In this section, based on Mawhin’s continuation theorem, we study the existence of at least one periodic solution of (1). To do so, we shall make some preparations.
Let \(\mathbb{X}\), \(\mathbb{Y}\) be two Banach space, \(L:\operatorname{Dom}L\subset \mathbb{X}\rightarrow\mathbb{Y}\) be a linear mapping, and \(N:\mathbb{X}\rightarrow\mathbb{Y}\) be a continuous mapping. Then L is called a Fredholm mapping of index zero if \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L<+\infty \) and ImL is closed in \(\mathbb{Y}\). If L is a Fredholm mapping of index zero and there exist continuous projectors \(P:\mathbb{X}\rightarrow\mathbb{X}\) and \(Q:\mathbb {Y}\rightarrow\mathbb{Y}\) such that \(\operatorname{Im}P=\operatorname{Ker}L\), \(\operatorname{Ker}Q=\operatorname{Im}(I-Q)\), then the mapping \(L|_{\operatorname{Dom}L\cap\operatorname{Ker}P}: (I-P)\mathbb{X}\rightarrow \operatorname{Im}L\) is invertible. We denote its inverse by \(K_{p}\). If Ω is an open bounded subset of \(\mathbb{X}\), then the mapping N is called L-compact on Ω̅ if \(QN(\overline{\Omega})\) is bounded and \(K_{p}(I-Q)N: \overline{\Omega}\rightarrow\mathbb{X}\) is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism \(J: \operatorname{Im}Q\rightarrow\operatorname{Ker}L\).
Lemma 3.1
[30]
Let
\(\mathbb{X}\), \(\mathbb{Y}\)
be two Banach spaces, and let
\(\Omega\subset\mathbb{X}\)
be open bounded. Suppose that
\(L:\operatorname{Dom}L\subset\mathbb {X}\rightarrow\mathbb{Y}\)
is a linear Fredholm operator of index zero with
\(\operatorname{Dom}L\cap\overline{\Omega}\neq\phi\)
and
\(N:\overline{\Omega}\rightarrow\mathbb{Y}\)
is
L-compact. Furthermore, suppose that:
-
(a)
for each
\(\lambda\in(0,1)\), \(x\in\partial\Omega\cap\operatorname{Dom}L\), \(Lx\neq\lambda Nx\);
-
(b)
for each
\(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNx\neq 0\);
-
(c)
\(\operatorname{deg}\{JQNx,\Omega\cap\operatorname{Ker}L,0\}\neq0\).
Then the equation
\(Lx=Nx\)
has at least one solution in
\(\overline{\Omega}\cap\operatorname{Dom}L\), where Ω̅ is the closure of Ω, and
∂Ω is the boundary of Ω.
Definition 3.1
A real matrix \(A=(a_{ij})_{n\times n}\) is said to be a nonsingular M-matrix if \(a_{ij}\le0\), \(i,j=1,2,\ldots,n\) and all successive principal minors of A are positive.
Theorem 3.1
Under conditions (A1)-(A5), let
H
be a nonsingular
M-matrix of the form
$$ H=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} H_{1}& H_{2} \\ H_{3}& H_{4} \end{array}\displaystyle \right ), $$
where
$$\begin{aligned}& H_{1}=\operatorname{diag}\{\underline{a}_{1}- \underline{a}_{1}\omega\overline {a}_{1} \delta_{1},\ldots, \underline{a}_{n}-\underline{a}_{n} \omega\overline{a}_{n}\delta_{n}\}, \\& H_{2}=(h_{ij})_{m\times n},\qquad H_{3}= \bigl(h_{ji}'\bigr)_{n\times m}, \\& h_{ij}=- \biggl(\frac {1}{\varrho_{i}} +\underline{a}_{i}\omega \biggr)\frac{\omega}{\sqrt{2}} \overline{a}_{i}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji})\kappa _{j} \overline{b}_{j}\delta_{j}', \\& h_{ji}'=- \biggl(\frac{1}{\varrho_{i}'} + \underline{b}_{j}\omega \biggr)\frac{\omega}{\sqrt{2}} \overline{b}_{j}( \overline{p}_{ij}+\overline{q}_{ij})\nu_{i}\overline {a}_{i}\delta_{i}, \\& H_{4}=\operatorname{diag}\bigl\{ \underline{b}_{1}- \underline{b}_{1}\omega\overline {b}_{1} \delta_{1}',\ldots, \underline{b}_{m}- \underline{b}_{m}\omega\overline{b}_{m}\delta_{m}' \bigr\} . \end{aligned}$$
Then system (1) has at least one
ω-periodic solution.
Proof
Let \(C[0,\omega;t_{1},\ldots,t_{q}]_{\mathbb{T}}\) = {\(u:[0,\omega]_{\mathbb{T}}\rightarrow R^{n+m}\) is a piecewise continuous map with first-class discontinuity points in \([0,\omega]_{\mathbb{T}}\cap\{t_{k}\}\), and at each discontinuity point, it is continuous on the left}. Take
$$\mathbb{X}=\bigl\{ u\in C[0,\omega;t_{1},\ldots,t_{q}]_{\mathbb{T}}|u(t+ \omega )=u(t)\bigr\} , \qquad \mathbb{Y}=\mathbb{X}\times\mathbb{R}^{(n+m)\times(q+1)} $$
with the norm \(\|u\|_{\mathbb{X}}=\sum_{i=1}^{n}|x_{i}|_{0}+\sum_{j=1}^{m}|y_{j}|_{0}\), where \(|x_{i}|_{0}=\max_{t\in[0,\omega]_{T}}|x_{i}(t)|\) and \(|y_{j}|_{0}=\max_{t\in[0,\omega]_{T}}|y_{j}(t)|\). Then \(\mathbb{X}\) is a Banach space.
