Lemma 3.1
For any
\(\lambda\in(0, 1)\), we are concerned with the following system:
$$ \begin{gathered} \begin{aligned} \frac{d [K_{n}u_{n}(t)]}{dt}={}&\lambda \Biggl\{ -d_{n}(t)u_{n}(t)+\sum _{j=1}^{l}b^{R}_{nj}(t)F_{j}^{R} \bigl(u_{j}(t), v_{j}(t)\bigr) \\ & -\sum_{j=1}^{l}b_{nj}^{I}(t)F_{j}^{I} \bigl(u_{j}(t), v_{j}(t)\bigr) \\ &+\sum_{j=1}^{l}e_{nj}^{R}(t)G_{j}^{R} \bigl(u_{j}^{t}, v_{j}^{t}\bigr)-\sum _{j=1}^{l}e_{nj}^{I}(t)G_{j}^{I} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P_{n}^{R}(t) \Biggr\} , \end{aligned} \\ \begin{aligned}\frac{d [K_{n}v_{n}(t)]}{dt}={}&\lambda \Biggl\{ -d_{n}(t)v_{n}(t)+\sum_{j=1}^{l}b^{R}_{nj}(t)F_{j}^{I} \bigl(u_{j}(t), v_{j}(t)\bigr) \\ &+\sum_{j=1}^{l} b_{nj}^{I}(t)F_{j}^{R} \bigl(u_{j}(t), v_{j}(t)\bigr) \\ &+\sum_{j=1}^{l}e_{nj}^{R}(t)G_{j}^{I} \bigl(u_{j}^{t}, v_{j}^{t}\bigr)+\sum _{j=1}^{l}e_{nj}^{I}(t)G_{j}^{R} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P_{n}^{I}(t) \Biggr\} . \end{aligned} \end{gathered} $$
(3)
Then the periodic solutions of system (3) are bounded and the boundary must be independent of the choice of
λ
under assumptions
\((h_{1})\)–\((h_{4})\)
if the periodic solutions of system (3) exist. Namely, there exists a positive constant
M
such that
$$\big\| \bigl(u(t), v(t)\bigr)^{T}\big\| = \big\| \bigl(u_{1}(t), u_{2}(t),\ldots, u_{l}(t), v_{1}(t), v_{2}(t),\ldots, v_{l}(t)\bigr)^{T} \big\| \leq M,$$
where
$$\big\| \bigl(u(t), v(t)\bigr)^{T}\big\| =\sum_{m=1}^{l}\max_{t\in[0, \omega]}\bigl(\big|u_{m}(t)\big|+\big|v_{m}(t)\big|\bigr).$$
Proof
From \((h_{3})\) and \((h_{4})\), it follows that there exists a positive number δ such that
-
\((h_{5})\)
:
-
\((1+|c_{n}|)(U_{nj}+\frac{E_{nj}}{1-\sigma})<\frac{2\underline {d_{j}}}{l}-\frac{|c_{j}|}{l}\overline{d_{j}} -A_{jn\delta}-|c_{j}|B_{jn\delta}-l\delta^{2}\overline{P^{R}_{n}}-\delta\).
-
\((h_{6})\)
:
-
\((1+|c_{n}|)(V_{nj}+\frac{F_{nj}}{1-\sigma})<\frac{2\underline {d_{j}}}{l}-\frac{|c_{j}|}{l}\overline{d_{j}} -A^{\ast}_{jn\delta}-|c_{j}|B^{\ast}_{jn\delta}-l\delta^{2}\overline {P^{I}_{n}}-\delta\).
