In this section, we prove under some conditions that system (2.1) has a unique equilibrium state by Banach’s contraction fixed point argument and prove under some additional conditions that system (2.1) is mean-square exponentially stable. The main ingredient in proving the stability result is a well-chosen Lyapunov functional. We first state the first main result concerned with the existence of equilibrium states of system (2.1).
Theorem 3.1
Let Assumptions 1, 2, and
3
be fulfilled. Suppose in addition that the following two properties hold:
-
(i)
If
\((u_{1},\ldots,u_{n},v_{1},\ldots,v_{m})\)
solves the system
$$\begin{aligned} \textstyle\begin{cases} \textstyle\begin{array}{rl} \mu_{1i}u_{i} =&\sum_{j=1}^{m}a^{1}_{ij}f_{1j}(v_{j}) +\sum_{j=1}^{m}b^{1}_{ij}f_{1j}(v_{j}) +\bigwedge_{j=1}^{m} \alpha^{1}_{ij}\check{K}_{1j}(0) f_{1j}(v_{j})\\ &{}+\bigvee_{j=1}^{m}\beta^{1}_{ij} \check{K}_{1j}(0) f_{1j}(v_{j}) +\sum_{j=1}^{m}c^{1}_{ij}w^{1}_{j} +\bigwedge_{j=1}^{m}T^{1}_{ij}w^{1}_{ij} +\bigvee_{j=1}^{m}H^{1}_{ij}w^{1}_{ij} +I_{i}, \end{array}\displaystyle \\ \textstyle\begin{array}{rl} \mu_{2j}v_{j} =&\sum_{i=1}^{n}a^{2}_{ji}f_{2i}(u_{i}) +\sum_{i=1}^{n}b^{2}_{ji}f_{2i}(u_{i}) +\bigwedge_{i=1}^{n} \alpha^{2}_{ji}\check{K}_{2i}(0) f_{2i}(u_{i})\\ &{}+\bigvee_{i=1}^{n}\beta^{2}_{ji} \check{K}_{2i}(0) f_{2i}(u_{i}) +\sum_{i=1}^{n}c^{2}_{ji} w^{2}_{i}+\bigwedge_{i=1}^{n}T^{2}_{ji}w^{2}_{ji} +\bigvee_{i=1}^{n}H^{2}_{ji}w^{2}_{ji} +J_{j}, \end{array}\displaystyle \end{cases}\displaystyle \end{aligned}$$
then
\((u_{1},\ldots,u_{n},v_{1},\ldots,v_{m})\)
satisfies
$$\begin{aligned} \textstyle\begin{cases} \sum_{j=1}^{m} \tilde{a}^{1}_{ij}\tilde{f}_{1j}(v_{j}) +\sum_{j=1}^{m} \tilde{b}^{1}_{ij} \tilde{f}_{1j}(v_{j}) +\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}\check{\tilde{K}}_{1j}(0) \tilde{f}_{1j}(v_{j}) +\bigvee_{j=1}^{m}\tilde{\beta}^{1}_{ij} \check{\tilde{K}}_{1j}(0) \tilde{f}_{1j}(v_{j})=0,\\ \sum_{i=1}^{n} \tilde{a}^{2}_{ji}\tilde{f}_{2i}(u_{i}) +\sum_{i=1}^{n} \tilde{b}^{2}_{ji} \tilde{f}_{2i}(u_{i}) +\bigwedge_{i=1}^{n} \tilde{\alpha}^{2}_{ji}\check{\tilde{K}}_{2i}(0) \tilde{f}_{2i}(u_{i}) +\bigvee_{i=1}^{n}\tilde{\beta}^{2}_{ji} \check{\tilde{K}}_{2i}(0) \tilde{f}_{2i}(u_{i})=0; \end{cases}\displaystyle \end{aligned}$$
-
(ii)
\(\lambda_{1}<1\)
and
\(\lambda_{2}<1\)
with
$$\begin{aligned} \left. \textstyle\begin{array}{l} \lambda_{1}=\max_{1\leqslant j\leqslant m}\sum_{i=1}^{n} \frac{L_{1j} ( \vert a^{1}_{ij} \vert + \vert b^{1}_{ij} \vert +\check{K}_{1j}(0) \vert \alpha^{1}_{ij} \vert +\check{K}_{1j}(0) \vert \beta^{1}_{ij} \vert )}{\mu_{1i}},\\ \lambda_{2}=\max_{1\leqslant i\leqslant n}\sum_{j=1}^{m} \frac{L_{2i} ( \vert a^{2}_{ji} \vert + \vert b^{2}_{ji} \vert +\check{K}_{2i}(0) \vert \alpha^{2}_{ji} \vert +\check{K}_{2i}(0) \vert \beta^{2}_{ji} \vert ) }{\mu_{2j}}. \end{array}\displaystyle \right\} \end{aligned}$$
(3.1)
Then system (2.1) admits a unique equilibrium state.
Proof
Let us define the nonlinear mapping on \(\mathbb{R}^{n+m}\) by
$$\begin{aligned} &\varPsi(u_{1},\ldots, u_{n},v_{1}, \ldots,v_{m}) \\ &\quad = \Biggl(\sum_{j=1}^{m} \frac{a^{1}_{1j}f_{1j}(v_{j})}{\mu_{11}} +\sum_{j=1}^{m} \frac{b^{1}_{1j}f_{1j}(v_{j})}{\mu_{11}} +\bigwedge_{j=1}^{m} \frac{\alpha^{1}_{1j}\check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{11}}+\bigvee_{j=1}^{m} \frac{\beta^{1}_{1j} \check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{11}} \\ &\qquad {} +\sum_{j=1}^{m} \frac{c^{1}_{1j}w^{1}_{j} }{\mu_{11}} +\bigwedge_{j=1}^{m} \frac{T^{1}_{1j}w^{1}_{1j}}{\mu_{11}}+\bigvee_{j=1}^{m} \frac{H^{1}_{1j}w^{1}_{1j}}{\mu_{11}} +\frac{I_{1}}{\mu_{11}},\ldots, \sum_{j=1}^{m} \frac{a^{1}_{nj}f_{1j}(v_{j})}{\mu_{1n}} +\sum_{j=1}^{m} \frac{b^{1}_{nj}f_{1j}(v_{j})}{\mu_{1n}} \\ &\qquad {}+\bigwedge_{j=1}^{m} \frac{\alpha^{1}_{nj}\check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1n}} +\bigvee_{j=1}^{m} \frac{\beta^{1}_{nj} \check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1n}} +\sum_{j=1}^{m} \frac{c^{1}_{nj}w^{1}_{j} }{\mu_{1n}} +\bigwedge_{j=1}^{m} \frac{T^{1}_{nj}w^{1}_{nj}}{\mu_{1n}} \\ &\qquad {} +\bigvee_{j=1}^{m} \frac{H^{1}_{nj}w^{1}_{nj}}{\mu_{1n}} +\frac{I_{n}}{\mu_{1n}}, \sum_{i=1}^{n} \frac{a^{2}_{1i}f_{2i}(u_{i})}{\mu_{21}} + \sum_{i=1}^{n} \frac{b^{2}_{1i}f_{2i}(u_{i})}{\mu_{21}} +\bigwedge_{i=1}^{n} \frac{\alpha^{2}_{1i}\check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{21}} \\ &\qquad {} +\bigvee_{i=1}^{n} \frac{\beta^{2}_{1i} \check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{21}} +\sum_{i=1}^{n} \frac{c^{2}_{1i}w^{2}_{i} }{\mu_{21}} +\bigwedge_{i=1}^{n} \frac{T^{2}_{1i}w^{2}_{1i}}{\mu_{21}} +\bigvee_{i=1}^{n} \frac{H^{2}_{1i}w^{2}_{1i}}{\mu_{21}} +\frac{J_{1}}{\mu_{21}} ,\ldots, \\ &\qquad \sum_{i=1}^{n} \frac{a^{2}_{mi}f_{2i}(u_{i})}{\mu_{2m}} + \sum_{i=1}^{n}\frac{b^{2}_{mi}f_{2i}(u_{i})}{\mu_{2m}} +\bigwedge _{i=1}^{n} \frac{\alpha^{2}_{mi}\check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2m}} +\bigvee _{i=1}^{n} \frac{\beta^{2}_{mi} \check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2m}} \\ &\qquad {} +\sum_{i=1}^{n} \frac{c^{2}_{mi}w^{2}_{i} }{\mu_{2m}} +\bigwedge_{i=1}^{n} \frac{T^{2}_{mi}w^{2}_{mi}}{\mu_{2m}} +\bigvee_{j=1}^{m} \frac{H^{2}_{mi}w^{2}_{mi}}{\mu_{2m}} +\frac{J_{m}}{\mu_{2m}} \Biggr) \\ &\quad \mbox{for } (u_{1},\ldots,u_{n},v_{1}, \ldots,v_{m}) \in\mathbb{R}^{n+m}. \end{aligned}$$
For every \(X_{1}\in\mathbb{R}^{n+m}\) and every \(X_{2}\in\mathbb{R}^{n+m}\) with \(X_{1}=(u_{1},\ldots,u_{n},v_{1}, \ldots,v_{m})\) and \(X_{2}=(x_{1},\ldots,x_{n},y_{1} \ldots,y_{m})\), by Lemma 2.2 and by conducting some routine calculations we obtain the following sequence of inequalities:
$$\begin{aligned} & \bigl\Vert \varPsi(X_{1})-\varPsi(X_{2}) \bigr\Vert \\ &\quad = \sum_{i=1}^{n} \Biggl\vert \sum _{j=1}^{m} \frac{a^{1}_{ij}f_{1j}(v_{j})}{\mu_{1i}} + \sum _{j=1}^{m}\frac{b^{1}_{ij}f_{1j}(v_{j})}{\mu_{1i}} + \bigwedge _{j=1}^{m} \frac{\alpha^{1}_{ij}\check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1i}} +\bigvee _{j=1}^{m} \frac{\beta^{1}_{ij} \check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1i}} \\ &\qquad {} -\sum_{j=1}^{m} \frac{a^{1}_{ij}f_{1j}(y_{j})}{\mu_{1i}} -\sum_{j=1}^{m} \frac{b^{1}_{ij}f_{1j}(y_{j})}{\mu_{1i}} -\bigwedge_{j=1}^{m} \frac{\alpha^{1}_{ij}\check{K}_{1j}(0) f_{1j}(y_{j})}{\mu_{1i}} -\bigvee_{j=1}^{m} \frac{\beta^{1}_{ij} \check{K}_{1j}(0) f_{1j}(y_{j})}{\mu_{1i}} \Biggr\vert \\ &\qquad {} +\sum_{j=1}^{m} \Biggl\vert \sum_{i=1}^{n} \frac{a^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} +\sum _{i=1}^{n}\frac{b^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} +\bigwedge _{i=1}^{n} \frac{\alpha^{2}_{ji}\check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2j}} +\bigvee _{i=1}^{n} \frac{\beta^{2}_{ji} \check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2j}} \\ &\qquad {} -\sum_{i=1}^{n} \frac{a^{2}_{ji}f_{2i}(x_{i})}{\mu_{2j}} -\sum_{i=1}^{n} \frac{b^{2}_{ji}f_{2i}(x_{i})}{\mu_{2j}} -\bigwedge_{i=1}^{n} \frac{\alpha^{2}_{ji}\check{K}_{2i}(0) f_{2i}(x_{i})}{\mu_{2j}} -\bigvee_{i=1}^{n} \frac{\beta^{2}_{ji} \check{K}_{2i}(0) f_{2i}(x_{i})}{\mu_{2j}} \Biggr\vert \\ &\quad \leqslant \sum_{i=1}^{n} \Biggl[ \Biggl\vert \sum_{j=1}^{m} \frac{a^{1}_{ij}f_{1j}(v_{j})}{\mu_{1i}} -\sum_{j=1}^{m} \frac{a^{1}_{ij}f_{1j}(y_{j})}{\mu_{1i}} \Biggr\vert + \Biggl\vert \sum _{j=1}^{m}\frac{b^{1}_{ij}f_{1j}(v_{j})}{\mu_{1i}} -\sum _{j=1}^{m}\frac{b^{1}_{ij}f_{1j}(y_{j})}{\mu_{1i}} \Biggr\vert \\ &\qquad {} + \Biggl\vert \bigwedge_{j=1}^{m} \frac{\alpha^{1}_{ij}\check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1i}} -\bigwedge_{j=1}^{m} \frac{\alpha^{1}_{ij}\check{K}_{1j}(0) f_{1j}(y_{j})}{\mu_{1i}} \Biggr\vert \\ &\qquad {} + \Biggl\vert \bigvee_{j=1}^{m} \frac{\beta^{1}_{ij} \check{K}_{1j}(0) f_{1j}(v_{j})}{\mu_{1i}} -\bigvee_{j=1}^{m} \frac{\beta^{1}_{ij} \check{K}_{1j}(0) f_{1j}(y_{j})}{\mu_{1i}} \Biggr\vert \Biggr] \\ &\qquad {}+\sum_{j=1}^{m} \Biggl[ \Biggl\vert \sum_{i=1}^{n} \frac{a^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} - \sum_{i=1}^{n} \frac{a^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} \Biggr\vert + \Biggl\vert \sum_{i=1}^{n} \frac{b^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} -\sum_{i=1}^{n} \frac{b^{2}_{ji}f_{2i}(u_{i})}{\mu_{2j}} \Biggr\vert \\ &\qquad {} + \Biggl\vert \bigwedge_{i=1}^{n} \frac{\alpha^{2}_{ji}\check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2j}} -\bigwedge_{i=1}^{n} \frac{\alpha^{2}_{ji}\check{K}_{2i}(0) f_{2i}(x_{i})}{\mu_{2j}} \Biggr\vert \\ &\qquad {} + \Biggl\vert \bigvee_{i=1}^{n} \frac{\beta^{2}_{ji} \check{K}_{2i}(0) f_{2i}(u_{i})}{\mu_{2j}} -\bigvee_{i=1}^{n} \frac{\beta^{2}_{ji} \check{K}_{2i}(0) f_{2i}(x_{i})}{\mu_{2j}} \Biggr\vert \Biggr] \\ &\quad \leqslant \lambda_{1}\sum_{j=1}^{m} \vert v_{j}-y_{j} \vert +\lambda_{2}\sum _{i=1}^{n} \vert u_{i}-x_{i} \vert \leqslant\max(\lambda_{1},\lambda_{2}) \Vert X_{1}-X_{2} \Vert , \end{aligned}$$
where \(\lambda_{1}\) and \(\lambda_{2}\) are given by (3.1). Since \(\lambda_{1}<1\) and \(\lambda_{2}<1\) by assumption, this means that Ψ is a (strict) contraction on \(\mathbb{R}^{n+m}\). By Banach’s contraction fixed point argument, Ψ admits a unique fixed point \((u_{1}^{*}, \ldots,u^{*}_{n},v_{1}^{*}, \ldots,v^{*}_{m})\) in \(\mathbb{R}^{n+m}\). By the definition of Ψ, \((u_{1}^{*}, \ldots,u^{*}_{n},v_{1}^{*}, \ldots,v^{*}_{m})\) is indeed an equilibrium state of system (2.1).
The uniqueness can be obtained readily, since Ψ is a strict contraction. The proof is complete. □
Theorem 3.2
Let Assumptions 1, 2, and 3
be fulfilled. If
\(M_{1i}(0)<0\), \(M_{2j}(0)<0\) (\(i=1,\ldots,n\), \(j=1,\ldots,m\)), and the hypothesis of Theorem 3.1
are satisfied, then system (2.1) is mean-square exponentially stable. Here
\(M_{1i}\)
and
\(M_{2j}\)
are continuous functions on
\([0,\bar{\varepsilon})\)
given by
$$\begin{aligned} M_{1i}(\varepsilon)={}&{-}2 \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr) +\sum_{j=1}^{m} \bigl\vert a^{1}_{ij} \bigr\vert L_{1j} +\sum _{j=1}^{m} \bigl\vert a^{2}_{ji} \bigr\vert L_{2i} +\sum_{j=1}^{m} \bigl\vert b^{1}_{ij} \bigr\vert L_{1j}e^{\varepsilon \overline{\sigma}_{1j}} \\ &{}+\sum_{j=1}^{m} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j} \check{K}_{1j}(\varepsilon) +\sum_{j=1}^{m} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i} \check{K}_{2i}(\varepsilon) +\sum_{j=1}^{m} \bigl\vert \beta^{1}_{ij} \bigr\vert L_{1j} \check{K}_{1j}(\varepsilon) \\ &{}+\sum_{j=1}^{m} \bigl\vert \beta^{2}_{ji} \bigr\vert L_{2i} \check{K}_{2i}(\varepsilon) +2\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl( \mu_{1i}e^{\varepsilon\tau_{1i}} -\varepsilon \bigr)\tau_{1i} +\sum _{j=1}^{m}\frac{ \vert b^{2}_{ji} \vert L_{2i}e^{\varepsilon \overline{\sigma}_{2i}}\overline{\sigma}_{2i}}{1-\hat{\sigma}_{2i}} \\ &{}+ \sum_{j=1}^{m}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl\vert a^{1}_{ij} \bigr\vert L_{1j} \tau_{1i} + \sum_{j=1}^{m} \mu_{2j}e^{\varepsilon\tau_{2j}} \bigl\vert a^{2}_{ji} \bigr\vert L_{2i} \tau_{2j} + \sum _{j=1}^{m}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl\vert b^{1}_{ij} \bigr\vert L_{1j}e^{\varepsilon\overline{\sigma}_{1j}} \tau_{1i} \\ &{}+ \sum_{j=1}^{m} \frac{\mu_{2j}e^{\varepsilon\tau_{2j}} \vert b^{2}_{ji} \vert L_{2i} e^{\varepsilon\overline{\sigma}_{2i}} \tau_{2j}\overline{\sigma}_{2i}}{1-\hat{\sigma}_{2i}} + \sum_{j=1}^{m} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j} \mu_{1i}e^{\varepsilon\tau_{1i}} \check{K}_{1j}(\varepsilon) \tau_{1i} \\ &{}+\sum_{j=1}^{m} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i} \mu_{2j}e^{\varepsilon\tau_{2j}} \check{K}_{2i}(\varepsilon) \tau_{2j} +4\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{a}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \\ &{}+4\sum_{i=1}^{n}\sum _{j=1}^{m}\frac{ \vert \tilde{b}^{2}_{ji} \vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \overline{\tilde{\sigma}}_{2i}}{1-\hat{\tilde{\sigma}}_{2i}} +4 \Biggl(\sum _{i=1}^{n} \sum_{j=1}^{m} \bigl\vert \tilde{\alpha}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \check{ \tilde{K}}_{2i}(\varepsilon) \Biggr)\check{\tilde{K}}_{2i}( \varepsilon) \\ &{}+4 \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{\beta}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \check{\tilde{K}}_{2i}(\varepsilon) \Biggr)\check{ \tilde{K}}_{2i}(\varepsilon), \end{aligned}$$
(m1)
and
$$\begin{aligned} M_{2j}(\varepsilon)={}&{-}2 \bigl(\mu_{2j} e^{\varepsilon\tau_{2j}}- \varepsilon \bigr) +\sum_{i=1}^{n} \bigl\vert a^{1}_{ij} \bigr\vert L_{1j} +\sum _{i=1}^{n} \bigl\vert a^{2}_{ji} \bigr\vert L_{2i} +\sum_{i=1}^{n} \frac{ \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon \overline{\sigma}_{1j}}\overline{\sigma}_{1j} }{1-\hat{\sigma}_{1j}} \\ &{} +\sum_{i=1}^{n} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j} \check{K}_{1j}(\varepsilon) +\sum_{i=1}^{n} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i} \check{K}_{2i}(\varepsilon) +\sum_{i=1}^{n} \bigl\vert \beta^{1}_{ij} \bigr\vert L_{1j} \check{K}_{1j}(\varepsilon) \\ &{}+\sum_{i=1}^{n} \bigl\vert \beta^{2}_{ji} \bigr\vert L_{2i} \check{K}_{2i}(\varepsilon) +2\mu_{2j}e^{\varepsilon\tau_{2j}} \bigl( \mu_{2j}e^{\varepsilon\tau_{2j}} -\varepsilon \bigr)\tau_{2j} +\sum _{i=1}^{n} \bigl\vert b^{2}_{ji} \bigr\vert L_{2i}e^{\varepsilon \overline{\sigma}_{2i}} \\ &{} + \sum_{i=1}^{n}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl\vert a^{1}_{ij} \bigr\vert L_{1j} \tau_{1i} + \sum_{i=1}^{n} \mu_{2j}e^{\varepsilon\tau_{2j}} \bigl\vert a^{2}_{ji} \bigr\vert L_{2i} \tau_{2j}\\ &{} + \sum _{i=1}^{n} \frac{\mu_{1i}e^{\varepsilon\tau_{1i}} \vert b^{1}_{ij} \vert L_{1j} e^{\varepsilon\overline{\sigma}_{1j}} \tau_{1i}\overline{\sigma}_{1j}}{1-\hat{\sigma}_{1j}} \\ &{}+ \sum_{i=1}^{n}\mu_{2j}e^{\varepsilon\tau_{2j}} \bigl\vert b^{2}_{ji} \bigr\vert L_{2i} e^{\varepsilon\overline{\sigma}_{2i}} \tau_{2j} +\sum_{i=1}^{n} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j} \mu_{1i}e^{\varepsilon\tau_{1i}} \check{K}_{1j}(\varepsilon) \tau_{1i} \\ &{}+\sum_{i=1}^{n} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i} \mu_{2j}e^{\varepsilon\tau_{2j}} \check{K}_{2i}(\varepsilon) \tau_{2j} +4\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{a}^{1}_{ij} \bigr\vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} \\ &{}+4\sum_{i=1}^{n}\sum _{j=1}^{m}\frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} \overline{\tilde{\sigma}}_{1j}}{1-\hat{\tilde{\sigma}}_{1j}} +4 \Biggl(\sum _{i=1}^{n} \sum_{j=1}^{m} \bigl\vert \tilde{\alpha}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \check{ \tilde{K}}_{1j}(\varepsilon) \Biggr)\check{\tilde{K}}_{1j}( \varepsilon) \\ &{}+4 \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{\beta}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} \check{\tilde{K}}_{1j}(\varepsilon) \Biggr)\check{ \tilde{K}}_{1j}(\varepsilon). \end{aligned}$$
(m2)
Proof