Set
$$L: \operatorname{Dom}L\cap\mathbb{X}\rightarrow\mathbb{Y},\qquad u\rightarrow \bigl(u^{\Delta}, \Delta u(t_{1}),\ldots,\Delta u(t_{q}), 0\bigr)\quad \mbox{and}\quad N:\mathbb {X}\rightarrow \mathbb{X}, $$
where
$$\begin{aligned}& Nu=\left ( \left( \textstyle\begin{array}{@{}c@{}} A_{1}(t)\\ \vdots\\ \vdots\\ \vdots\\ A_{n+m}(t) \end{array}\displaystyle \right), \left( \textstyle\begin{array}{@{}c@{}} I_{11}( x_{1}(t_{1}))\\ \vdots\\ I_{n1}(x_{n}(t_{1}))\\ J_{11}( y_{1}(t_{1}))\\ \vdots\\ J_{m1}(y_{m}(t_{1})) \end{array}\displaystyle \right), \ldots, \left( \textstyle\begin{array}{@{}c@{}} I_{1q}( x_{1}(t_{q}))\\ \vdots\\ I_{nq}(x_{n}(t_{q}))\\ J_{1q}( y_{1}(t_{q}))\\ \vdots\\ J_{mq}(y_{m}(t_{q})) \end{array}\displaystyle \right), \left( \textstyle\begin{array}{@{}c@{}} 0\\ \vdots\\ 0\\ 0\\ \vdots\\ 0 \end{array}\displaystyle \right) \right ), \\& A_{i}(t)=-a_{i}\bigl(x_{i}(t)\bigr) \Biggl[c_{i}\bigl(x_{i}(t)\bigr) -\bigwedge _{j=1}^{m}\alpha_{ji}(t)f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr) -\bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t-\tau _{ji})\bigr)+E_{i}(t)\Biggr], \\& A_{n+j}(t)=-b_{j}\bigl(y_{j}(t)\bigr) \Biggl[d_{j}\bigl(y_{j}(t)\bigr) -\bigwedge _{i=1}^{n}p_{ij}(t)g_{i} \bigl(x_{i}(t-\sigma_{ij})\bigr) -\bigvee_{i=1}^{n}q_{ij}(t)g_{i} \bigl(x_{i}(t-\sigma _{ij})\bigr)+F_{j}(t)\Biggr]. \end{aligned}$$
It is easy to see that
$$\begin{aligned}& \operatorname{Ker}L=\bigl\{ x\in\mathbb{X}:x=h\in\mathbb{R}^{n+m}\bigr\} , \\& \operatorname{Im}L= \Biggl\{ z=(f,c_{1},\ldots,c_{q},d)\in \mathbb{Y}: \int_{0}^{\omega}f(s)\Delta s+\sum _{k=1}^{q}C_{k}+d=0 \Biggr\} . \end{aligned}$$
Thus, \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L=n+m\). So, ImL is closed in \(\mathbb{Y}\), and L is a Fredholm mapping of index zero. Define the project operators P and Q as
$$\begin{aligned}& Px=\frac{1}{\omega} \int_{0}^{\omega}u(t)\Delta t, \quad x\in\mathbb{X}, \\& Qz=Q(f,C_{1},\ldots,C_{q},d)= \Biggl(\frac{1}{\omega} \Biggl[ \int_{0}^{\omega}f(s)\Delta s+\sum _{k=1}^{q}C_{k}+d \Biggr],0,\ldots,0,0 \Biggr). \end{aligned}$$
Obviously, P and Q are continuous projectors and satisfy
$$\operatorname{Im}P=\operatorname{Ker}L,\qquad \operatorname{Im}L= \operatorname{Ker}Q=\operatorname{Im}(I-Q). $$
Denoting \(L_{p}^{-1}=L|_{\operatorname{Dom}L\cap\operatorname{Ker}P}\) and generalized inverse by \(K_{p}=L_{P}^{-1}\), we have
$$(K_{p}z) (t)= \int_{0}^{t}f(s)\Delta s+\sum _{t>t_{k}}C_{k}-\frac{1}{\omega} \int_{0}^{\omega}\int_{0}^{t}f(s)\Delta s\Delta t-\sum _{k=1}^{q}C_{k}. $$
Similarly to [31], it is not difficult to show that \(QN(\overline{\Omega})\) and \(K_{P} (I-Q)N(\overline{\Omega})\) are relatively compact for any open bounded set \(\Omega\subset\mathbb{X}\). Therefore, N is L-compact on Ω̅ for any open bounded set \(\Omega\subset\mathbb{X}\).