Suppose that \((u(t), v(t))^{T}=(u_{1}(t), u_{2}(t),\ldots, u_{l}(t), v_{1}(t), v_{2}(t),\ldots, v_{l}(t))^{T}\) is one ω-periodic solution of system (3) for some \(\lambda\in(0, 1)\). Let \(V_{n}(t)=V_{1n}(t)+V_{2n}(t)\),
$$V_{1n}(t)=\bigl[K_{n}u_{n}(t)\bigr]^{2}+ \bigl[K_{n}v_{n}(t)\bigr]^{2}, $$
where
$$ \begin{aligned} V_{2n}(t)={}&\lambda \Biggl\{ \vert c_{n} \vert \int_{t-\tau}^{t} \Biggl(\sum _{j=1}^{l}B_{nj\delta}+\delta^{2} \overline{P^{R}_{n}} \Biggr)u_{n}^{2}(s) \,ds \\ &+\frac{(1+ \vert c_{n} \vert )}{1-\sigma}\sum_{j=1}^{l}E_{nj} \int_{t-\tau _{nj}(t)}^{t}u_{j}^{2}(s)\,ds \\ & +\frac{(1+ \vert c_{n} \vert )}{1-\sigma}\sum_{j=1}^{l}F_{nj} \int_{t-\tau _{nj}(t)}^{t}v_{j}^{2}(s)\,ds \\ &+ \vert c_{n} \vert \int_{t-\tau}^{t} \Biggl(\sum _{j=1}^{l}B^{\ast}_{nj\delta}+ \delta^{2}\overline{P^{I}_{n}} \Biggr)v_{n}^{2}(s) \,ds \Biggr\} . \end{aligned} $$
Then we have, along with the solutions of system (3),
$$\begin{aligned} \frac{dV_{1n}(t)}{dt}={}&\lambda \bigl[u_{n}(t)-c_{n}u_{n}(t-\tau)\bigr] \Biggl(-d_{n}(t)u_{n}(t)+\sum_{j=1}^{l}b_{nj}^{R}(t) F_{j}^{R}\bigl(u_{j}(t), v_{j}(t) \bigr) \\ &-\sum_{j=1}^{l}b_{nj}^{I}(t) F_{j}^{I}\bigl(u_{j}(t), v_{j}(t) \bigr)+\sum_{j=1}^{l}e_{nj}^{R}(t)G_{j}^{R} \bigl(u_{j}^{t}, v_{j}^{t}\bigr) \\ &-\sum_{j=1}^{l}e_{nj}^{I}(t)G_{j}^{I} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P_{n}^{R}(t) \Biggr) \\ &+\bigl[v_{n}(t)-c_{n} v_{n}(t-\tau)\bigr] \Biggl(-d_{n}(t)v_{n}(t)+ \sum_{j=1}^{l}b_{nj}^{R}(t) F_{j}^{I}\bigl(u_{j}(t), v_{j}(t) \bigr) \\ &+\sum_{j=1}^{l}b_{nj}^{I}(t)F_{j}^{R} \bigl(u_{j}(t), v_{j}(t)\bigr)+\sum_{j=1}^{l}e_{nj}^{R}(t)G_{j}^{I} \bigl(u_{j}^{t}, v_{j}^{t}\bigr) \\ &+\sum _{j=1}^{l}e_{nj}^{I}(t)G_{j}^{R} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P_{n}^{I}(t) \Biggr) \\ \leq{}&\lambda \Biggl\{ \bigl(-2\underline{d_{n}}+ \vert c_{n} \vert \overline {d_{n}}\bigr)u_{n}^{2}(t)+ \vert c_{n} \vert \overline{d_{n}}u_{n}^{2}(t- \tau) \\ &+ 2 \bigl[ \bigl\vert u_{n}(t) \bigr\vert + \vert c_{n} \vert \bigl\vert u_{n}(t-\tau) \bigr\vert \bigr] \Biggl(\sum_{j=1}^{l} \overline{b_{nj}^{R}} \bigl[l_{j}^{R} \bigl\vert