Let \((u_{1}^{*}, \ldots,u^{*}_{n},v_{1}^{*}, \ldots,v^{*}_{m})\) be the unique equilibrium state of system (2.1), and let \((u_{1}, \ldots,u_{n},v_{1}, \ldots,v_{m})\) be the solution to system (2.1) with initial data given by (2.2). Observe that \((\bar{u}_{1}(t), \ldots,\bar{u}_{n}(t),\bar{v}_{1}(t), \ldots,\bar{v}_{m}(t))=(u_{1}(t)-u_{1}^{*}, \ldots,u_{n}(t)-u_{n}^{*},v_{1}(v_{1})-v_{1}^{*}, \ldots,v_{m}(t)-v_{m}^{*})\) is the unique solution to the following initial value problem
$$\begin{aligned} \textstyle\begin{cases} d [\bar{u}_{i}(t)- \mu_{1i} \int_{t-\tau_{1i}}^{t}\bar{u}_{i}(\theta)\,d\theta ]\\ \quad = [-\mu_{1i}\bar{u}_{i}(t) +\sum_{j=1}^{m} a^{1}_{ij} (f_{1j}(v^{*}_{j}+\bar{v}_{j}(t)) -f_{1j}(v^{*}_{j}))\\ \qquad {}+\sum_{j=1}^{m}b^{1}_{ij} ( f_{1j}(v^{*}_{j}+\bar{v}_{j}(t-\sigma_{1j}(t))) -f_{1j}(v^{*}_{j}) ) \\ \qquad {}+\bigwedge_{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t} K_{1j}(t-s) f_{1j}(v^{*}_{j}+\bar{v}_{j}(s)) \,ds\\ \qquad {} -\bigwedge_{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t} K_{1j}(t-s) f_{1j}(v^{*}_{j}) \,ds\\ \qquad {} +\bigvee_{j=1}^{m}\beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}+\bar{v}_{j}(s)) \,ds \\ \qquad {}-\bigvee_{j=1}^{m}\beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}) \,ds]\,dt\\ \qquad {}+ [\sum_{j=1}^{m} \tilde{a}^{1}_{ij} (\tilde{f}_{1j}(v_{j}^{*}+\bar{v}_{j}(t))- \tilde{f}_{1j}(v_{j}^{*}))\\ \qquad {}+\sum_{j=1}^{m} \tilde{b}^{1}_{ij} (\tilde{f}_{1j}(v_{j}^{*}+\bar{v}_{j}(t-\tilde{\sigma}_{1j}(t))) -\tilde{f}_{1j}(v_{j}^{*}) )\\ \qquad {}+\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}\int_{-\infty}^{t} \tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*}+\bar{v}_{j}(s))\,ds\\ \qquad {}-\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}\int_{-\infty}^{t} \tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*})\,ds\\ \qquad {} + \bigvee_{j=1}^{m}\tilde{\beta}^{1}_{ij} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*}+\bar{v}_{j}(s))\,ds \\ \qquad {}-\bigvee_{j=1}^{m}\tilde{\beta}^{1}_{ij} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*})\,ds ]\,dB(t),\\ d [\bar{v}_{j}(t)- \mu_{2j} \int_{t-\tau_{2j}}^{t}\bar{v}_{j}(\theta)\,d\theta ]\\ \quad = [-\mu_{2j}\bar{v}_{j}(t) +\sum_{i=1}^{n} a^{2}_{ji} (f_{2i}(u^{*}_{i}+\bar{u}_{i}(t)) -f_{2i}(u^{*}_{i}))\\ \qquad {}+\sum_{i=1}^{n}b^{2}_{ji} ( f_{2i}(u^{*}_{i}+\bar{u}_{i}(t-\sigma_{2i}(t))) -f_{2i}(u^{*}_{i}) ) \\ \qquad {}+\bigwedge_{i=1}^{n}\alpha^{2}_{ji} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}+\bar{u}_{i}(s)) \,ds \\ \qquad {}-\bigwedge_{i=1}^{n}\alpha^{2}_{ji} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}) \,ds\\ \qquad {}+\bigvee_{i=1}^{n}\beta^{2}_{ji} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}+\bar{u}_{i}(s)) \,ds \\ \qquad {}-\bigvee_{i=1}^{n}\beta^{2}_{ji} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}) \,ds]\,dt\\ \qquad {}+ [\sum_{i=1}^{n} \tilde{a}^{2}_{ji} (\tilde{f}_{2i}(u_{i}^{*}+\bar{u}_{i}(t))- \tilde{f}_{2i}(u_{i}^{*})) \\ \qquad {}+\sum_{i=1}^{n} \tilde{b}^{2}_{ji} (\tilde{f}_{2i}(u_{i}^{*}+\bar{u}_{i}(t-\tilde{\sigma}_{2i}(t))) -\tilde{f}_{2i}(u_{i}^{*}) )\\ \qquad {}+\bigwedge_{i=1}^{n} \tilde{\alpha}^{2}_{ji}\int_{-\infty}^{t} \tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*}+\bar{u}_{i}(s))\,ds \\ \qquad {}-\bigwedge_{i=1}^{n} \tilde{\alpha}^{2}_{ji}\int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*})\,ds\\ \qquad {}+\bigvee_{i=1}^{n}\tilde{\beta}^{2}_{ji} \int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*}+\bar{u}_{i}(s))\,ds \\ \qquad {}-\bigvee_{i=1}^{n}\tilde{\beta}^{2}_{ji} \int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*})\,ds ]\,dB(t),\\ \bar{u}_{i}(s)=\phi_{i}(s) -u_{i}^{*},\qquad \bar{v}_{j}(s)=\psi_{j}(s)-v_{j}^{*},\quad \forall s\in(-\infty,0], \mathbb{P}\text{-a.s.},\\ \quad i=1,\ldots,n, j=1,\ldots,m. \end{cases}\displaystyle \end{aligned}$$
(3.2)
Let
$$U_{i}(t) = \textstyle\begin{cases} \bar{u}_{i}(t), &t< 0,\\ e^{\varepsilon t} \bar{u}_{i}(t), &t\geqslant0, \end{cases} $$
for \(i=1,\ldots,n\), and let
$$V_{j}(t) = \textstyle\begin{cases} \bar{v}_{j}(t), &t< 0,\\ e^{\varepsilon t} \bar{v}_{j}(t), &t\geqslant0, \end{cases} $$
for \(j=1,\ldots,m\). By Itô’s rule, we have
$$d\bar{u}_{i}(t)= e^{-\varepsilon t} \bigl(dU_{i}(t) - \varepsilon U_{i}(t)\,dt \bigr), $$
and
$$\begin{aligned} d \int_{t-\tau_{1i}}^{t}\bar{u}_{i}(s) \,ds&= d \int_{t-\tau_{1i}}^{t}e^{-\varepsilon s}U_{i}(s) \,ds \\ &= \bigl[e^{-\varepsilon t}U_{i}(t) -e^{-\varepsilon (t-\tau_{1i})}U_{i}(t- \tau_{1i}) \bigr]\,dt \\ &= e^{-\varepsilon t} \bigl[U_{i}(t) -e^{\varepsilon\tau_{1i}}U_{i}(t- \tau_{1i}) \bigr]\,dt \\ &= e^{-\varepsilon t} \biggl[ \bigl(1-e^{\varepsilon\tau_{1i}} \bigr)U_{i}(t) \,dt+ e^{\varepsilon\tau_{1i}}d \int_{t-\tau_{1i}}^{t} U_{i}(s) \,ds \biggr]. \end{aligned}$$
Therefore
$$\begin{aligned} &d\biggl[\bar{u}_{i}(t)- \mu_{1i} \int_{t-\tau_{1i}}^{t}\bar{u}_{i}(\theta)\,d\theta \biggr]+ \mu_{1i} \bar{u}_{i}(t)\,dt \\ &\quad = d \bar{u}_{i}(t)- \mu_{1i} d \int_{t-\tau_{1i}}^{t}\bar{u}_{i}(\theta)\,d\theta + \mu_{1i} \bar{u}_{i}(t)\,dt \\ &\quad = e^{-\varepsilon t} \bigl(dU_{i}(t) -\varepsilon U_{i}(t)\,dt \bigr)+ \mu_{1i}e^{-\varepsilon t} U_{i}(t)\,dt \\ &\qquad {}-\mu_{1i}e^{-\varepsilon t} \biggl[ \bigl(1-e^{\varepsilon\tau_{1i}} \bigr)U_{i}(t)\,dt+ e^{\varepsilon\tau_{1i}}d \int_{t-\tau_{1i}}^{t} U_{i}(s) \,ds \biggr] \\ &\quad =e^{-\varepsilon t} \biggl\{ d \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr] + \bigl( \mu_{1i} e^{\varepsilon\tau_{1i}}-\varepsilon \bigr)U_{i}(t) \,dt \biggr\} ,\quad i=1,\ldots,n. \end{aligned}$$
Similarly, we have
$$\begin{aligned} &d\biggl[\bar{v}_{j}(t)- \mu_{2j} \int_{t-\tau_{2j}}^{t}\bar{v}_{j}(\theta)\,d\theta \biggr]+ \mu_{2j} \bar{v}_{j}(t)\,dt \\ &\quad =e^{-\varepsilon t} \biggl\{ d \biggl[V_{j}(t)- \mu_{2j} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr] + \bigl( \mu_{2j} e^{\varepsilon\tau_{2j}}-\varepsilon \bigr)V_{j}(t) \,dt \biggr\} ,\quad j=1,\ldots,m. \end{aligned}$$
With these preparations in hand, we can deduce that, for every solution \((\bar{u}_{1},\ldots,\bar{u}_{n}, \bar{v}_{1},\ldots, \bar{v}_{m})\) to IVP (3.2), \((U_{1},\ldots,U_{n}, V_{1},\ldots,V_{m})\) is the unique solution to the initial value problem
$$\begin{aligned} \textstyle\begin{cases} d [U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds ]\\ \quad = [- (\mu_{1i} e^{\varepsilon\tau_{1i}}-\varepsilon )U_{i}(t) +\sum_{j=1}^{m} a^{1}_{ij}e^{\varepsilon t} (f_{1j}(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t)) -f_{1j}(v^{*}_{j}) ) \\ \qquad {}+\sum_{j=1}^{m}b^{1}_{ij} e^{\varepsilon t} ( f_{1j}(v^{*}_{j}+e^{-\varepsilon (t-\sigma_{1j}(t))}V_{j}(t-\sigma_{1j}(t))) -f_{1j}(v^{*}_{j}) ) \\ \qquad {}+\bigwedge_{j=1}^{m}\alpha^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)) \,ds\\ \qquad {} -\bigwedge_{j=1}^{m}\alpha^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}) \,ds\\ \qquad {}+\bigvee_{j=1}^{m}\beta^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)) \,ds\\ \qquad {}-\bigvee_{j=1}^{m}\beta^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j}(v^{*}_{j}) \,ds ]\,dt\\ \qquad {}+ [\sum_{j=1}^{m} \tilde{a}^{1}_{ij}e^{\varepsilon t} (\tilde{f}_{1j}(v_{j}^{*}+e^{-\varepsilon t}V_{j}(t))- \tilde{f}_{1j}(v_{j}^{*}) )\\ \qquad {}+\sum_{j=1}^{m} \tilde{b}^{1}_{ij}e^{\varepsilon t} ( \tilde{f}_{1j}(v_{j}^{*}+e^{-\varepsilon (t-\tilde{\sigma}_{1j}(t))}V_{j}(t-\tilde{\sigma}_{1j}(t))) -\tilde{f}_{1j}(v_{j}^{*}) ) \\ \qquad {}+\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}e^{\varepsilon t}\int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*}+e^{-\varepsilon s}V_{j}(s))\,ds\\ \qquad {}-\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}e^{\varepsilon t}\int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*})\,ds\\ \qquad {}+\bigvee_{j=1}^{m}\tilde{\beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*}+e^{-\varepsilon s}V_{j}(s))\,ds\\ \qquad {}-\bigvee_{j=1}^{m}\tilde{\beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}(v_{j}^{*})\,ds ]\,dB(t),\\ d [V_{j}(t)- \mu_{2j} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds ]\\ \quad = [- (\mu_{2j} e^{\varepsilon\tau_{2j}}-\varepsilon )V_{j}(t) +\sum_{i=1}^{n} a^{2}_{ji}e^{\varepsilon t} (f_{2i}(u^{*}_{i}+e^{-\varepsilon t}U_{i}(t)) -f_{2i}(u^{*}_{i}) ) \\ \qquad {}+\sum_{i=1}^{n}b^{2}_{ji}e^{\varepsilon t} (f_{2i}(u^{*}_{i}+e^{-\varepsilon (t-\sigma_{2i}(t))}U_{i}(t-\sigma_{2i}(t))) -f_{2i}(u^{*}_{i}) ) \\ \qquad {}+\bigwedge_{i=1}^{n}\alpha^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}+e^{-\varepsilon s}U_{i}(s)) \,ds\\ \qquad {} -\bigwedge_{i=1}^{n}\alpha^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}) \,ds\\ \qquad {} +\bigvee_{i=1}^{n}\beta^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}+e^{-\varepsilon s}U_{i}(s)) \,ds\\ \qquad {} -\bigvee_{i=1}^{n}\beta^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}K_{2i}(t-s) f_{2i}(u^{*}_{i}) \,ds ]\,dt\\ \qquad {}+ [\sum_{i=1}^{n} \tilde{a}^{2}_{ji}e^{\varepsilon t} (\tilde{f}_{2i}(u_{i}^{*}+e^{-\varepsilon t}U_{i}(t))- \tilde{f}_{2i}(u_{i}^{*}) )\\ \qquad {}+\sum_{i=1}^{n} \tilde{b}^{2}_{ji}e^{\varepsilon t} (\tilde{f}_{2i}(u_{i}^{*}+e^{-\varepsilon (t-\tilde{\sigma}_{2i}(t))}U_{i}(t-\tilde{\sigma}_{2i}(t))) -\tilde{f}_{2i}(u_{i}^{*}) ) \\ \qquad {}+\bigwedge_{i=1}^{n} \tilde{\alpha}^{2}_{ji}e^{\varepsilon t}\int_{-\infty}^{t} \tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*}+e^{-\varepsilon s}U_{i}(s))\,ds\\ \qquad {}-\bigwedge_{i=1}^{n} \tilde{\alpha}^{2}_{ji}e^{\varepsilon t}\int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*})\,ds\\ \qquad {} +\bigvee_{i=1}^{n}\tilde{\beta}^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*}+e^{-\varepsilon s}U_{i}(s))\,ds\\ \qquad {}-\bigvee_{i=1}^{n}\tilde{\beta}^{2}_{ji}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{2i}(t-s) \tilde{f}_{2i}(u_{i}^{*})\,ds ]\,dB(t), \\ U_{i}(s)=\phi_{i}(s) -u_{i}^{*},\qquad V_{j}(s)=\psi_{j}(s)-v_{j}^{*},\quad \forall s\in(-\infty,0],\ \mathbb{P}\text{-a.s.},\\ \quad i=1,\ldots,n, j=1,\ldots,m. \end{cases}\displaystyle \end{aligned}$$
(3.3)
Let \(\mathscr{V}(t;\varepsilon) =\mathbb{E}\sum_{k=1}^{4}\mathscr{V}_{k}(t;\varepsilon)\), where \(\mathscr{V}_{k}(t;\varepsilon)\) (\(k=1,2,3,4\)) are defined by
$$\begin{aligned} \mathscr{V}_{1}(t;\varepsilon)={}& \sum_{i=1}^{n} \biggl\vert U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr\vert ^{2}, \\ \mathscr{V}_{2}(t;\varepsilon)={}& \sum_{i=1}^{n} \sum_{j=1}^{m}\frac{ \vert b^{2}_{ji} \vert L_{2i}e^{\varepsilon \overline{\sigma}_{2i}}}{1-\hat{\sigma}_{2i}} \int^{t}_{t-\sigma_{2i}(t)} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i} \int^{+\infty}_{0}e^{\varepsilon s} K_{2i}(s) \int_{t-s}^{t} \bigl\vert U_{i}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \beta^{2}_{ji} \bigr\vert L_{2i} \int^{+\infty}_{0}e^{\varepsilon s} K_{2i}(s) \int_{t-s}^{t} \bigl\vert U_{i}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{} +\sum_{i=1}^{n}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i}e^{\varepsilon\tau_{1i}} -\varepsilon \bigr) \int^{t}_{t-\tau_{1i}} \int^{t}_{\tau} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{} +\sum_{i=1}^{n}\sum _{j=1}^{m}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl\vert a^{1}_{ij} \bigr\vert L_{1j} \int^{t}_{t-\tau_{1i}} \int^{t}_{\tau} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m}\mu_{1i}e^{\varepsilon\tau_{1i}} \bigl\vert b^{1}_{ij} \bigr\vert L_{1j}e^{\varepsilon\overline{\sigma}_{1j}} \int^{t}_{t-\tau_{1i}} \int^{t}_{\tau} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+ \sum_{i=1}^{n}\sum _{j=1}^{m} \frac{\mu_{2j}e^{\varepsilon\tau_{2j}} \vert b^{2}_{ji} \vert L_{2i} e^{\varepsilon\overline{\sigma}_{2i}} \tau_{2j}}{1-\hat{\sigma}_{2i}} \int^{t}_{t-\sigma_{2i}(t)} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j}\mu_{1i}e^{\varepsilon\tau_{1i}} \check{K}_{1j}(\varepsilon) \int^{t}_{t-\tau_{1i}} \int_{\tau}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i}\mu_{2j}e^{\varepsilon\tau_{2j}} \tau_{2j} \int^{+\infty}_{0}e^{\varepsilon s} K_{2i}(s) \int_{t-s}^{t} \bigl\vert U_{i}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \beta^{1}_{ij} \bigr\vert L_{1j}\mu_{1i}e^{\varepsilon\tau_{1i}} \check{K}_{1j}(\varepsilon) \int^{t}_{t-\tau_{1i}} \int_{\tau}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \beta^{2}_{ji} \bigr\vert L_{2i}\mu_{2j}e^{\varepsilon\tau_{2j}} \tau_{2j} \int^{+\infty}_{0}e^{\varepsilon s} K_{2i}(s) \int_{t-s}^{t} \bigl\vert U_{i}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+4 \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{m} \frac{ \vert \tilde{b}^{2}_{ji} \vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} }{1-\hat{\tilde{\sigma}}_{2i}} \Biggr)\sum _{i=1}^{n} \int^{t}_{t- \tilde{\sigma}_{2i}(t)} \bigl\vert U_{i}(s) \bigr\vert ^{2} \,ds \\ &{}+4 \Biggl(\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{\alpha}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2}\check{\tilde{K}}_{2i}(\varepsilon) \Biggr)\sum _{i=1}^{n} \int_{0}^{+\infty}\tilde{K}_{2i}(s) e^{\varepsilon s} \int^{t}_{t-s} \bigl\vert U_{i}(\theta) \bigr\vert ^{2} \,d\theta \,ds \\ &{}+4 \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{\beta}^{2}_{ji} \bigr\vert ^{2} \vert \tilde{L}_{2i} \vert ^{2}\check{\tilde{K}}_{2i}(\varepsilon) \Biggr)\sum _{i=1}^{n} \int_{0}^{+\infty}\tilde{K}_{2i}(s) e^{\varepsilon s} \int^{t}_{t-s} \bigl\vert U_{i}(\theta) \bigr\vert ^{2} \,d\theta \,ds, \\ \mathscr{V}_{3}(t;\varepsilon)={}& \sum_{j=1}^{m} \biggl\vert V_{j}(t)- \mu_{2j} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr\vert ^{2} , \end{aligned}$$
and
$$\begin{aligned} \mathscr{V}_{4}(t;\varepsilon) ={}& \sum_{j=1}^{m} \sum_{i=1}^{n}\frac{ \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon \overline{\sigma}_{1j}}}{1-\hat{\sigma}_{1j}} \int^{t}_{t-\sigma_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2}\,ds \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int_{t-s}^{t} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \beta^{1}_{ij} \bigr\vert L_{1j} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int_{t-s}^{t} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{j=1}^{m}\mu_{2j}e^{\varepsilon\tau_{2j}} \bigl(\mu_{2j}e^{\varepsilon\tau_{2j}} -\varepsilon\bigr) \int^{t}_{t-\tau_{2j}} \int^{t}_{\tau} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n}\mu_{2j}e^{\varepsilon\tau_{2j}} \bigl\vert a^{2}_{ji} \bigr\vert L_{2i} \int^{t}_{t-\tau_{2j}} \int^{t}_{\tau} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+ \sum_{j=1}^{m}\sum _{i=1}^{n}\frac{\mu_{1i}e^{\varepsilon\tau_{1i}} \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon\overline{\sigma}_{1j}} \tau_{1i}}{1-\hat{\sigma}_{1j}} \int^{t}_{t-\sigma_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2}\,ds \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n}\mu_{2j}e^{\varepsilon\tau_{2j}} \bigl\vert b^{2}_{ji} \bigr\vert