Now, to apply Lemma 3.1, we only need to look for an appropriate open bounded subset Ω. Correspondingly to the operator equation \(Lx= \lambda Nx\), \(\lambda\in(0, 1)\), we have
$$ \left \{ \textstyle\begin{array}{l} x_{i}^{\Delta}(t)= \lambda\{-a_{i}(x_{i}(t))[c_{i}(x_{i}(t)) -\bigwedge_{j=1}^{m}\alpha_{ji}(t)f_{j}(y_{j}(t-\tau_{ji})) \\ \hphantom{x_{i}^{\Delta}(t)={}}{}-\bigvee_{j=1}^{m}\beta_{ji}(t)f_{j}(y_{j}(t-\tau_{ji}))+E_{i}(t) ]\},\quad t\in\mathbb{T}^{+}, t\neq t_{k}, \\ \Delta_{i}(x_{i}(t_{k})) = \lambda I_{ik}(x_{i}(t_{k})),\quad k\in\mathbb{N}, i=1,2,\ldots,n; \\ y_{j}^{\Delta}(t)= \lambda\{-b_{j}(y_{j}(t))[d_{j}(y_{j}(t)) -\bigwedge_{i=1}^{n}p_{ij}(t)g_{i}(x_{i}(t-\sigma_{ij})) \\ \hphantom{y_{j}^{\Delta}(t)={}}{}-\bigvee_{i=1}^{n}q_{ij}(t)g_{i}(x_{i}(t-\sigma_{ij}))+F_{j}(t)] \},\quad t\in\mathbb{T}^{+}, t\neq t_{k}, \\ \Delta_{j}(y_{j}(t_{k})) = \lambda J_{jk}(y_{j}(t_{k})),\quad k\in\mathbb{N}, j=1,2,\ldots,m. \end{array}\displaystyle \right . $$
(3)
Suppose that \(u = (x_{1}, \ldots, x_{n},y_{1},\ldots,y_{m})^{T}\) is a solution of system (3) for a certain \(\lambda\in(0,1)\). Multiplying both sides of the first and third equations in system (3) by \(x_{i}^{\Delta}\) and \(y_{j}^{\Delta}\), respectively, and integrating over \([0,\omega]_{\mathbb{T}}\), we get
$$\begin{aligned}& \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert ^{2}\Delta t \\& \quad \le \overline{a}_{i} \int_{0}^{\omega}\Biggl\vert c_{i} \bigl(x_{i}(t)\bigr) -\bigwedge_{j=1}^{m} \alpha_{ji}(t)f_{j}\bigl(y_{j}(t- \tau_{ji})\bigr) \\& \qquad {}-\bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t-\tau _{ji})\bigr)+E_{i}(t)\Biggr\vert x_{i}^{\Delta}(t) \Delta t \\& \quad \le \overline{a}_{i}\Biggl[ \int_{0}^{\omega}\bigl\vert c_{i} \bigl(x_{i}(t)\bigr)-c_{i}(0)\bigr\vert x_{i}^{\Delta}(t)\Delta t \\& \qquad {}+ \int_{0}^{\omega}\Biggl\vert \bigwedge _{j=1}^{m}\alpha_{ji}(t)f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr) -\bigwedge _{j=1}^{m}\alpha_{ji}(t)g_{j}(0) \Biggr\vert x_{i}^{\Delta}(t)\Delta t \\& \qquad {}+ \int_{0}^{\omega}\Biggl\vert \bigvee _{j=1}^{m}\beta_{ji}(t)f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr) -\bigvee _{j=1}^{m}\beta_{ji}(t)g_{j}(0) \Biggr\vert x_{i}^{\Delta}(t)\Delta t \\& \qquad {}+\overline{E}_{i} \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t\Biggr]. \end{aligned}$$
In view of Lemma 2.4, Lemma 2.7, and (A2)-(A4), we have
$$\begin{aligned}& \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert ^{2}\Delta t \\& \quad \le \overline{a}_{i}\Biggl[\delta_{i} \int_{0}^{\omega}\bigl\vert x_{i}(t)\bigr\vert \bigl\vert x_{i}^{\Delta}(t)\bigr\vert \Delta t+\sum _{j=1}^{m}(\overline{\alpha}_{ji}+ \overline{\beta}_{ji}) \biggl( \int_{0}^{\omega}\bigl\vert f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr)\bigr\vert ^{2} \Delta t \biggr)^{1/2} \\& \qquad {}\times \biggl( \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert ^{2}\Delta t \biggr)^{1/2} +\overline{E}_{i} \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t\Biggr] \\& \quad \le \overline{a}_{i} \Biggl[\delta_{i} \|x_{i}\|_{2}\bigl\Vert x_{i}^{\Delta}\bigr\Vert _{2} +\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji})\sqrt{\omega }M_{j}\bigl\Vert x_{i}^{\Delta}\bigr\Vert _{2} +\overline{E}_{i}\sqrt{\omega}\bigl\Vert x_{i}^{\Delta}\bigr\Vert _{2} \Biggr], \end{aligned}$$
namely,
$$ \bigl\Vert x_{i}^{\Delta}\bigr\Vert _{2}\le\overline{a}_{i} \Biggl[\delta_{i} \|x_{i}\|_{2} +\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji})\sqrt{ \omega}M_{j} +\overline{E}_{i}\sqrt{\omega} \Biggr]:= \overline{a}_{i}\delta_{i}\|x_{i} \|_{2}+B_{i}, $$
(4)
where \(B_{i}=\overline{a}_{i} [\sum_{j=1}^{m}(\overline{\alpha }_{ji}+\overline{\beta}_{ji})\sqrt{\omega}M_{j} +\overline{E}_{i}\sqrt{\omega} ]\). Similarly, we obtain that
$$ \bigl\Vert y_{j}^{\Delta}\bigr\Vert _{2}\le\overline{b}_{j} \Biggl[\delta_{j}' \|y_{j}\|_{2} +\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij})\sqrt{ \omega}N_{i} +\overline{F}_{j}\sqrt{\omega} \Biggr]:= \overline{b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}', $$
(5)
where \(B_{j}'=\overline{b}_{j} [\sum_{i=1}^{n}(\overline{p}_{ij}+\overline {q}_{ij})\sqrt{\omega}N_{i} +\overline{F}_{j}\sqrt{\omega} ]\). Setting \(t_{0}=t_{0}^{+}=0\) and \(t_{q+1}=\omega\), in view of (3), (A2)-(A5), Lemma 2.4, and Lemma 2.7, we have
