Let
$$ X= \bigl\{ u=(x,y,z)=(x_{1},\ldots ,x_{n},y_{1}, \ldots ,y_{n},z_{1}, \ldots ,z_{n},)^{\top } \in C\bigl(\mathbb{R},\mathbb{R}^{3n}\bigr),u(t+\omega )=-u(t) \bigr\} $$
with the norm
$$ \Vert u \Vert _{X}=\sum_{i=1}^{n} \bigl( \vert x_{i} \vert _{\infty }+ \vert y_{i} \vert _{\infty }+ \vert z_{i} \vert _{ \infty }\bigr), \qquad \vert f \vert _{\infty }=\sup_{t\in \mathbb{R}} \bigl\vert f(t) \bigr\vert . $$
Clearly, X is a Banach space. Let
$$ L:D(L)\subset X\rightarrow X,\qquad Lu=\bigl((A_{1}x_{1})', \ldots ,(A_{n}x_{n})',y _{1}', \ldots ,y_{n}',z_{1}',\ldots ,z_{n}',\bigr)^{\top }, $$
(3.1)
where \(D(L)=\{u:u\in X,(A_{i}x_{i})',y_{i}',z_{i}'\in X\}\). Let there be a nonlinear operator \(N:X\rightarrow X\):
$$ (Nu) (t)= \bigl(F_{1}(\cdot ),\ldots ,F_{n}(\cdot ), G_{1}(\cdot ), \ldots ,G_{n}(\cdot ),H_{1}(\cdot ), \ldots ,H_{n}(\cdot ) \bigr)^{ \top }. $$
(3.2)
Clearly,
$$ \operatorname{Ker}L=\mathbb{R}^{3n}, \qquad \operatorname{Im}L=\biggl\{ u:u\in X, \int _{0}^{2\omega }u(s)\,ds=\mathbf{0}\biggr\} . $$
For \(u=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n},z_{1},\ldots ,z_{n})^{ \top }\in \operatorname{Im}L\), let the inverse operator of L be \(L^{-1}\) as follows:
$$\begin{aligned} \bigl(L^{-1}u\bigr) (t) =&\bigl(\bigl(A_{1}^{-1}F_{1}x_{1} \bigr) (t),\ldots ,\bigl(A_{n}^{-1}F_{n}x_{n} \bigr) (t), (F_{1}y_{1}) (t),\ldots , \\ &{}(F_{n}y_{n}) (t),(F_{1}z_{1}) (t),\ldots ,(F _{n}z_{n}) (t)\bigr)^{\top }, \end{aligned}$$
where
$$ (F_{i}u_{i}) (t)= \int _{0}^{T}G(t,s)u_{i}(s)\,ds,\qquad G(t,s)=\textstyle\begin{cases} \frac{s}{T},& 0\leq s< t\leq T, \\ \frac{s-T}{T},& 0\leq t< s\leq T. \end{cases} $$
Theorem 3.1
Assume that the assumptions (H1)–(H3) hold. Furthermore, the following assumption holds:
- (H4):
-
$$\begin{aligned} \varTheta =& \min_{1\leq i\leq n} \Biggl\{ 1-\sum _{i=1}^{n} \frac{\omega \vert 1- \alpha _{i}^{-} \vert }{ \vert 1- \vert c_{i} \vert \vert \pi } \\ &{}-\sum_{i=1}^{n} \Biggl[\beta _{i} ^{+}+\sum_{j=1}^{n} \biggl(b_{ij}^{+}L_{j} +c_{ij}^{+}L_{j} \max_{s\in \mathbb{R}}\frac{1}{ \vert 1-\gamma _{ij}'(s) \vert } \biggr) \Biggr] \frac{ \omega ^{2}}{ \vert 1- \vert c_{i} \vert \vert \pi ^{2}} \\ &{}-\sum_{i=1}^{n} \frac{B_{i}^{+}dL _{i}\omega ^{3}}{ \vert 1- \vert c_{i} \vert \vert \pi ^{3}} \Biggr\} >0. \end{aligned}$$
Then system (2.1) has at least one anti-periodic solution.