L_{2i}e^{\varepsilon\overline{\sigma}_{2i}} \int^{t}_{t-\tau_{2j}} \int^{t}_{\tau} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \alpha^{1}_{ij} \bigr\vert L_{1j}\mu_{1i}e^{\varepsilon\tau_{1i}} \tau_{1i} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int_{t-s}^{t} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \alpha^{2}_{ji} \bigr\vert L_{2i}\mu_{2j}e^{\varepsilon\tau_{2j}} \check{K}_{2i}(\varepsilon) \int^{t}_{t-\tau_{2j}} \int_{\tau}^{t} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \beta^{1}_{ij} \bigr\vert L_{1j}\mu_{1i}e^{\varepsilon\tau_{1i}} \tau_{1i} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int_{t-s}^{t} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &{}+\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert \beta^{2}_{ji} \bigr\vert L_{2i}\mu_{2j}e^{\varepsilon\tau_{2j}} \check{K}_{2i}(\varepsilon) \int^{t}_{t-\tau_{2j}} \int_{\tau}^{t} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds\,d\tau \\ &{}+4 \Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}\frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} }{1-\hat{\tilde{\sigma}}_{1j}}\Biggr)\sum _{j=1}^{m} \int^{t}_{t-\tilde{\sigma}_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds \\ &{}+4 \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert \tilde{\alpha}^{1}_{ij} \bigr\vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} \check{\tilde{K}}_{1j}(\varepsilon)\Biggr) \sum _{j=1}^{m} \int_{0}^{+\infty}\tilde{K}_{1j}(s) e^{\varepsilon s} \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2} \,d\theta \,ds \\ &{}+4 \Biggl(\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{\beta}^{1}_{1j} \bigr\vert ^{2} \vert \tilde{L}_{1j} \vert ^{2}\check{\tilde{K}}_{1j}(\varepsilon) \Biggr)\sum _{j=1}^{m} \int_{0}^{+\infty}\tilde{K}_{1j}(s) e^{\varepsilon s} \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds. \end{aligned}$$
Let us denote
$$ \textstyle\begin{cases} d [U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds ]= \mathfrak{a}_{1i}(t)\,dt +\mathfrak{a}_{2i}(t)\,dB(t),& i=1,\ldots,n,\\ d [V_{j}(t)- \mu_{2j} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds ]= \mathfrak{b}_{1j}(t)\,dt +\mathfrak{b}_{2j}(t)\,dB(t),& j=1,\ldots,m. \end{cases} $$
(3.4)
By Itô’s formula,
$$\begin{aligned} d\mathscr{V}_{1}(t;\varepsilon)= {}&d\sum _{i=1}^{n} \biggl\vert U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr\vert ^{2} \\ ={}&\sum_{i=1}^{n} \biggl\{ 2 \mathfrak{a}_{1i}(t) \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr]+ \bigl\vert \mathfrak{a}_{2i}(t) \bigr\vert ^{2} \biggr\} \,dt \\ &{}+2\sum_{i=1}^{n} \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr] \mathfrak{a}_{2i}(t)\,dB(t). \end{aligned}$$
(3.5)
Thanks to (3.3) and (3.4), we have
$$\begin{aligned} & \sum_{i=1}^{n}\mathfrak{a}_{1i}(t) \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr] \\ &\quad = - \sum_{i=1}^{n} \bigl( \mu_{1i} e^{\varepsilon\tau_{1i}}-\varepsilon \bigr) \bigl\vert U_{i}(t) \bigr\vert ^{2} \\ &\qquad {}+\sum _{i=1}^{n} \sum_{j=1}^{m} a^{1}_{ij}e^{\varepsilon t}U_{i}(t) \bigl(f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t) \bigr) -f_{1j}\bigl(v^{*}_{j}\bigr) \bigr) \\ &\qquad {} +\sum_{i=1}^{n} \sum _{j=1}^{m}b^{1}_{ij} e^{\varepsilon t}U_{i}(t) \bigl( f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon (t-\sigma_{1j}(t))}V_{j} \bigl(t-\sigma_{1j}(t)\bigr)\bigr) -f_{1j} \bigl(v^{*}_{j}\bigr) \bigr) \\ &\qquad {} +\sum_{i=1}^{n} e^{\varepsilon t}U_{i}(t) \Biggl[ \bigwedge _{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \\ & \qquad {}+\sum_{i=1}^{n}e^{\varepsilon t}U_{i}(t) \Biggl[ \bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \\ &\qquad {} +\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr)U_{i}(t) \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \\ &\qquad {} -\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} a^{1}_{ij} e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \bigl(f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t) \bigr) -f_{1j}\bigl(v^{*}_{j}\bigr) \bigr) \\ &\qquad {} -\sum_{i=1}^{n} \sum _{j=1}^{m}\mu_{1i} e^{\varepsilon\tau_{1i}} b^{1}_{ij} e^{\varepsilon t} \\ &\qquad {} \times \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \bigl( f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon (t-\sigma_{1j}(t))}V_{j}\bigl(t- \sigma_{1j}(t)\bigr)\bigr)-f_{1j}\bigl(v^{*}_{j} \bigr) \bigr) \\ &\qquad {} -\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}}e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \\ &\qquad {} \times \Biggl[ \bigwedge_{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds -\bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \\ &\qquad {} -\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \\ &\qquad {} \times \Biggl[ \bigvee _{j=1}^{m}\beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr]. \end{aligned}$$
(3.6)
Let us spare some space to analyze the right-hand side of (3.6) term-by-term. Firstly, we have
$$\begin{aligned} &2 \Biggl\vert \sum_{i=1}^{n} \sum _{j=1}^{m} a^{1}_{ij}e^{\varepsilon t}U_{i}(t) \bigl(f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t) \bigr) -f_{1j}\bigl(v^{*}_{j}\bigr) \bigr) \Biggr\vert \\ &\quad \leqslant2\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert a^{1}_{ij} \bigr\vert e^{\varepsilon t} \bigl\vert U_{i}(t) \bigr\vert \bigl\vert f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t) \bigr) -f_{1j}\bigl(v^{*}_{j}\bigr) \bigr\vert \\ &\quad \leqslant\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum _{j=1}^{m}\sum_{i=1}^{n} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \bigl\vert V_{j}(t) \bigr\vert ^{2}, \end{aligned}$$
(3.7)
where the first inequality follows from the Cauchy–Schwarz inequality. We have
$$\begin{aligned} &2 \Biggl\vert \sum_{i=1}^{n} \sum _{j=1}^{m}b^{1}_{ij} e^{\varepsilon t}U_{i}(t) \bigl( f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon (t-\sigma_{1j}(t))}V_{j} \bigl(t-\sigma_{1j}(t)\bigr)\bigr) -f_{1j} \bigl(v^{*}_{j}\bigr) \bigr) \Biggr\vert \\ &\quad \leqslant2\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert e^{\varepsilon \sigma_{1j}(t)} \bigl\vert U_{i}(t) \bigr\vert \bigl\vert V_{j}\bigl(t-\sigma_{1j}(t)\bigr) \bigr\vert \\ &\quad \leqslant\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert e^{\varepsilon \bar{\sigma}_{1j} } \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert e^{\varepsilon \bar{\sigma}_{1j}} \bigl\vert V_{j}\bigl(t- \sigma_{1j}(t)\bigr) \bigr\vert ^{2} \\ &\quad =\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert e^{\varepsilon \bar{\sigma}_{1j} } \bigl\vert U_{i}(t) \bigr\vert ^{2} -\sum_{j=1}^{m}\sum _{i=1}^{n} \frac{ \vert L_{1j}b^{1}_{ij} \vert e^{\varepsilon \bar{\sigma}_{1j}} }{1-\hat{\sigma}_{1j}}\,\frac{d}{dt} \int^{t}_{t-\sigma_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds \\ &\qquad {}+\sum_{j=1}^{m}\sum _{i=1}^{n}\frac{ \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon \overline{\sigma}_{1j}} (\dot{\sigma}_{1j}(t)- \hat{\sigma}_{1j} )}{1-\hat{\sigma}_{1j}} \bigl\vert V_{j}\bigl(t- \sigma_{1j}(t)\bigr) \bigr\vert ^{2}\,dt \\ &\qquad {}+\sum_{j=1}^{m}\sum _{i=1}^{n} \frac{ \vert L_{1j}b^{1}_{ij} \vert e^{\varepsilon \bar{\sigma}_{1j}} \bar{\sigma}_{1j}}{1-\hat{\sigma}_{1j}} \bigl\vert V_{j}(t) \bigr\vert ^{2}, \end{aligned}$$
(3.8)