$$\begin{aligned}& \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t \\& \quad = \sum_{k=1}^{q+1} \int_{t_{k-1}^{+}}^{t_{k}}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t+\sum_{k=1}^{q}\bigl\vert I_{k}\bigl(x_{i}(t_{k})\bigr)\bigr\vert \\& \quad \le \overline{a}_{i}\Biggl[ \int_{0}^{\omega}\bigl\vert c_{i} \bigl(x_{i}(t)\bigr)\bigr\vert \Delta t + \int_{0}^{\omega}\Biggl\vert \bigwedge _{j=1}^{m}\alpha _{ji}(t)f_{j} \bigl(y_{j}(t)\bigr)-\bigwedge_{j=1}^{m} \alpha_{ji}(t)f_{j}(0)\Biggr\vert \Delta t \\& \qquad {}+ \int_{0}^{\omega}\Biggl\vert \bigwedge _{j=1}^{m}\alpha_{ji}(t)f_{j} \bigl(y_{j}(t-\tau _{ji})\bigr)-\bigwedge _{j=1}^{m}\alpha_{ji}(t)f_{j}(t) \Biggr\vert \Delta t \\& \qquad {}+ \int_{0}^{\omega}\Biggl\vert \bigvee _{j=1}^{m}\beta_{ji}(t)f_{j} \bigl(y_{j}(t)\bigr)-\bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}(0)\Biggr\vert \Delta t \\& \qquad {}+ \int_{0}^{\omega}\Biggl\vert \bigvee _{j=1}^{m}\beta_{ji}(t)f_{j} \bigl(y_{j}(t-\tau _{ji})\bigr)-\bigvee _{j=1}^{m}\beta_{ji}(t)f_{j}(t) \Biggr\vert \Delta t+\overline{E}_{i}\omega\Biggr]+q \overline{I}_{k} \\& \quad \le \overline{a}_{i}\Biggl[ \int_{0}^{\omega}\bigl\vert c_{i} \bigl(x_{i}(t)\bigr)\bigr\vert \Delta t+\sum _{j=1}^{m}(\overline{\alpha}_{ji}+\overline{ \beta}_{ji}) \int_{0}^{\omega}\bigl\vert f_{j} \bigl(y_{j}(t)\bigr)\bigr\vert \Delta t \\& \qquad {}+\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta }_{ji}) \int_{0}^{\omega}\bigl\vert f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr)-f_{j} \bigl(y_{j}(t)\bigr)\bigr\vert \Delta t+\overline{E}_{i} \omega\Biggr]+q\overline{I}_{k} \\& \quad \le \overline{a}_{i}\Biggl[\delta_{i}\sqrt{\omega} \|x_{i}\|_{2}+\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji})\omega M_{j} \\& \qquad {} +\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta }_{ji})\frac{\kappa_{j}}{\sqrt{2}} \omega^{3/2}\bigl\Vert y_{j}^{\Delta}\bigr\Vert _{2}+\overline {E}_{i}\omega\Biggr]+q\overline{I}_{k} \\& \quad \le \overline{a}_{i}\Biggl[\delta_{i}\sqrt{\omega} \|x_{i}\|_{2}+\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji})\omega M_{j} \\& \qquad {}+\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta }_{ji})\frac{\kappa_{j}}{\sqrt{2}} \omega^{3/2}\bigl(\overline{b}_{j}\delta_{j}' \| y_{j}\|_{2}+B_{j}'\bigr) + \overline{E}_{i}\omega\Biggr]+q\overline{I}_{k} \end{aligned}$$
(6)
and
$$\begin{aligned}& \int_{0}^{\omega}\bigl\vert y_{j}^{\Delta}(t) \bigr\vert \Delta t \\& \quad \le \overline{b}_{j}\Biggl[\delta_{j}' \sqrt{\omega}\|y_{j}\|_{2}+\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij})\omega N_{i} \\& \qquad {}+\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij})\frac{\nu _{i}}{\sqrt{2}} \omega^{3/2}\bigl(\overline{a}_{i}\delta_{i}\Vert x_{i}\Vert _{2}+B_{i}\bigr) + \overline{F}_{j}\omega\Biggr]+q\overline{J}_{k}. \end{aligned}$$
(7)
Integrating both sides of (3) from 0 to ω, we have
$$\begin{aligned}& \biggl\vert \int_{0}^{\omega}a_{i}\bigl(x_{i}(t) \bigr)c_{i}\bigl(x_{i}(t)\bigr)\Delta t \biggr\vert \\& \quad =\Biggl\vert \int_{0}^{\omega}a_{i}\bigl(x_{i}(t) \bigr)\Biggl[\bigwedge_{j=1}^{m}\alpha _{ji}(t)f_{j}\bigl(y_{j}(t-\tau_{ji}) \bigr) \\& \qquad {}+\bigwedge_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t-\tau _{ji})\bigr)+E_{i}(t)\Biggr]\Delta t + \sum _{k=1}^{q}I_{k}\bigl(x_{i}(t_{k}) \bigr)\Biggr\vert \\& \quad \le\overline{a}_{i} \int_{0}^{\omega}\Biggl[\Biggl\vert \bigwedge _{j=1}^{m}\alpha _{ji}(t)f_{j} \bigl(y_{j}(t)\bigr)-\bigwedge_{j=1}^{m} \alpha_{ji}(t)f_{j}(0)\Biggr\vert \\& \qquad {}+\Biggl\vert \bigwedge_{j=1}^{m} \alpha_{ji}(t)f_{j}\bigl(y_{j}(t-\tau _{ji})\bigr)-\bigwedge_{j=1}^{m} \alpha_{ji}(t)f_{j}\bigl(y_{j}(t)\bigr)\Biggr\vert \\& \qquad {}+\Biggl\vert \bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t)\bigr)-\bigvee _{j=1}^{m}\beta _{ji}(t)f_{j}(0) \Biggr\vert \\& \qquad {}+\Biggl\vert \bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t-\tau _{ji})\bigr)-\bigvee_{j=1}^{m} \beta_{ji}(t)f_{j}\bigl(y_{j}(t)\bigr)\Biggr\vert +E_{i}(t)\Biggr]\Delta t +\sum_{k=1}^{q}\bigl\vert I_{k}\bigl(x_{i}(t_{k})\bigr)\bigr\vert \\& \quad \le\overline{a}_{i}\Biggl[\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline {\beta}_{ji}) \int_{0}^{\omega}\bigl\vert f_{j} \bigl(y_{j}(t)\bigr)\bigr\vert \Delta t \\& \qquad {}+\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta }_{ji}) \int_{0}^{\omega}\bigl\vert f_{j} \bigl(y_{j}(t-\tau_{ji})\bigr)-f_{j} \bigl(y_{j}(t)\bigr)\bigr\vert \Delta t+\overline{E}_{i} \omega\Biggr] +\sum_{k=1}^{q}\bigl\vert I_{k}\bigl(x_{i}(t_{k})\bigr)\bigr\vert \\& \quad \le\overline{a}_{i}\Biggl[\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j}+\sum_{j=1}^{m}(\overline{ \alpha}_{ji} +\overline{\beta}_{ji})\frac{\kappa_{j}}{\sqrt{2}} \omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}' \|y_{j}\|_{2}+B_{j}'\bigr) +\overline{E}_{i}\omega\Biggr]+q\rho_{k} \end{aligned}$$