u_{j}(t) \bigr\vert +l_{j}^{I} \bigl\vert v_{j}(t) \bigr\vert + \bigl\vert F_{j}^{R}(0,0) \bigr\vert \bigr] \\ & +\sum_{j=1}^{l}\overline{b_{nj}^{I}} \bigl[ \bigl\vert F_{j}^{I}(0, 0) \bigr\vert +k_{j}^{R} \bigl\vert u_{j}(t) \bigr\vert +k_{j}^{I} \bigl\vert v_{j}(t) \bigr\vert \bigr] \\ & +\sum_{j=1}^{l}\overline{e_{nj}^{R}} \bigl[G_{j}^{R} \bigl\vert u_{j}^{t} \bigr\vert +q_{j}^{I} \bigl\vert v_{j}^{t} \bigr\vert + \bigl\vert G_{j}^{R}(0,0) \bigr\vert \bigr] \\ &+\sum_{j=1}^{l}\overline {e_{nj}^{I}} \bigl[p_{j}^{R} \bigl\vert u_{j}^{t} \bigr\vert +p_{j}^{I} \bigl\vert v_{j}^{t} \bigr\vert + \bigl\vert G_{j}^{I}(0,0) \bigr\vert \bigr]+\overline{P_{n}^{R}} \Biggr) \\ &+ \bigl(-2 \underline{d_{n}}+ \vert c_{n} \vert \overline{d_{n}} \bigr)v_{n}^{2}(t)+ \vert c_{n} \vert \overline {d_{n}}v_{n}^{2}(t-\tau)+ 2 \bigl[ \bigl\vert v_{n}(t) \bigr\vert + \vert c_{n} \vert \bigl\vert v_{n}(t-\tau) \bigr\vert \bigr] \\ &\times \Biggl(\sum_{j=1}^{l}\overline {b_{nj}^{R}} \bigl[k_{j}^{R} \bigl\vert u_{j}(t) \bigr\vert +k_{j}^{I} \bigl\vert v_{j}(t) \bigr\vert + \bigl\vert F_{j}^{I}(0,0) \bigr\vert \bigr] \\ &+\sum _{j=1}^{l}\overline{b_{nj}^{I}} \bigl[ \bigl\vert F_{j}^{R}(0, 0) \bigr\vert +l_{j}^{R} \bigl\vert u_{j}(t) \bigr\vert +l_{j}^{I} \bigl\vert v_{j}(t) \bigr\vert \bigr] \\ & +\sum_{j=1}^{l}\overline{e_{nj}^{R}} \bigl[p_{j}^{R} \bigl\vert u_{j}^{t} \bigr\vert +p_{j}^{I} \bigl\vert v_{j}^{t} \bigr\vert + \bigl\vert G_{j}^{I}(0,0) \bigr\vert \bigr] \\ &+\sum_{j=1}^{l}\overline{e_{nj}^{I}} \bigl[q_{j}^{R} \bigl\vert u_{j}^{t} \bigr\vert +q_{j}^{I} \bigl\vert v_{j}^{t} \bigr\vert + \bigl\vert G_{j}^{R}(0,0) \bigr\vert \bigr]+\overline{P_{n}^{I}} \Biggr) \Biggr\} . \end{aligned}$$
(4)
From (4), by using the inequality \(2|ab|\leq a^{2}+b^{2}\) (\(a, b=u_{n}(t), u_{n}(t-\tau), v_{n}(t), v_{n}(t-\tau), u_{j}(t), v_{j}(t), u_{j}^{t}, v_{j}^{t} \)), \(2u_{n}(t)|F^{R}_{j}(0,0)|\leq |F^{R}_{j}(0,0)|[\delta^{2}u_{n}^{2}(t)+\frac{1}{\delta^{2}}]\), \(2u_{n}(t)|F^{I}_{j}(0,0)|\leq |F^{I}_{j}(0,0)|[\delta^{2}u_{n}^{2}(t)+\frac{1}{\delta^{2}}]\), \(2u_{n}(t)|G^{I}_{j}(0,0)|\leq |G^{I}_{j}(0,0)|[\delta^{2}u_{n}^{2}(t)+\frac{1}{\delta ^{2}}]\), \(2u_{n}(t)|G^{R}_{j}(0,0)|\leq |G^{R}_{j}(0,0)|[\delta^{2}u_{n}^{2}(t)+\frac{1}{\delta^{2}}]\), \(2u_{n}(t)\overline{P^{R}_{n}}\leq \overline{P^{R}_{n}}[u_{n}^{2}(t)\delta^{2}+\frac{1}{\delta^{2}}]\), it follows that