where the first inequality follows from the triangle inequality and the definition of \(L_{1j}\), the second inequality follows from the Cauchy–Schwarz inequality, and the equality follows from some routine calculations. Further, we have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n}e^{\varepsilon t}U_{i}(t) \Biggl[ \bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {}-\bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds\Biggr] \Biggr\vert \\ &\quad \leqslant 2 \sum_{i=1}^{n}e^{\varepsilon t} \bigl\vert U_{i}(t) \bigr\vert \Biggl\vert \bigwedge _{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr\vert \\ &\quad \leqslant 2 \sum_{i=1}^{n}\sum _{j=1}^{m}L_{1j} \bigl\vert U_{i}(t) \bigr\vert \bigl\vert \alpha^{1}_{ij} \bigr\vert \int_{-\infty}^{t}e^{\varepsilon (t-s)}K_{1j}(t-s) \bigl\vert V_{j}(s) \bigr\vert \,ds \\ &\quad \leqslant \sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert L_{1j} \alpha^{1}_{ij} \bigr\vert \check{K}_{1j}( \varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum _{j=1}^{m}\sum_{i=1}^{n} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \bigl\vert V_{j}(t-s) \bigr\vert ^{2}\,ds \\ &\quad = \sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert L_{1j} \alpha^{1}_{ij} \bigr\vert \check{K}_{1j}( \varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum _{j=1}^{m}\sum_{i=1}^{n} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon) \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert L_{1j} \alpha^{1}_{ij} \bigr\vert \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds, \end{aligned}$$
(3.9)
where the second inequality follows from Lemma 2.2, the third inequality follows from the Cauchy–Schwarz inequality, and the equality follows from some routine calculations. Similarly, we have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n}e^{\varepsilon t}U_{i}(t) \Biggl[ \bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad -\bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \Biggr\vert \\ &\quad \leqslant \sum_{i=1}^{n}\sum _{j=1}^{m} \bigl\vert L_{1j} \beta^{1}_{ij} \bigr\vert \check{K}_{1j}( \varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum _{j=1}^{m}\sum_{i=1}^{n} \bigl\vert L_{1j}\beta^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon) \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{j=1}^{m}\sum _{i=1}^{n} \bigl\vert L_{1j} \beta^{1}_{ij} \bigr\vert \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds. \end{aligned}$$
(3.10)
We further have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr)U_{i}(t) \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \Biggr\vert \\ &\quad \leqslant 2\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr) \bigl\vert U_{i}(t) \bigr\vert \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert \,ds \\ &\quad \leqslant \sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr) \int_{t-\tau_{1i}}^{t} \bigl( \bigl\vert U_{i}(s) \bigr\vert ^{2}+ \bigl\vert U_{i}(t) \bigr\vert ^{2} \bigr)\,ds \\ &\quad = 2\sum_{i=1}^{n} \tau_{1i} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}-\varepsilon \bigr) \bigl\vert U_{i}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl(\mu_{1i} e^{\varepsilon\tau_{1i}}- \varepsilon \bigr) \,\frac{d}{dt} \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds, \end{aligned}$$
(3.11)
where the second inequality follows from the Cauchy–Schwarz inequality, and the equality follows from some routine calculations. We have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} a^{1}_{ij} e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \bigl(f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon t}V_{j}(t) \bigr) -f_{1j}\bigl(v^{*}_{j}\bigr) \bigr) \Biggr\vert \\ &\quad \leqslant 2\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert \,ds \bigl\vert V_{j}(t) \bigr\vert \\ & \quad \leqslant \sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds +\sum_{j=1}^{m} \sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}}\tau_{1i} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\quad = \sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds +\sum _{j=1}^{m}\sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}}\tau_{1i} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}a^{1}_{ij} \bigr\vert \, \frac{d}{dt} \int_{t-\tau_{1i}}^{t} \int^{t}_{\tau} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds\,d\tau, \end{aligned}$$
(3.12)
where the first inequality follows from the triangle inequality, the second inequality follows from the Cauchy–Schwarz inequality, and the equality follows from some routine calculations. By some calculations as in (3.8) and (3.12), we obtain
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n} \sum _{j=1}^{m}\mu_{1i} e^{\varepsilon\tau_{1i}} b^{1}_{ij} e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \bigl( f_{1j}\bigl(v^{*}_{j}+e^{-\varepsilon (t-\sigma_{1j}(t))}V_{j}\bigl(t- \sigma_{1j}(t)\bigr)\bigr) -f_{1j}\bigl(v^{*}_{j} \bigr) \bigr) \Biggr\vert \\ &\quad \leqslant \sum_{i=1}^{n} \sum _{j=1}^{m}\mu_{1i} e^{\varepsilon\tau_{1i}}e^{\varepsilon\bar{\sigma}_{1j}} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert \tau_{1i} \bigl\vert U_{i}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{i=1}^{n} \sum _{j=1}^{m}\mu_{1i} e^{\varepsilon\tau_{1i}}e^{\varepsilon\bar{\sigma}_{1j}} \bigl\vert L_{1j}b^{1}_{ij} \bigr\vert \, \frac{d}{dt} \int_{t-\tau_{1i}}^{t} \int^{t}_{\tau} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds\,d\tau \\ &\qquad {} +\sum_{j=1}^{m}\sum _{i=1}^{n} \frac{\mu_{1i} e^{\varepsilon\tau_{1i}}e^{\varepsilon\bar{\sigma}_{1j}}\tau_{1i} \vert L_{1j}b^{1}_{ij} \vert (\dot{\sigma}_{1j} -\hat{\sigma}_{1j} )}{1-\hat{\sigma}_{1j}} \bigl\vert V_{j}\bigl(t-\sigma_{1j}(t)\bigr) \bigr\vert ^{2} \\ &\qquad {} - \sum_{j=1}^{m}\sum _{i=1}^{n}\frac{\mu_{1i}e^{\varepsilon\tau_{1i}} \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon\overline{\sigma}_{1j}} \tau_{1i}}{1-\hat{\sigma}_{1j}} \,\frac{d}{dt} \int^{t}_{t-\sigma_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2}\,ds \\ &\qquad {} + \sum_{j=1}^{m}\sum _{i=1}^{n}\frac{\mu_{1i}e^{\varepsilon\tau_{1i}} \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon\overline{\sigma}_{1j}} \tau_{1i}\overline{\sigma}_{1j}}{1-\hat{\sigma}_{1j}} \bigl\vert V_{j}(t) \bigr\vert ^{2}. \end{aligned}$$
(3.13)
We have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}}e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \Biggl[ \bigwedge_{j=1}^{m}\alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigwedge_{j=1}^{m} \alpha^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \Biggr\vert \\ &\quad \leqslant 2\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert \,ds \int_{-\infty}^{t}e^{\varepsilon (t-s)} K_{1j}(t-s) \bigl\vert V_{j}(s) \bigr\vert \,ds \\ &\quad \leqslant \sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon) \int_{t-\tau_{1i}}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds \\ &\qquad {} +\sum_{j=1}^{m}\sum _{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \tau_{1i} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \bigl\vert V_{j}(t-s) \bigr\vert ^{2}\,ds \\ &\quad = \sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon) \tau_{1i} \bigl\vert U_{i}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{1i} e^{\varepsilon\tau_{1i}} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon)\,\frac{d}{dt} \int_{t-\tau_{1i}}^{t} \int_{\tau}^{t} \bigl\vert U_{i}(s) \bigr\vert ^{2}\,ds\,d\tau \\ & \qquad {}-\sum_{j=1}^{m}\sum _{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \tau_{1i} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s} K_{1j}(s) \int_{t-s}^{t} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &\qquad {} +\sum_{j=1}^{m}\sum _{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} \tau_{1i} \bigl\vert L_{1j}\alpha^{1}_{ij} \bigr\vert \check{K}_{1j}(\varepsilon) \bigl\vert V_{j}(t) \bigr\vert ^{2}, \end{aligned}$$