and
$$\begin{aligned}& \biggl\vert \int_{0}^{\omega}b_{i}\bigl(y_{j}(t) \bigr)d_{j}\bigl(y_{j}(t)\bigr)\Delta t \biggr\vert \\& \quad \le\overline{b}_{j}\Biggl[\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij})\omega N_{i}+ \sum_{i=1}^{n}(\overline{p}_{ij} + \overline{q}_{ij})\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {a}_{i}\delta_{i}\|x_{i}\|_{2}+B_{i}\bigr) +\overline{F}_{j}\omega\Biggr]+q\rho_{k}'. \end{aligned}$$
Applying Lemma 2.6 and (A3), we obtain
$$\begin{aligned}& \biggl\vert \int_{0}^{\omega}a_{i}\bigl(x_{i}(t) \bigr)x_{i}(t)\Delta t\biggr\vert \\& \quad \le \frac{\overline{a}_{i}}{\varrho_{i}}\Biggl[\sum_{j=1}^{m}( \overline {\alpha}_{ji}+\overline{\beta}_{ji})\omega M_{j}+\sum_{j=1}^{m}(\overline { \alpha}_{ji} +\overline{\beta}_{ji})\frac{\kappa_{j}}{\sqrt{2}} \omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}' \|y_{j}\|_{2}+B_{j}'\bigr) +\overline{E}_{i}\omega\Biggr]+\frac{1}{\varrho}_{i}q \rho_{k} \end{aligned}$$
(8)
and
$$\begin{aligned}& \biggl\vert \int_{0}^{\omega}b_{i}\bigl(y_{j}(t) \bigr)y_{j}(t)\Delta t \biggr\vert \\& \quad \le\frac{\overline{b}_{j}}{\varrho_{j}'}\Biggl[\sum_{i=1}^{n}( \overline {p}_{ij}+\overline{q}_{ij})\omega N_{i}+ \sum_{i=1}^{n}(\overline{p}_{ij} + \overline{q}_{ij})\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {a}_{i}\delta_{i}\Vert x_{i}\Vert _{2}+B_{i}\bigr) +\overline{F}_{j}\omega\Biggr]+\frac{1}{\varrho_{j}'}q \rho_{k}'. \end{aligned}$$
(9)
From Lemma 2.3, for any \(t_{1}^{i},t_{2}^{i}, t_{3}^{j},t_{4}^{j}\in[0,\omega]_{\mathbb{T}}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,m\), we have
$$ \left \{ \textstyle\begin{array}{l} \int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t\le\int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t_{1}^{i})\Delta t \\ \hphantom{\int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t\le{}}{}+\int_{0}^{\omega}a_{i}(x_{i}(t)) (\int_{0}^{\omega}|x_{i}^{\Delta}(t)|\Delta t )\Delta t, \\ \int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t \le \int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t_{2}^{i})\Delta t \\ \hphantom{\int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t \le{}}{}-\int_{0}^{\omega}a_{i}(x_{i}(t)) (\int_{0}^{\omega}|x_{i}^{\Delta}(t)|\Delta t )\Delta t \end{array}\displaystyle \right . $$
(10)
and
$$ \left \{ \textstyle\begin{array}{l} \int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t \le \int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t_{3}^{j})\Delta t \\ \hphantom{\int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t \le{}}{}+\int_{0}^{\omega}b_{j}(y_{j}(t)) (\int_{0}^{\omega}|y_{j}^{\Delta}(t)|\Delta t )\Delta t, \\ \int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t \le \int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t_{4}^{j})\Delta t \\ \hphantom{\int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t \le{}}{}-\int_{0}^{\omega}b_{j}(y_{j}(t)) (\int_{0}^{\omega}|y_{j}^{\Delta}(t)|\Delta t )\Delta t. \end{array}\displaystyle \right . $$
(11)
Dividing by \(\int_{0}^{\omega}a_{i}(x_{i}(t))\Delta t\) and \(\int_{0}^{\omega}b_{j}(y_{j}(t))\Delta t\) both sides of (10) and (11), respectively, we obtain, for \(i=1,2,\ldots,n\),
$$ \left \{ \textstyle\begin{array}{l} x_{i}(t_{1}^{i}) \ge \frac{1}{\int_{0}^{\omega}a_{i}(x_{i}(t))\Delta t}\int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t-\int_{0}^{\omega}|x_{i}^{\Delta}(t)|\Delta t, \\ x_{i}(t_{2}^{i}) \le \frac{1}{\int_{0}^{\omega}a_{i}(x_{i}(t))\Delta t}\int_{0}^{\omega}a_{i}(x_{i}(t))x_{i}(t)\Delta t+\int_{0}^{\omega}|x_{i}^{\Delta}(t)|\Delta t, \end{array}\displaystyle \right . $$
(12)
and, for \(j=1,2,\ldots,m\),
