_{p}(u) \,\mathrm{d}u}-e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u} \bigr\vert \bigl\vert \varOmega _{p1}^{l}(s+\tau ) \bigr\vert \,\mathrm{d}s \\& \qquad {}+ \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u} \bigl\vert \varOmega _{p1}^{l}(s+\tau )-\varOmega _{p1}^{l}(s) \bigr\vert \,\mathrm{d}s. \end{aligned}$$
(3)
By \((e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u} )_{t}'=-c_{p}(t)e ^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\), we have
$$\begin{aligned}& \bigl\vert e^{-\int _{s+\tau }^{t+\tau }c_{p}(u)\,\mathrm{d}u}-e^{-\int _{s} ^{t}c_{p}(u)\,\mathrm{d}u} \bigr\vert \\& \quad = \bigl\vert - \bigl(e^{-\int _{\theta }^{t}c_{p}(u)\,\mathrm{d}u}e^{- \int _{s+\tau }^{\theta +\tau }c_{p}(u)\,\mathrm{d}u} \bigr) \big|_{\theta =t}^{s} \bigr\vert \\& \quad = \biggl\vert - \biggl[ \int _{t}^{s}e^{-\int _{\theta }^{t}c_{p}(u)\,\mathrm{d}u} \bigl(e^{-\int _{s+\tau }^{\theta +\tau }c_{p}(u)\,\mathrm{d}u} \bigr)_{ \theta }'\,\mathrm{d}\theta \\& \qquad {}+ \int _{t}^{s} \bigl(e^{-\int _{\theta }^{t}c_{p}(u)\,\mathrm{d}u} \bigr)_{ \theta }' e^{-\int _{s+\tau }^{\theta +\tau }c_{p}(u)\,\mathrm{d}u} \,\mathrm{d}\theta \biggr] \biggr\vert \\& \quad = \biggl\vert \int _{t}^{s}e^{-\int _{\theta }^{t}c_{p}(u)\,\mathrm{d}u} \bigl(c _{p}(\theta +\tau )-c_{p}(\theta ) \bigr) e^{-\int _{s+\tau }^{\theta + \tau }c_{p}(u)\,\mathrm{d}u}\,\mathrm{d}\theta \biggr\vert \\& \quad \leq \int _{t}^{s}e^{-\int _{\theta }^{t}c_{p}(u)\,\mathrm{d}u} \bigl\vert c _{p}(\theta +\tau )-c_{p}(\theta ) \bigr\vert \,\mathrm{d}\theta \leq \frac{1}{c _{p}^{-}} e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}\epsilon . \end{aligned}$$
(4)
Since \(\varOmega _{p1}^{l}\in \mathit{AP}(\mathbb{R},\mathbb{R})\), it is a uniformly continuous and bounded function. Denote \(G(t):=\int _{- \infty }^{t} |\varOmega _{p1}^{l}(s) |\,\mathrm{d}s\) and substitute (4) into (3), we have
$$\begin{aligned} \bigl\vert \varTheta _{p1}^{l}(t+\tau )-\varTheta _{p1}^{l}(t) \bigr\vert \leq \frac{ \epsilon }{(c_{p}^{-})^{2}} \bigl(c_{p}^{-}+ \Vert G \Vert _{\mathbb{B}} \bigr), \end{aligned}$$
which implies that \(\varTheta _{p1}^{l}\in \mathit{AP}(\mathbb{R}, \mathbb{R})\), \(p\in S\), \(l\in T\).
Next, for \(p\in S\), \(l\in T\), set
$$\begin{aligned} \varLambda _{p}^{l}=\frac{1}{\nu (Q_{r})} \int _{Q_{r}} \biggl\vert \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u} \varOmega _{p2}^{l}(s) \,\mathrm{d}s \biggr\vert \nu (t) \,\mathrm{d}t. \end{aligned}$$
To prove that \(\varTheta _{p2}^{l}\in \mathit{PAP}_{0}(\mathbb{R}, \mathbb{R},\nu )\), we only need to show that \(\lim_{r\rightarrow \infty }\varLambda _{p}^{l}=0\), \(p\in S\), \(l\in T\). By a similar argument as that in the proof of Lemma 3.4 in [47], one can see that \(\varTheta _{p2}^{l}\in \mathit{PAP}_{0}( \mathbb{R},\mathbb{R},\nu )\), \(p\in S\), \(l\in T\). Therefore, we have \((x^{\varphi })^{l}_{p}\in \mathit{PAP}(\mathbb{R},\mathbb{R},\nu )\), that is, Φ maps \(\mathbb{B}\) into itself. This completes the proof. □
Remark 2
It is easy to check that, for \(p\in S\), \(l\in T\),
$$\begin{aligned} \bigl(x^{\varphi }\bigr)^{l}_{p}(t)= \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u) \,\mathrm{d}u} \varOmega _{p}^{l}(s)\,\mathrm{d}s \end{aligned}$$
satisfy the following equations:
$$\begin{aligned} \bigl(x_{p}^{l} \bigr)'(t)=-c_{p}(t)x_{p}^{l}(t)+ \varOmega _{p}^{l}(t), \quad p\in S, l\in T. \end{aligned}$$
Theorem 1
Assume that
\((H_{1})\)–\((H_{3})\)
hold, then system (2) has a unique weighted pseudo-almost periodic solution in
\(\mathbb{B}^{ \ast }= \{\varphi |\varphi \in \mathbb{B}, \|\varphi -\varphi ^{0} \|_{\mathbb{B}}\leq \frac{\rho \kappa }{1-\rho } \}\).