(3.14)
where the first inequality follows from Lemma 2.2, the second inequality follows from the Cauchy–Schwarz inequality, and the equality follows from some routine calculations. Similarly, we have
$$\begin{aligned} & 2 \Biggl\vert \sum_{i=1}^{n} \mu_{1i} e^{\varepsilon\tau_{1i}} e^{\varepsilon t} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \\ &\qquad {}\times\Biggl[ \bigvee _{j=1}^{m}\beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}+e^{-\varepsilon s}V_{j}(s)\bigr) \,ds \\ &\qquad {} -\bigvee_{j=1}^{m} \beta^{1}_{ij} \int_{-\infty}^{t}K_{1j}(t-s) f_{1j} \bigl(v^{*}_{j}\bigr) \,ds \Biggr] \Biggr\vert \\ &\quad \leqslant \sum_{j=1}^{m}\sum _{i=1}^{n} \mu_{2j} e^{\varepsilon\tau_{2j}} \bigl\vert L_{2i}\alpha^{2}_{ji} \bigr\vert \check{K}_{2i}(\varepsilon) \tau_{2j} \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ & \qquad {}-\sum_{j=1}^{m}\sum _{i=1}^{n} \mu_{2j} e^{\varepsilon\tau_{2j}} \bigl\vert L_{2i}\alpha^{2}_{ji} \bigr\vert \check{K}_{2i}(\varepsilon)\,\frac{d}{dt} \int_{t-\tau_{2j}}^{t} \int_{\tau}^{t} \bigl\vert V_{j}(s) \bigr\vert ^{2}\,ds\,d\tau \\ &\qquad {}-\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{2j} e^{\varepsilon\tau_{2j}} \tau_{2j} \bigl\vert L_{2i}\alpha^{2}_{ji} \bigr\vert \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s} K_{2i}(s) \int_{t-s}^{t} \bigl\vert U_{i}(\theta) \bigr\vert ^{2}\,d\theta \,ds \\ &\qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{m} \mu_{2j} e^{\varepsilon\tau_{2j}} \tau_{2j} \bigl\vert L_{2i}\alpha^{2}_{ji} \bigr\vert \check{K}_{2i}(\varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2}. \end{aligned}$$
(3.15)
Now, we estimate the term \(\sum_{i=1}^{n}|\mathfrak{u}_{2i}(t)|^{2} \). By the inequality \((a+b+c+d)^{2}\leqslant 4(a^{2}+b^{2}+c^{2}+d^{2})\) we have
$$\begin{aligned} \sum_{i=1}^{n} \bigl\vert \mathfrak{u}_{2i}(t) \bigr\vert ^{2} \leqslant{}& 4\sum _{i=1}^{n} \Biggl\vert \sum _{j=1}^{m} \tilde{a}^{1}_{ij}e^{\varepsilon t} \bigl(\tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon t}V_{j}(t) \bigr)- \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr) \bigr) \Biggr\vert ^{2} \\ &{} +4\sum_{i=1}^{n} \Biggl\vert \sum _{j=1}^{m} \tilde{b}^{1}_{ij}e^{\varepsilon t} \bigl( \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon (t-\tilde{\sigma}_{1j}(t))}V_{j} \bigl(t-\tilde{\sigma}_{1j}(t)\bigr)\bigr) -\tilde{f}_{1j} \bigl(v_{j}^{*}\bigr) \bigr) \Biggr\vert ^{2} \\ &{} +4\sum_{i=1}^{n} \Biggl\vert \bigwedge_{j=1}^{m} \tilde{ \alpha}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon s}V_{j}(s) \bigr)\,ds \\ &{} -\bigwedge_{j=1}^{m} \tilde{ \alpha}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr)\,ds \Biggr\vert ^{2} \\ &{} +4\sum_{i=1}^{n} \Biggl\vert \bigvee _{j=1}^{m}\tilde{\beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon s} V_{j}(s)\bigr)\,ds \\ &{} -\bigvee_{j=1}^{m}\tilde{ \beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr)\,ds \Biggr\vert ^{2}. \end{aligned}$$
(3.16)
We have
$$\begin{aligned} & \sum_{i=1}^{n} \Biggl\vert \sum _{j=1}^{m} \tilde{a}^{1}_{ij}e^{\varepsilon t} \bigl(\tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon t}V_{j}(t) \bigr)- \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr) \bigr) \Biggr\vert ^{2} \\ &\quad \leqslant \sum_{i=1}^{n} \Biggl(\sum _{j=1}^{m} \bigl\vert \tilde{a}^{1}_{ij} \bigr\vert \tilde{L}_{1j} \bigl\vert V_{j}(t) \bigr\vert \Biggr)^{2} \\ &\quad \leqslant \Biggl(\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{L}_{1j} \tilde{a}^{1}_{ij} \bigr\vert ^{2} \Biggr)\sum _{j=1}^{m} \bigl\vert V_{j}(t) \bigr\vert ^{2}, \end{aligned}$$
(3.17)
where the first inequality follows from the triangle inequality and the definition of \(\tilde{L}_{1j}\), and the second inequality follows from the Cauchy–Schwarz inequality. By mimicking the steps in (3.8) we have
$$\begin{aligned} & \sum_{i=1}^{n} \Biggl\vert \sum _{j=1}^{m} \tilde{b}^{1}_{ij}e^{\varepsilon t} \bigl( \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon (t-\tilde{\sigma}_{1j}(t))}V_{j} \bigl(t-\tilde{\sigma}_{1j}(t)\bigr)\bigr) -\tilde{f}_{1j} \bigl(v_{j}^{*}\bigr) \bigr) \Biggr\vert ^{2} \\ &\quad \leqslant \sum_{j=1}^{m} \sum _{i=1}^{n} \bigl\vert \tilde{L}_{1j} \tilde{b}^{1}_{ij} \bigr\vert ^{2} e^{2\varepsilon \bar{\tilde{\sigma}}_{1j} }\sum_{j=1}^{m} \bigl\vert V_{j}\bigl(t-\tilde{\sigma}_{1j}(t)\bigr) \bigr\vert ^{2} \\ &\quad = \Biggl( \sum_{j=1}^{m} \sum _{i=1}^{n}\frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} (\dot{\tilde{\sigma}}_{1j}(t) -\hat{\tilde{\sigma}}_{1j} )}{1-\hat{\tilde{\sigma}}_{1j}} \Biggr)\sum _{j=1}^{m} \bigl\vert V_{j} \bigl(t-\tilde{\sigma}_{1j}(t)\bigr) \bigr\vert ^{2} \\ &\qquad {}- \Biggl( \sum_{j=1}^{m} \sum _{i=1}^{n}\frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} }{1-\hat{\tilde{\sigma}}_{1j}} \Biggr) \sum _{j=1}^{m}\,\frac{d}{dt} \int^{t}_{t- \tilde{\sigma}_{1j}(t)} \bigl\vert V_{j}(s) \bigr\vert ^{2} \,ds \\ &\qquad {}+ \Biggl( \sum_{j=1}^{m} \sum _{i=1}^{n}\frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} \overline{\tilde{\sigma}}_{1j} }{1-\hat{\tilde{\sigma}}_{1j}} \Biggr) \sum _{j=1}^{m} \bigl\vert V_{j}(t) \bigr\vert ^{2}, \end{aligned}$$
(3.18)
where the inequality follows from the triangle inequality and the definition of \(\tilde{L}_{1j}\). We have
$$\begin{aligned} & \sum_{i=1}^{n} \Biggl\vert \bigwedge _{j=1}^{m} \tilde{\alpha}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon s}V_{j}(s) \bigr)\,ds \\ & \qquad{}-\bigwedge_{j=1}^{m} \tilde{\alpha}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr)\,ds \Biggr\vert ^{2} \\ &\quad \leqslant \sum_{i=1}^{n} \Biggl[\sum _{j=1}^{m}e^{\varepsilon t} \bigl\vert \tilde{\alpha}^{1}_{ij} \bigr\vert \biggl\vert \int_{-\infty}^{t} \tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon s}V_{j}(s) \bigr) \,ds \\ & \qquad {}- \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr)\,ds \biggr\vert \Biggr]^{2} \\ &\quad \leqslant \sum_{i=1}^{n} \Biggl[ \sum_{j=1}^{m} \bigl\vert \tilde{L}_{1j}\tilde{\alpha}^{1}_{ij} \bigr\vert \int_{-\infty}^{t}e^{\varepsilon(t-s)}\tilde{K}_{1j}(t-s) \bigl\vert V_{j}(s) \bigr\vert \,ds \Biggr]^{2} \\ &\quad \leqslant \sum_{i=1}^{n} \Biggl[ \sum_{j=1}^{m} \bigl\vert \tilde{L}_{1j}\tilde{\alpha}^{1}_{ij} \bigr\vert \biggl(\check{\tilde{K}}_{1j}(\varepsilon) \int^{+\infty}_{0}e^{\varepsilon s}\tilde{K}_{1j}(s) \bigl\vert V_{j}(t-s) \bigr\vert ^{2}\,ds \biggr)^{\frac{1}{2}} \Biggr]^{2} \\ &\quad \leqslant \sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{L}_{1j} \tilde{\alpha}^{1}_{ij} \bigr\vert ^{2} \Biggl[ \sum_{j=1}^{m} \check{\tilde{K}}_{1j}( \varepsilon) \int^{+\infty}_{0}e^{\varepsilon s}\tilde{K}_{1j}(s) \bigl\vert V_{j}(t-s) \bigr\vert ^{2}\,ds \Biggr] \\ &\quad = \sum_{j=1}^{m} \check{ \tilde{K}}_{1j}(\varepsilon) \Biggl(\sum_{i=1}^{n} \sum_{j=1}^{m} \bigl\vert \tilde{L}_{1j}\tilde{\alpha}^{1}_{ij} \bigr\vert ^{2}\check{\tilde{K}}_{1j}(\varepsilon) \Biggr) \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\qquad {}- \Biggl(\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{L}_{1j} \tilde{\alpha}^{1}_{ij} \bigr\vert ^{2}\check{ \tilde{K}}_{1j}(\varepsilon) \Biggr)\sum_{j=1}^{m} \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s}\tilde{K}_{1j}(s) \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds, \end{aligned}$$