$$ \left \{ \textstyle\begin{array}{l} y_{j}(t_{3}^{j}) \ge \frac{1}{\int_{0}^{\omega}b_{j}(y_{j}(t))\Delta t}\int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t-\int_{0}^{\omega}|y_{j}^{\Delta}(t)|\Delta t, \\ y_{j}(t_{4}^{j}) \le \frac{1}{\int_{0}^{\omega}b_{j}(y_{j}(t))\Delta t}\int_{0}^{\omega}b_{j}(y_{j}(t))y_{j}(t)\Delta t+\int_{0}^{\omega}|y_{j}^{\Delta}(t)|\Delta t. \end{array}\displaystyle \right . $$
(13)
Let \(\overline{t}_{i},\underline{t}_{i},\overline{t}_{j}',\underline{t}_{j}'\in [0,\omega]_{\mathbb{T}}\) be such that \(x_{i}(\overline{t}_{i})=\max_{t\in[0,\omega]_{\mathbb{T}}}x_{i}(t)\), \(x_{i}(\underline{t}_{i})=\min_{t\in[0,\omega]_{\mathbb{T}}}x_{i}(t)\), \(y_{j}(\overline{t}_{j}')=\max_{t\in[0,\omega]_{\mathbb{T}}}y_{j}(t)\), \(y_{j}(\underline{t}_{j}')=\min_{t\in[0,\omega]_{\mathbb{T}}}y_{j}(t)\). From (12) we have, for \(i=1,2,\ldots,n\),
$$\begin{aligned} x_{i}(\underline{t}_{i}) \ge&-\frac{1}{\underline{a}_{i}\omega}\biggl\vert \int _{0}^{\omega}a_{i}\bigl(x_{i}(t) \bigr)x_{i}(t)\Delta t\biggr\vert - \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t \\ \ge&-\frac{1}{\underline{a}_{i}\varrho_{i}\omega}\Biggl[q\rho_{k} +\overline{a}_{i} \Biggl(\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{b}_{j} \delta _{j}'\|y_{j}\|_{2}+B_{j}' \bigr)+\overline{E}_{i}\omega\Biggr)\Biggr] \\ &{}-\overline{a}_{i} \Biggl[\delta_{i}\sqrt {\omega}\|x_{i}\|_{2} +\sum _{j=1}^{m}(\overline{\alpha}_{ji}+ \overline{\beta}_{ji}) \omega M_{j} +\sum _{j=1}^{m}(\overline{\alpha}_{ji}+\overline{ \beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr) +\overline{E}_{i} \omega\Biggr]-q\rho_{k} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} x_{i}(\overline{t}_{i})\le{}&\frac{1}{\underline{a}_{i}\omega}\biggl\vert \int _{0}^{\omega}a_{i}\bigl(x_{i}(t) \bigr)x_{i}(t)\Delta t\biggr\vert - \int_{0}^{\omega}\bigl\vert x_{i}^{\Delta}(t) \bigr\vert \Delta t \\ \le{}&\frac{1}{\underline{a}_{i}\varrho_{i}\omega}\Biggl[q\rho_{k} +\overline{a}_{i} \Biggl(\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{b}_{j} \delta _{j}'\|y_{j}\|_{2}+B_{j}' \bigr)+\overline{E}_{i}\omega\Biggr)\Biggr] \\ &{}+\overline{a}_{i} \Biggl[\delta_{i}\sqrt {\omega}\|x_{i}\|_{2} +\sum _{j=1}^{m}(\overline{\alpha}_{ji}+ \overline{\beta}_{ji}) \omega M_{j} +\sum _{j=1}^{m}(\overline{\alpha}_{ji}+\overline{ \beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr) +\overline{E}_{i} \omega\Biggr]+q\rho_{k}. \end{aligned} \end{aligned}$$
Similarly, from (13) we obtain, for \(j=1,2,\ldots,m\),
$$\begin{aligned} y_{j}\bigl(\underline{t}_{j}'\bigr) \ge&- \frac{1}{\underline{b}_{j}\omega}\biggl\vert \int _{0}^{\omega}b_{j}\bigl(y_{j}(t) \bigr)y_{j}(t)\Delta t\biggr\vert - \int_{0}^{\omega}\bigl\vert y_{j}^{\Delta}(t) \bigr\vert \Delta t \\ \ge&-\frac{1}{\underline{b}_{j}\varrho_{j}'\omega}\Biggl[q\rho_{k}' + \overline{b}_{j}\Biggl(\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij})\omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta_{i}\Vert x_{i}\Vert _{2}+B_{i} \bigr)+\overline{F}_{j}\omega\Biggr)\Biggr] \\ &{}-\overline{b}_{j} \Biggl[\delta _{j}'\sqrt{\omega}\|y_{j} \|_{2} +\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij}) \omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta _{i}\Vert x_{i}\Vert _{2}+B_{i} \bigr) +\overline{F}_{j}\omega\Biggr]-q\rho_{k}' \end{aligned}$$
and
$$\begin{aligned} y_{j}\bigl(\underline{t}_{j}'\bigr) \le& \frac{1}{\underline{b}_{j}\omega}\biggl\vert \int _{0}^{\omega}b_{j}\bigl(y_{j}(t) \bigr)y_{j}(t)\Delta t\biggr\vert - \int_{0}^{\omega}\bigl\vert y_{j}^{\Delta}(t) \bigr\vert \Delta t \\ \le&\frac{1}{\underline{b}_{j}\varrho_{j}'\omega}\Biggl[q\rho_{k}' + \overline{b}_{j}\Biggl(\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij})\omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta_{i}\Vert x_{i}\Vert _{2}+B_{i} \bigr)+\overline{F}_{j}\omega\Biggr)\Biggr] \\ &{}+\overline{b}_{j} \Biggl[\delta _{j}'\sqrt{\omega}\|y_{j} \|_{2} +\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij}) \omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta _{i}\|x_{i}\|_{2}+B_{i}\bigr) + \overline{F}_{j}\omega\Biggr]+q\rho_{k}'. \end{aligned}$$
Therefore, we obtain that, for \(i=1,2,\ldots,n\),