Proof
Consider the operator equation
$$ Lu=\lambda Nu,\qquad u\in D(L), \lambda \in (0,1), $$
(3.3)
where L and N are defined by (3.1) and (3.2). Let \(u\in D(L)\) be an arbitrary solution of (3.3), then
$$ \textstyle\begin{cases} (A_{i}x_{i})'(t)=\lambda F_{i}(\cdot ), \\ y_{i}'(t)=\lambda G_{i}(\cdot ), & i=1,2,\ldots ,n. \\ z_{i}'(t)=\lambda H_{i}(\cdot ), \end{cases} $$
(3.4)
Multiplying by \(y_{i}'(t)\) on both sides of the second equation of (3.4) and integrating it over \([0,2\omega ]\), we have
$$\begin{aligned}& \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \\& \quad =\lambda \int _{0}^{2\omega } \Biggl[-\bigl(\alpha _{i}(t)-1\bigr)x_{i}'(t)y_{i}'(t)- \beta _{i}(t)x_{i}(t)y_{i}'(t) +\sum _{j=1}^{n}b_{ij}(t)f_{j} \bigl(x_{j}(t)\bigr)y_{i}'(t) \\& \qquad {}+\sum_{j=1}^{n}c _{ij}(t)f_{j} \bigl(x_{j}\bigl(t-\gamma _{ij}(t)\bigr)\bigr)y_{i}'(t) +B_{i}(t)z_{i}(t)y _{i}'(t) \Biggr] \\& \quad =\lambda \int _{0}^{2\omega } \Biggl[-\bigl(\alpha _{i}(t)-1\bigr)x _{i}'(t)y_{i}'(t)- \beta _{i}(t)x_{i}(t)y_{i}'(t) +\sum _{j=1}^{n}b_{ij}(t) \bigl(f_{j}\bigl(x_{j}(t)\bigr)-f_{j}(0) \bigr)y_{i}'(t) \\& \qquad {}+\sum_{j=1}^{n}c_{ij}(t) \bigl(f_{j}\bigl(x_{j}\bigl(t-\gamma _{ij}(t) \bigr)\bigr)-f_{j}(0) \bigr)y_{i}'(t) \\& \qquad {}+\sum _{j=1} ^{n}\bigl(b_{ij}(t)+c_{ij}(t) \bigr)f_{j}(0)y_{i}'(t)+B_{i}(t)z_{i}(t)y_{i}'(t) \Biggr] \\& \quad \leq \bigl\vert 1-\alpha _{i}^{-} \bigr\vert \int _{0}^{2\omega } \bigl\vert x_{i}'(t)y_{i}'(t) \bigr\vert \,dt + \beta _{i}^{+} \int _{0}^{2\omega } \bigl\vert x_{i}(t)y_{i}'(t) \bigr\vert \,dt +\sum_{j=1}^{n}b _{ij}^{+}L_{j} \int _{0}^{2\omega } \bigl\vert x_{j}(t)y_{i}'(t) \bigr\vert \,dt \\& \qquad {}+\sum_{j=1} ^{n}c_{ij}^{+}L_{j} \int _{0}^{2\omega } \bigl\vert x_{j}\bigl(t- \gamma _{ij}(t)\bigr)y_{i}'(t) \bigr\vert \,dt + \sum_{j=1}^{n}\bigl(b_{ij}^{+}+c_{ij}^{+} \bigr) \bigl\vert f_{j}(0) \bigr\vert \int _{0}^{2\omega } \bigl\vert y _{i}'(t) \bigr\vert \,dt \\& \qquad {}+B_{i}^{+} \int _{0}^{2\omega } \bigl\vert z_{i}(t)y_{i}'(t) \bigr\vert \,dt. \end{aligned}$$
(3.5)
From (3.5) and the Hölder inequality, we have
$$\begin{aligned} \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \leq& \bigl\vert 1-\alpha _{i}^{-} \bigr\vert \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+\beta _{i} ^{+} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+ \sum_{j=1}^{n}b_{ij}^{+}L_{j} \biggl( \int _{0}^{2\omega } \bigl\vert x_{j}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\ &{}+\sum_{j=1}^{n}c_{ij}^{+}L_{j} \biggl( \int _{0}^{2\omega } \bigl\vert x_{j}\bigl(t- \gamma _{ij}(t)\bigr) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0} ^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+\sum_{j=1}^{n}\sqrt{2 \omega } \bigl(b_{ij}^{+}+c_{ij}^{+}\bigr) \bigl\vert f_{j}(0) \bigr\vert \biggl( \int _{0}^{2\omega } \bigl\vert y _{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+B_{i}^{+} \biggl( \int _{0}^{2 \omega } \bigl\vert z_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}, \end{aligned}$$
which results in