$$\begin{aligned} \frac{dV_{1n}(t)}{dt} \leq{}&\lambda \Biggl\{ \Biggl(-2\underline{d_{n}}+ \vert c_{n} \vert \overline{d_{n}}+\sum_{j=1}^{l}A_{nj\delta} \Biggr)u_{n}^{2}(t)+ \vert c_{n} \vert \Biggl( \sum_{j=1}^{l} B_{nj\delta}+\delta^{2}\overline{P^{R}_{n}} \Biggr)u^{2}_{n}(t-\tau) \\ &+\sum_{j=1}^{l} \bigl(1+ \vert c_{n} \vert \bigr)\bigl(\overline{b_{nj}^{R}}l_{j}^{R}+ \overline {b_{nj}^{I}}k_{j}^{R} \bigr)u_{j}^{2}(t)+\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l}\bigl( \overline{b_{nj}^{R}} l_{j}^{I}+\overline{b_{nj}^{I}}k_{j}^{I} \bigr)v_{j}^{2}(t) \\ & +\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l}\bigl( \overline{e_{nj}^{R}}q_{j}^{R}+ \overline{e_{nj}^{I}}p_{j}^{R}\bigr) \bigl(u_{j}^{t} \bigr)^{2} +\bigl(1+ \vert c_{n} \vert \bigr)\sum _{j=1}^{l}\bigl(\overline{e_{nj}^{R}}q_{j}^{I}+ \overline {e_{nj}^{I}}p_{j}^{I}\bigr) \bigl(v^{t}_{j}\bigr)^{2} \\ &+ \Biggl(-2\underline{d_{n}}+ \vert c_{n} \vert \overline{d_{n}}+\sum_{j=1}^{l}A^{\ast}_{nj\delta} \Biggr)v_{n}^{2}(t)+ \vert c_{n} \vert \Biggl(\sum_{j=1}^{l} B^{\ast}_{nj\delta}+\delta^{2}\overline{P^{I}_{n}} \Biggr)v^{2}_{n}(t-\tau) \\ &+\sum_{j=1}^{l} \bigl(1+ \vert c_{n} \vert \bigr) \bigl(\overline{b_{nj}^{R}}k_{j}^{R}+\overline{b_{nj}^{I}}l_{j}^{R} \bigr)v_{j}^{2}(t)+\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l}\bigl( \overline{b_{nj}^{R}} k_{j}^{I}+ \overline{b_{nj}^{I}}l_{j}^{I} \bigr)u_{j}^{2}(t) \\ & +\bigl(1+ \vert c_{n} \vert \bigr)\sum _{j=1}^{l}\bigl(\overline{e_{nj}^{R}}p_{j}^{R} +\overline {e_{nj}^{I}}q_{j}^{R}\bigr) \bigl(v_{j}^{t}\bigr)^{2} +\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l} \bigl(\overline{e_{nj}^{R}}p_{j}^{I}+\overline{e_{nj}^{I}}q_{j}^{I}\bigr) \bigl(u^{t}_{j}\bigr)^{2} \Biggr\} \\ &+\lambda N, \end{aligned}$$
(5)
where N is a positive constant.