Proof
For any \(\varphi \in \mathbb{B}\), by Lemma 4, Φ maps \(\mathbb{B}\) into itself. Obviously,
$$\begin{aligned} \bigl\Vert \varphi ^{0} \bigr\Vert _{\mathbb{B}} =&\max _{1\leq p\leq n} \biggl\{ \sup_{t\in \mathbb{R}}\max _{l\in T} \biggl\vert \int _{-\infty }^{t}e^{-\int _{s}^{t}c_{p}(u)\,\mathrm{d}u}J_{p}^{l}(s) \,\mathrm{d}s \biggr\vert \biggr\} \\ \leq & \max_{1\leq p\leq n} \biggl\{ \max_{l\in T} \biggl\{ \frac{J_{p}^{l^{+}}}{c_{p}^{-}} \biggr\} \biggr\} =\kappa . \end{aligned}$$
Hence, for all \(\varphi \in \mathbb{B}^{\ast }= \{\varphi |\varphi \in \mathbb{B}, \|\varphi -\varphi ^{0}\|_{\mathbb{B}}\leq \frac{ \rho \kappa }{1-\rho } \}\), we have
$$\begin{aligned} \Vert \varphi \Vert _{\mathbb{B}} \leq \bigl\Vert \varphi -\varphi ^{0} \bigr\Vert _{\mathbb{B}}+ \bigl\Vert \varphi ^{0} \bigr\Vert _{\mathbb{B}} \leq \frac{\rho \kappa }{1-\rho }+ \kappa =\frac{\kappa }{1-\rho }. \end{aligned}$$
Next, we show that Φ maps \(\mathbb{B}^{\ast }\) into itself. In fact, for any \(\varphi \in \mathbb{B}^{\ast }\), by \((H_{2})\), we have
$$\begin{aligned}& \sup_{t\in \mathbb{R}} \bigl\vert (\varPhi \varphi )_{p}^{R}(t)-\bigl( \varphi ^{0} \bigr)_{p}^{R}(t) \bigr\vert \\& \quad = \sup_{t\in \mathbb{R}} \Biggl\vert \int _{-\infty }^{t}e^{-\int _{s} ^{t}c _{p} (u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl({a}_{pq} ^{R}(s)f_{q}^{R}[t,\varphi ]- {a}_{pq}^{I}(s)f_{q}^{I}[t, \varphi ] \\& \qquad {}-{a}_{pq}^{J}(s)f_{q}^{J}[t, \varphi ]-{a}_{pq}^{K}(s)f_{q}^{K}[t, \varphi ] \bigr) +\sum_{q=1}^{n} \bigl({b}_{pq}^{R}(s)g_{q}^{R}[t, \tau ,\varphi ] \\& \qquad {} -{b}_{pq}^{I}(s)g_{q}^{I}[t, \tau ,\varphi ]-{b}_{pq}^{J}(s)g_{q} ^{J}[t,\tau ,\varphi ] -{b}_{pq}^{K}(s)g_{q}^{K}[t, \tau ,\varphi ] \bigr) \Biggr]\,\mathrm{d}s \Biggr\vert \\& \quad \leq \sup_{t\in \mathbb{R}} \int _{-\infty }^{t}e^{-\int _{s} ^{t}c_{p} (u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl( \bigl\vert a_{pq} ^{R}(s) \bigr\vert \bigl\vert f_{q}^{R}[t,\varphi ] \bigr\vert + \bigl\vert a_{pq}^{I}(s) \bigr\vert \bigl\vert f_{q}^{I}[t,\varphi ] \bigr\vert \\& \qquad {}+ \bigl\vert a_{pq}^{J}(s) \bigr\vert \bigl\vert f_{q}^{J}[t,\varphi ] \bigr\vert + \bigl\vert a_{pq} ^{K}(s) \bigr\vert \bigl\vert f_{q}^{K}[t,\varphi ] \bigr\vert \bigr) +\sum _{q=1} ^{n} \bigl( \bigl\vert {b}_{pq}^{R}(s) \bigr\vert \bigl\vert g_{q}^{R}[t,\tau ,\varphi ] \bigr\vert \\& \qquad {} + \bigl\vert {b}_{pq}^{I}(s) \bigr\vert \bigl\vert g_{q}^{I}[t,\tau ,\varphi ] \bigr\vert + \bigl\vert {b}_{pq}^{J}(s) \bigr\vert \bigl\vert g_{q}^{J}[t,\tau ,\varphi ] \bigr\vert + \bigl\vert {b}_{pq}^{K}(s) \bigr\vert \bigl\vert g_{q}^{K}[t,\tau ,\varphi ] \bigr\vert \bigr) \Biggr] \,\mathrm{d}s \\& \quad \leq \sup_{t\in \mathbb{R}} \int _{-\infty }^{t}e^{-\int _{s} ^{t}c _{p} (u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl(a_{pq}^{R ^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}} +a_{pq}^{K^{+}} \bigr) \\& \qquad {} \times \bigl(L_{f}^{R}+L_{f}^{I}+L_{f}^{J}+L_{f}^{K} \bigr) \Vert \varphi \Vert _{\mathbb{B}} +\sum _{q=1}^{n} \bigl(b_{pq}^{R^{+}}+b_{pq}^{I ^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}} \bigr) \\& \qquad {} \times \bigl(L_{g}^{R}+L_{g}^{I} +L_{g}^{J}+L_{g}^{K} \bigr) \Vert \varphi \Vert _{\mathbb{B}} \Biggr]\,\mathrm{d}s \\& \quad \leq \frac{1}{c_{p}^{-}} (A_{p}+B_{p} ) \Vert \varphi \Vert _{ \mathbb{B}}, \quad p\in S. \end{aligned}$$
(5)
Similarly, we can obtain
$$ \sup_{t\in \mathbb{R}} \bigl\vert (\varPhi \varphi )_{p}^{l}(t)-\bigl(\varphi ^{0} \bigr)_{p}^{l}(t) \bigr\vert \leq \frac{1}{c_{p}^{-}} (A_{p}+B_{p} ) \Vert \varphi \Vert _{\mathbb{B}}, \quad p\in S, l=I,J,K. $$
(6)
It follows from (5) and (6) that
$$\begin{aligned} \bigl\Vert \varPhi \varphi -\varphi ^{0} \bigr\Vert _{\mathbb{B}}\leq \rho \Vert \varphi \Vert _{ \mathbb{B}}\leq \frac{\rho \kappa }{1-\rho }, \end{aligned}$$
which implies that \(\varPhi \varphi \in \mathbb{B}^{\ast }\). So, the mapping Φ is a self mapping from \(\mathbb{B}^{\ast }\) to \(\mathbb{B}^{\ast }\). Finally, we prove that Φ is a contraction mapping. In fact, in view of \((H_{2})\), for any \(\varphi ,\psi \in \mathbb{B }\), we have
$$\begin{aligned}& \sup_{t\in \mathbb{R}} \bigl\vert (\varPhi \varphi )_{p}^{R}(t)-( \varPhi \psi )_{p}^{R}(t) \bigr\vert \\ & \quad = \sup_{t\in \mathbb{R}} \Biggl\vert \int _{-\infty }^{t}e^{-\int _{s} ^{t} c _{p}(u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl({a}_{pq} ^{R}(s) \bigl(f_{q}^{R}\{t,\varphi \} -f_{q}^{R}\{t,\psi \} \bigr) \\ & \qquad {} -{a}_{pq}^{I}(s) \bigl(f_{q}^{I} \{t,\varphi \}-f_{q}^{I}\{t,\psi \} \bigr)-{a}_{pq}^{J}(s) \bigl(f_{q}^{J}\{t,\varphi \}-f_{q}^{J} \{t,\psi \} \bigr) \\ & \qquad {} -{a}_{pq}^{K}(s) \bigl(f_{q}^{K} \{t,\varphi \} -f_{q}^{K}\{t,\psi \} \bigr) \bigr)+\sum _{q=1}^{n} \bigl({b}_{pq}^{R}(s) \bigl(g_{q}^{R}\{t, \tau ,\varphi \} \\ & \qquad {}-g_{q}^{R}\{t,\tau ,\psi \} \bigr) -{b}_{pq}^{I}(s) \bigl(g_{q}^{I} \{t, \tau ,\varphi \}-g_{q}^{I}\{t,\tau ,\psi \} \bigr) \\ & \qquad {} -{b}_{pq}^{J}(s) \bigl(g_{q}^{J} \{t,\tau ,\varphi \}-g_{q}^{J}\{t, \tau ,\psi \} \bigr) -{b}_{pq}^{K}(s) \bigl(g_{q}^{K} \{t,\tau ,\varphi \} \\ & \qquad {}-g_{q}^{K}\{t,\tau ,\psi \} \bigr) \bigr) \Biggr] \,\mathrm{d}s \Biggr\vert \\ & \quad \leq \sup_{t\in \mathbb{R}} \int _{-\infty }^{t}e^{-\int _{s} ^{t} c _{p}(u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl( \bigl\vert {a} _{pq}^{R}(s) \bigr\vert \bigl\vert f_{q}^{R}\{t,\varphi \} -f_{q}^{R} \{t,\psi \} \bigr\vert \\ & \qquad {} + \bigl\vert {a}_{pq}^{I}(s) \bigr\vert \bigl\vert f_{q}^{I}\{t,\varphi \}-f_{q}^{I} \{t, \psi \} \bigr\vert + \bigl\vert {a}_{pq}^{J}(s) \bigr\vert \bigl\vert f_{q}^{J}\{t,\varphi \}-f _{q}^{J}\{t,\psi \} \bigr\vert \\ & \qquad {} + \bigl\vert {a}_{pq}^{K}(s) \bigr\vert \bigl\vert f_{q}^{K}\{t,\varphi \} -f_{q}^{K} \{t,\psi \} \bigr\vert \bigr)+\sum_{q=1}^{n} \bigl( \bigl\vert {b}_{pq}^{R}(s) \bigr\vert \bigl\vert g_{q}^{R}\{t,\tau ,\varphi \} \\ & \qquad {}-g_{q}^{R}\{t,\tau ,\psi \} \bigr\vert + \bigl\vert {b}_{pq}^{I}(s) \bigr\vert \bigl\vert g _{q}^{I}\{t,\tau ,\varphi \}-g_{q}^{I} \{t,\tau ,\psi \} \bigr\vert \\ & \qquad {}+ \bigl\vert {b}_{pq}^{J}(s) \bigr\vert \bigl\vert g_{q}^{J}\{t,\tau ,\varphi \}-g_{q} ^{J}\{t,\tau ,\psi \} \bigr\vert + \bigl\vert {b}_{pq}^{K}(s) \bigr\vert \bigl\vert g_{q}^{K}\{t, \tau ,\varphi \} \\ & \qquad {}-g_{q}^{K}\{t,\tau ,\psi \} \bigr\vert \bigr) \Biggr]\,\mathrm{d}s \\ & \quad \leq \sup_{t\in \mathbb{R}} \int _{-\infty }^{t}e^{-\int _{s} ^{t} c _{p}(u)\,\mathrm{d}u} \Biggl[\sum _{q=1}^{n} \bigl(a_{pq}^{R ^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}} +a_{pq}^{K^{+}} \bigr) \\ & \qquad {} \times \bigl(L_{f}^{R}+L_{f}^{I}+L_{f}^{J}+L_{f}^{K} \bigr) \Vert \varphi -\psi \Vert _{\mathbb{B}} +\sum _{q=1}^{n} \bigl(b_{pq}^{R^{+}}+b_{pq} ^{I^{+}} \\ & \qquad {}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}} \bigr) \bigl(L_{g}^{R}+L_{g}^{I} +L_{g} ^{J}+L_{g}^{K} \bigr) \Vert \varphi -\psi \Vert _{\mathbb{B}} \Biggr]\,\mathrm{d}s \\& \quad \leq \frac{1}{c_{p}^{-}} (A_{p}+B_{p} ) \Vert \varphi -\psi \Vert _{\mathbb{B}}, \quad p\in S. \end{aligned}$$
(7)
Similarly, we can get
$$ \sup_{t\in \mathbb{R}} \bigl\vert (\varPhi \varphi )_{p}^{R}(t)-(\varPhi \psi )_{p}^{R}(t) \bigr\vert \leq \frac{1}{c_{p}^{-}} (A_{p}+B_{p} ) \Vert \varphi -\psi \Vert _{\mathbb{B}}, \quad p\in S, l=I,J,K. $$
(8)
From (7) and (8), we obtain
$$\begin{aligned} \Vert \varPhi \varphi -\varPhi \psi \Vert _{\mathbb{B}}\leq \rho \Vert \varphi -\psi \Vert _{\mathbb{B}}. \end{aligned}$$
Since \((H_{3})\), Φ is a contraction mapping. Hence, Φ has a fixed point in \(\mathbb{B}^{\ast }\). That is, system (2) has a unique weighted pseudo-almost periodic solution in \(\mathbb{B}^{ \ast }\). This completes the proof. □