$$\begin{aligned}& \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \quad \leq \bigl\vert 1- \alpha _{i}^{-} \bigr\vert \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} +\beta _{i}^{+} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{j=1}^{n}b_{ij}^{+}L_{j} \biggl( \int _{0} ^{2\omega } \bigl\vert x_{j}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} +\sum _{j=1}^{n}c_{ij} ^{+}L_{j} \biggl( \int _{0}^{2\omega } \bigl\vert x_{j}\bigl(t- \gamma _{ij}(t)\bigr) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{j=1}^{n}\sqrt{2\omega } \bigl(b_{ij}^{+}+c _{ij}^{+}\bigr) \bigl\vert f_{j}(0) \bigr\vert +B_{i}^{+} \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.6)
Consider the fourth term \(\sum_{j=1}^{n}c_{ij}^{+}L_{j} (\int _{0} ^{2\omega } |x_{j}(t-\gamma _{ij}(t))|^{2}\,dt )^{\frac{1}{2}}\) in (3.6). In view of Remark 2.2, we have
$$\begin{aligned}& \sum_{j=1}^{n}c_{ij}^{+}L_{j} \biggl( \int _{0}^{2\omega } \bigl\vert x_{j}\bigl(t- \gamma _{ij}(t)\bigr) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \quad =\sum_{j=1}^{n}c_{ij} ^{+}L_{j} \biggl( \int _{-\gamma _{ij}(0)}^{2\omega -\gamma _{ij}(0)} \frac{ \vert x _{j}(u_{ij}(t)) \vert ^{2}}{1-\gamma _{ij}'(\varGamma _{ij}(u_{ij}))}\,du_{ij} \biggr)^{\frac{1}{2}} \\& \quad \leq \sum_{j=1}^{n}c_{ij}^{+}L_{j} \max_{s\in \mathbb{R}}\frac{1}{ \vert 1-\gamma _{ij}'(s) \vert } \biggl( \int _{0} ^{2\omega } \bigl\vert x_{j}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.7)
From (3.6) and (3.7), we have
$$\begin{aligned}& \sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\& \quad \leq \sum _{i=1}^{n} \bigl\vert 1-\alpha _{i}^{-} \bigr\vert \biggl( \int _{0}^{2 \omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} +\sum _{i=1}^{n}\beta _{i}^{+} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{i=1}^{n}\sum _{j=1}^{n} \biggl(b_{ij}^{+}L_{j} +c_{ij}^{+}L _{j} \max_{s\in \mathbb{R}} \frac{1}{ \vert 1-\gamma _{ij}'(s) \vert } \biggr) \biggl( \int _{0}^{2\omega } \bigl\vert x_{j}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{i=1} ^{n}\sum _{j=1}^{n}\sqrt{2\omega }\bigl(b_{ij}^{+}+c_{ij}^{+} \bigr) \bigl\vert f_{j}(0) \bigr\vert + \sum _{i=1}^{n}B_{i}^{+} \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \quad = \sum_{i=1}^{n} \bigl\vert 1-\alpha _{i}^{-} \bigr\vert \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{i=1} ^{n} \Biggl[\beta _{i}^{+}+\sum_{j=1}^{n} \biggl(b_{ij}^{+}L_{j} +c_{ij} ^{+}L_{j} \max_{s\in \mathbb{R}}\frac{1}{ \vert 1-\gamma _{ij}'(s) \vert } \biggr) \Biggr] \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+\sum_{i=1}^{n}\sum _{j=1}^{n}\sqrt{2\omega }\bigl(b_{ij}^{+}+c_{ij} ^{+}\bigr) \bigl\vert f_{j}(0) \bigr\vert +\sum _{i=1}^{n}B_{i}^{+} \biggl( \int _{0}^{2\omega } \bigl\vert z _{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.8)
Multiplying by \(A_{i}x_{i}'(t)\) on both sides of the first equation of (3.4) and integrating it over \([0,2\omega ]\), we have
$$ \begin{aligned} \int _{0}^{2\omega } \bigl\vert A_{i}x_{i}'(t) \bigr\vert ^{2}\,dt & =-\lambda \int _{0}^{2 \omega }x_{i}(t)A_{i}x_{i}'(t)\,dt+ \lambda \int _{0}^{2\omega }y_{i}(t)A _{i}x_{i}'(t)\,dt \\ &=\lambda \int _{0}^{2\omega }y_{i}(t)A_{i}x_{i}'(t)\,dt \\ &\leq \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0}^{2\omega } \bigl\vert A_{i}x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}, \end{aligned} $$
which results in
$$ \biggl( \int _{0}^{2\omega } \bigl\vert A_{i}x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. $$