$$\begin{aligned} \max_{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert x_{i}(t)\bigr\vert \le&\frac{1}{\underline{a}_{i}\varrho_{i}\omega}\Biggl[q\rho_{k} +\overline{a}_{i} \Biggl(\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr)+\overline{E}_{i} \omega\Biggr)\Biggr] \\ &{}+\overline{a}_{i}\Biggl[\delta_{i}\sqrt{\omega} \|x_{i}\|_{2} +\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji}) \omega M_{j} +\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr)+\overline{E}_{i} \omega\Biggr]+q\rho_{k} \end{aligned}$$
(14)
and, for \(j=1,2,\ldots,m\),
$$\begin{aligned} \max_{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert y_{j}(t)\bigr\vert \le&\frac{1}{\underline{b}_{j}\varrho_{j}'\omega}\Biggl[q\rho_{k}' + \overline{b}_{j}\Biggl(\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij})\omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta _{i}\|x_{i}\|_{2}+B_{i}\bigr)+ \overline{F}_{j}\omega\Biggr)\Biggr] \\ &{}+\overline{b}_{j}\Biggl[\delta_{j}' \sqrt{\omega}\|y_{j}\|_{2} +\sum _{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \omega N_{i} +\sum _{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}\bigl(\overline{a}_{i} \delta _{i}\|x_{i}\|_{2}+B_{i}\bigr)+ \overline{F}_{j}\omega\Biggr]+q\rho_{k}'. \end{aligned}$$
(15)
In addition, we have that
$$\begin{aligned}& \|x_{i}\|_{2}= \biggl( \int_{0}^{\omega}\bigl\vert x_{i}(s)\bigr\vert ^{2}\Delta s \biggr)^{1/2}\le\sqrt {\omega}\max _{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert x_{i}(t)\bigr\vert , \quad i=1,2, \ldots,n, \\& \|y_{j}\|_{2}= \biggl( \int_{0}^{\omega}\bigl\vert y_{j}(s)\bigr\vert ^{2}\Delta s \biggr)^{1/2}\le\sqrt {\omega}\max _{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert y_{j}(t)\bigr\vert ,\quad j=1,2, \ldots,m. \end{aligned}$$
By (14) we obtain, for \(i=1,2,\ldots,n\),
$$\begin{aligned} \underline{a}_{i}\sqrt{\omega}\|x_{i}\|_{2} \le& \frac{1}{\varrho_{i}} \Biggl[q\rho_{k} +\overline{a}_{i}\Biggl( \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr)+\overline{E}_{i} \omega\Biggr)\Biggr] \\ &{}+\underline{a}_{i}\omega\overline{a}_{i}\Biggl[ \delta_{i}\sqrt{\omega}\| x_{i}\|_{2} +\sum _{j=1}^{m}(\overline{\alpha}_{ji}+\overline{ \beta}_{ji}) \omega M_{j} +\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'\|y_{j} \|_{2}+B_{j}'\bigr)+\overline{E}_{i} \omega\Biggr]+\underline {a}_{i}\omega q\rho_{k}, \end{aligned}$$
that is,
$$ (\underline{a}_{i}-\underline{a}_{i}\omega \overline{a}_{i}\delta_{i})\|x_{i}\| _{2}- \overline{a}_{i} \biggl(\overline{a}_{i}\omega+ \frac{1}{\varrho_{i}} \biggr) \sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji}) \frac{\kappa _{j}}{\sqrt{2}}\omega\overline{b}_{j}\delta_{j}' \|y_{j}\|_{2}\le\frac{1}{\sqrt {\omega}}\Upsilon_{i}, $$
(16)
where, for \(i=1,2,\ldots,n\),
$$\Upsilon_{i}=\biggl(\underline{a}_{i}\omega+ \frac{1}{\varrho_{i}}\biggr) \Biggl[q\varrho _{k}+\overline{a}_{i} \Biggl(\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji}) \omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji})\frac{\kappa _{j}}{\sqrt{2}} \omega^{3/2}B_{j}'+\overline{E}_{i} \omega \Biggr) \Biggr]. $$
Similarly, we have
$$ \bigl(\underline{b}_{j}-\underline{b}_{j} \omega\overline{b}_{j}\delta_{j}'\bigr) \|y_{j}\| _{2}-\overline{b}_{j} \biggl( \overline{b}_{j}\omega+\frac{1}{\varrho_{j}'} \biggr) \sum _{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij})\frac{\nu_{i}}{\sqrt {2}}\omega\overline{a}_{i} \delta_{i}\|x_{i}\|_{2}\le\frac{1}{\sqrt{\omega }} \Upsilon_{j}', $$
(17)
where, for \(j=1,2,\ldots,m\),
$$\Upsilon_{j}'=\biggl(\underline{b}_{j} \omega+\frac{1}{\varrho_{j}'}\biggr) \Biggl[q\varrho _{k}'+ \overline{b}_{j} \Biggl(\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij}) \omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij})\frac{\nu_{i}}{\sqrt {2}}\omega^{3/2}B_{i}+ \overline{F}_{j}\omega \Biggr) \Biggr]. $$
Denote \(\|u\|_{2}=(\|x_{1}\|_{2},\ldots,\|x_{n}\|_{2},\|y_{1}\|_{2},\ldots,\|y_{m}\| _{2})^{T}\) and
$$\Upsilon=\frac{1}{\sqrt{\omega}}\bigl(\Upsilon_{1},\ldots, \Upsilon_{n},\Upsilon _{1}',\ldots, \Upsilon_{m}'\bigr)^{T}. $$
Then (16) and (17) can be written in the matrix form
$$H\|u\|_{2}\le\Upsilon. $$
From the conditions of Theorem 3.1 we have that H is a nonsingular M-matrix, so that
$$ \|u\|_{2}\le H^{-1}\Upsilon:= \bigl(D_{1},\ldots,D_{n},D_{1}', \ldots,D_{m}'\bigr)^{T}, $$
(18)
that is, \(\|x_{i}\|_{2}\le D_{i}\), \(i=1,2,\ldots,n\), and \(|y_{j}\|_{2}\le D_{j}'\), \(j=1,2,\ldots,m\).