(3.19)
where the second inequality follows from the Lemma 2.2, the third inequality (resp., fourth) follows from the Cauchy–Schwarz inequality for functions (resp., for finite sequences), and the equality follows from some routine calculations. Similarly, we have
$$\begin{aligned} & \sum_{i=1}^{n} \Biggl\vert \bigvee _{j=1}^{m}\tilde{\beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}+e^{-\varepsilon s} V_{j}(s)\bigr)\,ds \\ & \qquad {}-\bigvee_{j=1}^{m}\tilde{ \beta}^{1}_{ij}e^{\varepsilon t} \int_{-\infty}^{t}\tilde{K}_{1j}(t-s) \tilde{f}_{1j}\bigl(v_{j}^{*}\bigr)\,ds \Biggr\vert ^{2} \\ &\quad \leqslant \sum_{j=1}^{m} \bigl\vert \check{\tilde{K}}_{1j}(\varepsilon) \bigr\vert ^{2} \Biggl(\sum_{i=1}^{n} \sum _{j=1}^{m} \bigl\vert \tilde{L}_{1j} \tilde{\beta}^{1}_{ij} \bigr\vert ^{2} \Biggr) \bigl\vert V_{j}(t) \bigr\vert ^{2} \\ &\qquad {} -\sum_{j=1}^{m} \check{ \tilde{K}}_{1j}(\varepsilon) \Biggl(\sum_{i=1}^{n} \sum_{j=1}^{m} \bigl\vert \tilde{L}_{1j}\tilde{\beta}^{1}_{ij} \bigr\vert ^{2} \Biggr) \,\frac{d}{dt} \int^{+\infty}_{0}e^{\varepsilon s}\tilde{K}_{1j}(s) \int^{t}_{t-s} \bigl\vert V_{j}(\theta) \bigr\vert ^{2}\,d\theta \,ds. \end{aligned}$$
(3.20)
Combining (3.4), (3.5), (3.6), (3.7), (3.8), (3.9), (3.10), (3.11), (3.12), (3.13), (3.14), (3.15), (3.16), (3.17), (3.18), (3.19), (3.20) and utilizing the symmetry of \(U_{i}(t)\) and \(V_{j}(t)\), by some calculations we obtain
$$\begin{aligned} &d \bigl[\mathscr{V}_{1}(t;\varepsilon) +\mathscr{V}_{2}(t; \varepsilon) +\mathscr{V}_{3}(t;\varepsilon) +\mathscr{V}_{4}(t; \varepsilon) \bigr] \\ &\quad \leqslant \sum_{i=1}^{n}M_{1i}( \varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2}\,dt + \sum_{j=1}^{m}M_{2j}(\varepsilon) \bigl\vert V_{j}(t) \bigr\vert ^{2}\,dt \\ &\qquad {}+\sum_{i=1}^{n}\sum _{j=1}^{m}\frac{ \vert b^{2}_{ji} \vert L_{2i}e^{\varepsilon \overline{\sigma}_{2i}} (\mu_{2j}e^{\varepsilon\tau_{2j}}+1 ) (\dot{\sigma}_{2i}(t)-\hat{\sigma}_{2i} ) }{1-\hat{\sigma}_{2i}} \bigl\vert U_{i}\bigl(t- \sigma_{2i}(t)\bigr) \bigr\vert ^{2}\,dt \\ &\qquad {}+\sum_{j=1}^{m}\sum _{i=1}^{n}\frac{ \vert b^{1}_{ij} \vert L_{1j}e^{\varepsilon \overline{\sigma}_{1j}} (\mu_{1i}e^{\varepsilon\tau_{1i}}+1 ) (\dot{\sigma}_{1j}(t)- \hat{\sigma}_{1j} )}{1-\hat{\sigma}_{1j}} \bigl\vert V_{j}\bigl(t- \sigma_{1j}(t)\bigr) \bigr\vert ^{2}\,dt \\ &\qquad {}+4 \Biggl[\sum_{i=1}^{n}\sum _{j=1}^{m} \frac{ \vert \tilde{b}^{2}_{ji} \vert ^{2} \vert \tilde{L}_{2i} \vert ^{2} (\dot{\tilde{\sigma}}_{2i}(t)-\hat{\tilde{\sigma}}_{2i} ) }{1-\hat{\tilde{\sigma}}_{2i}} \Biggr]\sum _{i=1}^{n} \bigl\vert U_{i} \bigl(t- \tilde{\sigma}_{2i}(t)\bigr) \bigr\vert ^{2}\,dt \\ &\qquad {}+4 \Biggl[\sum_{j=1}^{m}\sum _{i=1}^{n} \frac{ \vert \tilde{b}^{1}_{ij} \vert ^{2} \vert \tilde{L}_{1j} \vert ^{2} (\dot{\tilde{\sigma}}_{1j}(t)- \hat{\tilde{\sigma}}_{1j} ) }{1-\hat{\tilde{\sigma}}_{1j}} \Biggr]\sum _{j=1}^{m} \bigl\vert V_{j} \bigl(t- \tilde{\sigma}_{1j}(t)\bigr) \bigr\vert ^{2}\,dt \\ &\qquad {}+2\sum_{i=1}^{n} \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr] \mathfrak{a}_{2i}(t)\,dB(t) \\ &\qquad {}+2\sum_{j=1}^{m} \biggl[V_{j}(t)- \mu_{1i} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr] \mathfrak{b}_{2j}(t)\,dB(t) \\ &\quad \leqslant \sum_{i=1}^{n}M_{1i}( \varepsilon) \bigl\vert U_{i}(t) \bigr\vert ^{2}\,dt + \sum_{j=1}^{m}M_{2j}(\varepsilon) \bigl\vert V_{j}(t) \bigr\vert ^{2}\,dt \\ &\qquad {}+2\sum_{i=1}^{n} \biggl[U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr] \mathfrak{a}_{2i}(t)\,dB(t) \\ &\qquad {}+2\sum_{j=1}^{m} \biggl[V_{j}(t)- \mu_{1i} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr] \mathfrak{b}_{2j}(t)\,dB(t), \\ &\quad \forall t\in[0,+\infty),\mathbb{P}\text{-a.s.} \end{aligned}$$
Therefore
$$\begin{aligned} \frac{d}{dt}\mathscr{V}(t;\varepsilon) \leqslant \sum _{i=1}^{n}M_{1i}(\varepsilon)\mathbb{E} \bigl\vert U_{i}(t) \bigr\vert ^{2} +\sum _{j=1}^{m}M_{2j}(\varepsilon)\mathbb{E} \bigl\vert V_{j}(t) \bigr\vert ^{2}, \quad\forall t \in[0,+\infty). \end{aligned}$$
Since \(M_{1i}(\varepsilon)\) and \(M_{2j}(\varepsilon)\) are continuous and \(M_{1i}(0)<0\) and \(M_{2j}(0)<0\), there exists \(\varepsilon^{*}\in(0,\bar{\varepsilon})\) such that \(M_{1i}(\varepsilon^{*})\leqslant0\) and \(M_{2j}(\varepsilon^{*})\leqslant0\), and therefore \(\tilde{\mathscr{V}}(t;\varepsilon^{*})\) is decreasing. This, together with the definition of \(\tilde{\mathscr{V}}(t;\varepsilon^{*})\), directly implies
$$\begin{aligned} &\mathbb{E} \sum_{i=1}^{n} \biggl\vert U_{i}(t)- \mu_{1i} e^{\varepsilon\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr\vert ^{2}+\mathbb{E} \sum_{j=1}^{m} \biggl\vert V_{j}(t)- \mu_{2j} e^{\varepsilon\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr\vert ^{2} \\ &\quad \leqslant \mathscr{V}\bigl(t;\varepsilon^{*}\bigr) \leqslant \mathscr{V} \bigl(0;\varepsilon^{*}\bigr), \quad\forall t\in[0,+\infty). \end{aligned}$$
Applying the inequality \((a+b)^{2}\leqslant 2(a^{2}+b^{2})\), we have
$$\begin{aligned} &\mathbb{E} \Biggl( \sum_{i=1}^{n} \bigl\vert U_{i}(t) \bigr\vert ^{2} + \sum _{j=1}^{m} \bigl\vert V_{j}(t) \bigr\vert ^{2} \Biggr) \\ &\quad \leqslant 2\mathbb{E} \sum_{i=1}^{n} \biggl\vert U_{i}(t)- \mu_{1i} e^{ \varepsilon_{0}\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr\vert ^{2} +2\mathbb{E}\sum_{i=1}^{n} \biggl\vert \mu_{1i} e^{\varepsilon_{0}\tau_{1i}} \int_{t-\tau_{1i}}^{t}U_{i}(s)\,ds \biggr\vert ^{2} \\ &\qquad {}+2\mathbb{E} \sum_{j=1}^{m} \biggl\vert V_{j}(t)- \mu_{2j} e^{ \varepsilon_{0}\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr\vert ^{2} +2\mathbb{E}\sum_{j=1}^{m} \biggl\vert \mu_{2j} e^{\varepsilon_{0}\tau_{2j}} \int_{t-\tau_{2j}}^{t}V_{j}(s)\,ds \biggr\vert ^{2} \\ &\quad \leqslant 2\mathscr{V}\bigl(0;\varepsilon^{*}\bigr) +2 \max \bigl( ( \mu_{1i} )^{2}e^{2\varepsilon_{0}\tau_{1i}} \tau_{1i}, ( \mu_{2j} )^{2}e^{2\varepsilon_{0}\tau_{2j}} \tau_{2j} \bigr) \\ &\qquad {} \times \int_{t-\max (\bigvee _{i=1}^{n}\tau_{1i}, \bigvee _{j=1}^{m}\tau_{2j} )}^{t} \sum_{i=1}^{n} \mathbb{E} \Biggl( \sum_{i=1}^{n} \bigl\vert U_{i}(t) \bigr\vert ^{2} + \sum _{j=1}^{m} \bigl\vert V_{j}(t) \bigr\vert ^{2} \Biggr)\,ds, \quad \forall t\in[0,+\infty). \end{aligned}$$
By Gronwall’s lemma we have
$$\begin{aligned} &\mathbb{E} \Biggl( \sum_{i=1}^{n} \bigl\vert U_{i}(t) \bigr\vert ^{2} + \sum _{j=1}^{m} \bigl\vert V_{j}(t) \bigr\vert ^{2} \Biggr) \leqslant C,\quad\forall t\in[0,+\infty), \end{aligned}$$
or equivalently
$$\begin{aligned} &\mathbb{E} \Biggl( \sum_{i=1}^{n} \bigl\vert u_{i}(t) -u_{i}^{*} \bigr\vert ^{2} + \sum _{j=1}^{m} \bigl\vert v_{j}(t)-v_{j}^{*} \bigr\vert ^{2} \Biggr) \leqslant Ae^{-2\varepsilon_{0}t}, \quad\forall t \in[0,+\infty), \end{aligned}$$
where
$$\begin{aligned} A&= 2\mathscr{V}\bigl(0;\varepsilon^{*}\bigr)\exp \Biggl(\max \bigl( ( \mu_{1i} )^{2}e^{2\varepsilon_{0}\tau_{1i}} \tau_{1i}, ( \mu_{2j} )^{2}e^{2\varepsilon_{0}\tau_{2j}} \tau_{2j} \bigr) \max \Biggl(\bigvee_{i=1}^{n}\tau_{1i}, \bigvee_{j=1}^{m}\tau_{2j} \Biggr) \Biggr) \\ &\leqslant\tilde{A} \sup_{s\in(-\infty,0]}\mathbb{E} \bigl\vert \bigl( \phi_{1}(\cdot,s),\ldots,\phi_{n}(\cdot,s), \psi_{1}(\cdot,s),\ldots,\psi_{m}(\cdot,s)\bigr) \bigr\vert ^{2}, \end{aligned}$$
where à is independent of the initial data \((\phi_{1},\ldots,\phi_{n}, \psi_{1},\ldots,\psi_{m})\). The proof is complete. □