(3.9)
Using Lemma 2.3 and (3.9), we have
$$\begin{aligned} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} =& \biggl( \int _{0}^{2\omega } \bigl\vert A_{i}^{-1}A_{i}x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\ \leq& \frac{1}{ \vert 1- \Vert c_{i} \Vert \vert } \biggl( \int _{0}^{2\omega } \bigl\vert A _{i}x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.10)
The Wirtinger inequality, (3.9) and (3.10) result in
$$\begin{aligned} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq& \frac{1}{ \vert 1-c_{i} \vert } \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ \leq& \frac{\omega }{ \vert 1- \Vert c_{i} \Vert \vert \pi } \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.11)
Using again the Wirtinger inequality and (3.11), we have
$$ \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq \frac{ \omega }{\pi } \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \leq \frac{\omega ^{2}}{ \vert 1- \Vert c_{i} \Vert \vert \pi ^{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. $$
(3.12)
Multiplying by \(z_{i}'(t)\) on both sides of the third equation of (3.4) and integrating it over \([0,2\omega ]\), we have
$$\begin{aligned} \int _{0}^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt =&-\lambda \int _{0}^{2\omega }z _{i}(t)z_{i}'(t)\,dt+ \lambda \int _{0}^{2\omega }\,df_{i}\bigl(x_{i}(t) \bigr)z_{i}'(t)\,dt \\ \leq& \int _{0}^{2\omega }\,d\bigl\vert f_{i} \bigl(x_{i}(t)\bigr)-f_{i}(0) \bigr\vert \bigl\vert z_{i}'(t) \bigr\vert \,dt + \int _{0}^{2\omega }d \bigl\vert f_{i}(0) \bigr\vert \bigl\vert z_{i}'(t) \bigr\vert \,dt \\ \leq& dL_{i} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int _{0} ^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+d \bigl\vert f_{i}(0) \bigr\vert \sqrt{2 \omega } \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}, \end{aligned}$$
which results in
$$ \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq dL _{i} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} +d \bigl\vert f _{i}(0) \bigr\vert \sqrt{2\omega }. $$
(3.13)
From (3.12) and (3.13), we have
$$ \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq \frac{dL _{i}\omega ^{2}}{ \vert 1- \Vert c_{i} \Vert \vert \pi ^{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} +d \bigl\vert f_{i}(0) \bigr\vert \sqrt{2\omega } $$
(3.14)
which together with the Wirtinger inequality results in
$$ \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \leq \frac{dL _{i}\omega ^{3}}{ \vert 1- \Vert c_{i} \Vert \vert \pi ^{3}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} +\frac{\omega d \vert f_{i}(0) \vert \sqrt{2\omega }}{ \pi }. $$
(3.15)
From (3.8), (3.11), (3.12) and (3.15), we have
$$\begin{aligned}& \sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\& \quad \leq \sum _{i=1}^{n} \frac{\omega \vert 1-\alpha _{i}^{-} \vert }{ \vert 1- \Vert c _{i} \Vert \vert \pi } \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\& \qquad {}+\sum_{i=1}^{n} \Biggl[\beta _{i}^{+}+\sum_{j=1}^{n} \biggl(b_{ij}^{+}L_{j} +c_{ij}^{+}L_{j} \max_{s\in \mathbb{R}}\frac{1}{ \vert 1- \gamma _{ij}'(s) \vert } \biggr) \Biggr] \\& \qquad {}\times \frac{\omega ^{2}}{ \vert 1- \Vert c_{i} \Vert \vert \pi ^{2}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\& \qquad {}+ \sum_{i=1}^{n}\sum _{j=1}^{n}\sqrt{2\omega }\bigl(b_{ij}^{+}+c_{ij}^{+} \bigr) \bigl\vert f _{j}(0) \bigr\vert +\sum _{i=1}^{n} \frac{B_{i}^{+}dL_{i}\omega ^{3}}{ \vert 1- \Vert c_{i} \Vert \vert \pi ^{3}} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\& \qquad {}+\sum_{i=1}^{n}\frac{B_{i}^{+}\omega d \vert f_{i}(0) \vert \sqrt{2 \omega }}{\pi }. \end{aligned}$$