From (14) and (15) we have, for \(i=1,2,\ldots,n\),
$$\begin{aligned} \max_{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert x_{i}(t)\bigr\vert \le& \frac{1}{\underline{a}_{i}\varrho_{i}\omega}\Biggl[q\rho_{k} +\overline{a}_{i}\Biggl( \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline {\beta}_{ji})\omega M_{j} + \sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'D_{j}'+B_{j}' \bigr)+\overline{E}_{i}\omega\Biggr)\Biggr] \\ &{}+\overline{a}_{i}\Biggl[\delta_{i}\sqrt{ \omega}D_{i} +\sum_{j=1}^{m}( \overline{\alpha}_{ji}+\overline{\beta}_{ji}) \omega M_{j} +\sum_{j=1}^{m}(\overline{ \alpha}_{ji}+\overline{\beta}_{ji}) \\ &{}\times\frac{\kappa_{j}}{\sqrt{2}}\omega^{3/2}\bigl(\overline {b}_{j}\delta_{j}'D_{j}'+B_{j}' \bigr)+\overline{E}_{i}\omega\Biggr]+q\rho_{k}:=G_{i} \end{aligned}$$
and, for \(j=1,2,\ldots,m\),
$$\begin{aligned} \max_{t\in[0,\omega]_{\mathbb{T}}}\bigl\vert y_{j}(t)\bigr\vert \le& \frac{1}{\underline{b}_{j}\varrho_{j}'\omega}\Biggl[q\rho_{k}' + \overline{b}_{j}\Biggl(\sum_{i=1}^{n}( \overline{p}_{ij}+\overline {q}_{ij})\omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}(\overline{a}_{i} \delta _{i}D_{i}+B_{i})+\overline{F}_{j} \omega\Biggr)\Biggr] \\ &{}+\overline{b}_{j}\Biggl[\delta_{j}' \sqrt{\omega}D_{j}' +\sum_{i=1}^{n}( \overline{p}_{ij}+\overline{q}_{ij}) \omega N_{i} + \sum_{i=1}^{n}(\overline{p}_{ij}+ \overline{q}_{ij}) \\ &{}\times\frac{\nu_{i}}{\sqrt{2}}\omega^{3/2}(\overline{a}_{i} \delta _{i}D_{i}+B_{i})+\overline{F}_{j} \omega\Biggr]+q\rho_{k}':=G_{j}'. \end{aligned}$$
Let \(G=\sum_{i=1}^{n}G_{i}+\sum_{j=1}^{m}G_{j}'+G_{0}\), where \(G_{0}\) is a positive constant. Clearly, F is independent of λ. Take \(\Omega=\{u\in\mathbb{X}|\|u\|_{\mathbb{X}}< G\}\). Obviously, Ω satisfies condition (a) of Lemma 3.1.
When \(u(t)\in\partial\Omega\cap\operatorname{Ker}L=\partial\Omega\cap\mathbb{R}^{n+m}\), u is a constant vector with \(\|u\|=G\). Furthermore, take \(J:\operatorname{Im}Q\rightarrow\operatorname{Ker}L\). Then
$$JQN(x_{i})=-a_{i}(x_{i}) \Biggl[c_{i}(x_{i})- \bigwedge_{j=1}^{m}\hat{\alpha }_{ji}f_{j}(y_{j})-\bigvee _{j=1}^{m}\hat{\beta}_{ji}f_{j}(y_{j}) +\hat{E_{i}} \Biggr]+\frac{1}{\omega}\sum _{k=1}^{q}I_{ik}(x_{i}) $$
for \(i=1,2,\ldots,n\), and
$$JQN(y_{j})=-b_{j}(y_{j}) \Biggl[d_{j}(y_{j})- \bigwedge_{i=1}^{n}\hat {p}_{ij}g_{i}(x_{i})- \bigvee_{i=1}^{n}\hat{q}_{ij}g_{i}(x_{i}) +\hat{F_{j}} \Biggr]+\frac{1}{\omega}\sum _{k=1}^{q}J_{jk}(y_{j}) $$
for \(j=1,2,\ldots,m\).
We can take G large enough such that
$$\begin{aligned} u^{T}JQNu \le&\sum_{i=1}^{n} \Biggl\{ -x_{i}a_{i}(x_{i}) \Biggl[c_{i}(x_{i}) -\bigwedge_{j=1}^{m}\hat{ \alpha}_{ji}f_{j}(y_{j})-\bigvee _{j=1}^{m}\hat{\beta }_{ij}f_{j}(y_{j}) +\hat{E_{i}} \Biggr] \\ &{}+\frac{1}{\omega}x_{i}\sum_{k=1}^{q}I_{ik}(x_{i}) \Biggr\} \\ &{}+\sum_{j=1}^{m}\Biggl\{ -y_{j}b_{j}(y_{j}) \Biggl[d_{j}(y_{j}) -\bigwedge_{i=1}^{n}\hat{p}_{ij}g_{i}(x_{i})- \bigvee_{i=1}^{n}\hat{q}_{ij}g_{i}(x_{i}) +\hat{F_{j}} \Biggr] \\ &{}+\frac{1}{\omega}y_{j}\sum_{k=1}^{q}J_{jk}(y_{j}) \Biggr\} \\ < &0. \end{aligned}$$
Hence, for any \(x\in\partial\Omega\cap\operatorname{Ker}L\), \(QNu\neq0\), namely, condition (b) in Lemma 3.1 is satisfied.
Furthermore, let \(\Psi(\gamma;u)=-\gamma u+(1-\gamma)QNu\). Then, for any \(u\in\partial\Omega\cap\operatorname{Ker}L\), \(u^{T}\Psi(\gamma;u)<0\), and we get
$$\operatorname{deg}\{JQN,\Omega\cap\operatorname{Ker}L,0\}=\operatorname{deg} \{-u,\Omega\cap\operatorname {Ker}L,0\}\neq0. $$
This shows that condition (c) in Lemma 3.1 is satisfied. Thus, by Lemma 3.1 we conclude that \(Lu = Nu\) has at least one solution in \(\mathbb{X}\), that is, system (1) has at least one ω-periodic solution. This completes the proof. □