(3.16)
By using assumption (H4) and (3.16), we obtain
$$\begin{aligned}& \sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert y_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \\& \quad \leq \frac{1}{\varTheta } \Biggl(\sum_{i=1}^{n}\sum _{j=1}^{n}\sqrt{2 \omega }\bigl(b_{ij}^{+}+c_{ij}^{+} \bigr) \bigl\vert f_{j}(0) \bigr\vert +\sum _{i=1}^{n}\frac{B_{i} ^{+}\omega d \vert f_{i}(0) \vert \sqrt{2\omega }}{\pi } \Biggr) \\& \quad :=M_{1}. \end{aligned}$$
(3.17)
Equations (3.11) and (3.17) result in
$$ \sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \leq \sum _{i=1}^{n}\frac{M_{1}\omega }{ \vert 1- \Vert c_{i} \Vert \vert \pi }:=M_{2}. $$
(3.18)
Equations (3.14) and (3.17) result in
$$\begin{aligned} \sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert z_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}} \leq& \sum _{i=1}^{n}\frac{M_{1}dL_{i}\omega ^{2}}{ \vert 1- \Vert c _{i} \Vert \vert \pi ^{2}} +d \bigl\vert f_{i}(0) \bigr\vert \sqrt{2\omega } \\ :=&M_{3}. \end{aligned}$$
(3.19)
Since \(u\in X\) is an ω-anti-periodic function, there exist \(\xi _{i},\eta _{i},\zeta _{i}\in [0,2\omega ]\) such that
$$ x_{i}(\xi _{i})=y_{i}(\eta _{i})=z_{i}( \zeta _{i})=0, $$
which results in
$$ \vert x_{i} \vert _{\infty }\leq \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert \,dt\leq \sqrt{2 \omega } \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} $$
and
$$ \sum_{i=1}^{n} \vert x_{i} \vert _{\infty }\leq \sqrt{2\omega }\sum_{i=1}^{n} \biggl( \int _{0}^{2\omega } \bigl\vert x_{i}'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}:= \widetilde{M}_{1}. $$
Similarly, using (3.17) and (3.19), we also find that there exist positive constants \(\widetilde{M}_{2}\), \(\widetilde{M}_{3}\) such that
$$\begin{aligned}& \sum_{i=1}^{n} \vert y_{i} \vert _{\infty }\leq \widetilde{M}_{2}, \end{aligned}$$
(3.20)
$$\begin{aligned}& \sum_{i=1}^{n} \vert z_{i} \vert _{\infty }\leq \widetilde{M}_{3}. \end{aligned}$$
(3.21)
From (3.19)–(3.21), we get
$$ \Vert u \Vert _{X}=\sum_{i=1}^{n} \bigl( \vert x_{i} \vert _{\infty }+ \vert y_{i} \vert _{\infty }+ \vert z_{i} \vert _{ \infty }\bigr) \leq \widetilde{M}_{1}+\widetilde{M}_{2}+\widetilde{M}_{3}:= \widetilde{M}. $$
Let
$$ \varOmega =\bigl\{ u\in X: \Vert u \Vert _{X}< \widetilde{M}+1\bigr\} . $$
From Lemma 2.2, the operator equation \(Lu=Nu\) has at least one ω-anti-periodic solution in X. Thus, system (2.1) has at least one ω-anti-periodic solution. □
Remark 3.1
We very much want to obtain the globally exponential stability of system (2.1) with initial values conditions (2.3). But transforming system (2.6) of system (2.1) contains a neutral term \(A_{i}x_{i}\) which makes constructing the appropriate Lyapinov function very difficult. Hence, we wish that some authors will develop new methods to derive globally exponential stability of system (2.1) in the future.