Since
$$ \begin{aligned}[b] \frac{dV_{2n}(t)}{dt} ={}&\lambda \Biggl\{ \frac{1+ \vert c_{n} \vert }{1-\sigma}\sum_{j=1}^{l} \bigl\{ E_{nj}u_{j}^{2}(t)-E_{nj}x_{j}^{2} \bigl(t-\tau_{ij}(t)\bigr) \bigl(1-\tau'_{nj}(t) \bigr) \\ &+F_{nj}v_{j}^{2}(t)-F_{nj}v_{j}^{2} \bigl(t-\tau_{nj}(t)\bigr) \bigl(1-\tau'_{nj}(t) \bigr) \bigr\} \\ &+ \vert c_{n} \vert \sum_{j=1}^{l} \bigl(B_{nj\delta}+\delta^{2}\overline{P^{R}_{n}} \bigr)u_{n}^{2}(t) - \vert c_{n} \vert \sum_{j=1}^{l} \bigl(B_{nj\delta}+\delta^{2}\overline{P^{R}_{n}} \bigr)u_{n}^{2}(t-\tau) \\ &+ \vert c_{n} \vert \sum_{j=1}^{l} \bigl(B^{\ast}_{nj\delta}+\delta^{2}\overline{P^{I}_{n}} \bigr)v_{n}^{2}(t)- \vert c_{n} \vert \sum _{j=1}^{l}\bigl(B^{\ast}_{nj\delta}+ \delta^{2}\overline{P^{I}_{n}}\bigr)v_{n}^{2}(t- \tau ) \Biggr\} \\ \leq{}&\lambda\sum_{j=1}^{l} \biggl\{ \frac{E_{nj}(1+ \vert c_{n} \vert )}{1-\sigma }u_{j}^{2}(t)-E_{nj}\bigl(1+ \vert c_{n} \vert \bigr)u_{j}^{2}\bigl(t- \tau_{nj}(t)\bigr) \\ & +\frac{F_{nj}(1+ \vert c_{n} \vert )}{1-\sigma}v_{j}^{2}(t)-\bigl(1+ \vert c_{n} \vert \bigr) F_{nj}v_{j}^{2}\bigl(t- \tau_{nj}(t)\bigr) \biggr\} \\ &+ \vert c_{n} \vert \sum _{j=1}^{l}\bigl(B_{nj\delta}+\delta ^{2}\overline{P^{R}_{n}}\bigr)u_{n}^{2}(t)- \vert c_{n} \vert \sum_{j=1}^{l} \bigl(B_{nj\delta} +\delta^{2}\overline{P^{R}_{n}} \bigr)u_{n}^{2}(t-\tau) \\ & +\vert c_{n} \vert \sum_{j=1}^{l} \bigl(B^{\ast}_{nj\delta}+\delta^{2}\overline{P^{I}_{n}} \bigr)v_{n}^{2}(t)- \vert c_{n} \vert \sum _{j=1}^{l}\bigl(B^{\ast}_{nj\delta}+ \delta^{2}\overline{P^{I}_{n}}\bigr)v_{n}^{2}(t- \tau ) \}, \end{aligned} $$
(6)
from (5) and (6), we have
$$ \begin{aligned}[b] \frac{dV_{n}(t)}{dt} \leq{}&\lambda \Biggl\{ \Biggl[ -2\underline{d_{n}}+ \vert c_{n} \vert \overline{d_{n}}+\sum_{j=1}^{l}A_{nj\delta }+ \vert c_{n} \vert \Biggl(\sum_{j=1}^{l}B_{nj\delta}+\delta^{2}\overline{P^{R}_{n}}\Biggr) \Biggr]u_{n}^{2}(t) \\ & +\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l}\biggl(U_{nj}+ \frac{E_{nj}}{1-\sigma}\biggr)u_{j}^{2}(t)+\bigl(1+ \vert c_{n} \vert \bigr)\sum_{j=1}^{l} \biggl(V_{nj}+\frac{F_{nj}}{1-\sigma}\biggr)v_{j}^{2}(t) \\ &+ \Biggl[ -2\underline{d_{n}}+ \vert c_{n} \vert \overline{d_{n}}+\sum_{j=1}^{l}A^{\ast}_{nj\delta} + \vert c_{n} \vert \Biggl(\sum_{j=1}^{l} B^{\ast}_{nj\delta}+\delta^{2}\overline{P^{I}_{n}} \Biggr) \Biggr]v_{n}^{2}(t) \Biggr\} +\lambda N \\ ={}&\lambda\sum_{j=1}^{l} \Biggl\{ \biggl[ -2\frac{\underline{d_{n}}}{l}+ \vert c_{n} \vert \frac{\overline{d_{n}}}{l}+A_{nj\delta}+ \vert c_{n} \vert \bigl(B_{nj\delta }+l\delta^{2}\overline{P^{R}_{n}} \bigr)+\delta \biggr]u_{n}^{2}(t) \\ & + \vert c_{n} \vert )\sum_{j=1}^{l}(V_{nj}+ \bigl(1+ \vert c_{n} \vert \bigr) \biggl(U_{nj}+ \frac{E_{nj}}{1-\sigma}\biggr)u_{j}^{2}(t) \\ & +\bigl(1+ \vert c_{n} \vert \bigr) \biggl(V_{nj}+\frac{F_{nj}}{1-\sigma}\biggr)v_{j}^{2}(t)\\ &+ \biggl[ -2\frac{\underline{d_{n}}}{l}+ \vert c_{n} \vert \frac{\overline{d_{n}}}{l}+A^{\ast}_{nj\delta} + \vert c_{n} \vert \bigl( B^{\ast}_{nj\delta}+l \delta^{2}\overline{P^{I}_{n}}\bigr) +\delta \biggr]v_{n}^{2}(t)\\ &-\delta \bigl[u_{n}^{2}(t)+v_{n}^{2}(t) \bigr]+N \Biggr\} . \end{aligned} $$
(7)
By using \((h_{5})\) and \((h_{6})\), from (7), it follows that
$$ \begin{aligned}[b] \frac{dV_{n}(t)}{dt} \leq{}&\lambda \sum_{j=1}^{l} \biggl\{ \biggl[ 2 \frac{\underline{d_{j}}}{l}- \vert c_{j} \vert \frac{\overline{d_{j}}}{l}-A_{jn\delta}- \vert c_{j} \vert (B_{jn\delta }+l\delta^{2}\overline{P^{R}_{j}}-\delta \biggr]u_{j}^{2}(t) ) \\ &- \biggl[ 2\frac{\underline{d_{n}}}{l}- \vert c_{n} \vert \frac{\overline{d_{n}}}{l}-A_{nj\delta}- \vert c_{n} \vert \bigl(B_{nj\delta}+l \delta^{2}\overline{P^{R}_{n}}\bigr)-\delta \biggr]u_{n}^{2}(t) \biggr\} \\ & +\lambda\sum _{j=1}^{l} \biggl\{ \biggl[ 2\frac{\underline{d_{j}}}{l}- \vert c_{j} \vert \frac{\overline{d_{j}}}{l}-A^{\ast}_{jn\delta}- \vert c_{j} \vert \bigl(B^{\ast}_{jn\delta} +l \delta^{2}\overline{P^{I}_{j}}\bigr)-\delta \biggr]v_{j}^{2}(t) \\ &- \biggl[ 2\frac{\underline{d_{n}}}{l} - \vert c_{n} \vert \frac{\overline{d_{n}}}{l}-A^{\ast}_{nj\delta}- \vert c_{n} \vert \bigl(B^{\ast}_{nj\delta }+l \delta^{2}\overline{P^{I}_{n}}\bigr) -\delta \biggr]v_{n}^{2}(t) \biggr\} \\ &-\delta\bigl[u_{n}^{2}(t)+v_{n}^{2}(t) \bigr]+N \}. \end{aligned} $$
(8)
Letting \(b_{nj}=1\) (\(n\neq j\)), \(b_{nj}=0\), \(n=j\), \(G_{nj}(u^{2}_{n}(t), u^{2}_{j}(t))= [2\frac{\underline{d_{j}}}{l}-|c_{j}|\frac{\overline{d_{j}}}{l}-A_{jn\delta }-|c_{j}|(B_{jn\delta}+l\delta^{2}\overline{P^{R}_{j}})-\delta ] u_{j}^{2}(t)- [ 2\frac{\underline{d_{n}}}{l}-|c_{n}|\frac{\overline{d_{n}}}{l}-A_{nj\delta}-|c_{n}|(B_{nj\delta }+l\delta^{2}\overline{P^{R}_{n}})-\delta ]u_{n}^{2}(t)\) and \(p_{n}(u_{n}^{2}(t))= [ 2\frac{\underline{d_{n}}}{l}-|c_{n}|\frac{\overline{d_{n}}}{l}-A_{nj\delta}-|c_{n}|(B_{nj\delta }+l\delta^{2}\overline{P^{R}_{n}})-\delta ]u_{n}^{2}(t)\); \(b^{\ast}_{nj}=1\) (\(n\neq j\)), \(b_{nj}^{\ast}=0\), \(n=j\), \(G^{\ast}_{nj}(v^{2}_{n}(t), v^{2}_{j}(t))= [2\frac{\underline{d_{j}}}{l}-|c_{j}|\frac{\overline{d_{j}}}{l}-A^{\ast}_{jn\delta }-|c_{j}|(B^{\ast}_{jn\delta}+l\delta^{2}\overline{P^{I}_{j}})-\delta ] v_{j}^{2}(t)- [ 2\frac{\underline{d_{n}}}{l}-|c_{n}|\frac{\overline{d_{n}}}{l}-A^{\ast}_{nj\delta}-|c_{n}|(B^{\ast}_{nj\delta}+l\delta^{2}\overline{P^{I}_{n}}) -\delta ]v_{n}^{2}(t)\) and \(p^{\ast}_{n}(v_{n}^{2}(t))= [ 2\frac{\underline{d_{n}}}{l}-|c_{n}|\frac{\overline{d_{n}}}{l}-A^{\ast}_{nj\delta}-|c_{n}|(B^{\ast}_{nj\delta}+l\delta^{2}\overline{P^{I}_{n}})-\delta ]v_{n}^{2}(t)\), then we have, from (8),
$$\begin{aligned}& \begin{aligned}[b] \frac{dV_{n}(t)}{dt} \leq{}&\lambda \Biggl\{ \sum_{j=1}^{l}b_{nj}G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr) \\ &+\sum_{j=1}^{l} b^{\ast}_{nj}G^{\ast}_{nj} \bigl(v_{n}^{2}(t), v_{j}^{2}(t) \bigr)- \sum_{j=1}^{l}\delta\bigl[u_{n}^{2}(t)+v_{n}^{2}(t) \bigr]+\frac{N}{l} \Biggr\} , \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned}& G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr)=p_{j}\bigl(u_{j}^{2}(t) \bigr)-p_{n}\bigl(u_{n}^{2}(t)\bigr), \end{aligned}$$
(10)
and
$$ G^{\ast}_{nj} \bigl(v_{n}^{2}(t), v_{j}^{2}(t) \bigr)=p^{\ast}_{j} \bigl(v_{j}^{2}(t)\bigr)-p^{\ast}_{n} \bigl(v_{n}^{2}(t)\bigr). $$
(11)
We construct the following Lyapunov function for system (3):
$$V(t)= \sum_{n=1}^{l}c^{\ast}_{n}V_{n}(t), $$
where \(c^{\ast}_{n}>0 \) is the cofactor of the nth diagonal element of the Laplacian matrix of \((g, B)\). From (9), we have
$$ \begin{aligned}[b] \frac{dV(t)}{dt}={}&\sum _{n=1}^{l}c^{\ast}_{n} \frac{dV_{n}(t)}{dt} \\ ={}&\lambda\sum_{n=1}^{l}c^{\ast}_{n} \sum_{j=1}^{l} \bigl\{ b_{nj}G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr)+b_{nj}^{\ast}G^{\ast}_{nj} \bigl(v_{j}^{2}(t), v_{j}^{2}(t)\bigr) \\ & -\delta \bigl[u_{n}^{2}(t)+v_{n}^{2}(t) \bigr]+N \bigr\} . \end{aligned} $$
(12)
From Lemma 2.2, it follows that
$$\begin{aligned}& \sum_{n=1}^{l}\sum _{j=1}^{l}c^{\ast}_{n}b_{nj}G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr)= \sum_{Q\in\Omega}W(Q)\sum_{(n, j)\in K(C_{\Omega})}G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr), \end{aligned}$$
(13)
$$\begin{aligned}& \sum_{n=1}^{l}\sum _{j=1}^{l}c^{\ast}_{n}b^{\ast}_{nj}G^{\ast}_{nj} \bigl(v_{n}^{2}(t), v_{j}^{2}(t) \bigr)= \sum_{Q\in\Omega}W(Q)\sum_{(n, j)\in K(C_{\Omega})}G^{\ast}_{nj} \bigl(v_{n}^{2}(t), v_{j}^{2}(t) \bigr). \end{aligned}$$
(14)
By substituting (10) into (13) and substituting (11) into (14), it follows that, from the fact \(W(Q)>0\),
$$ \begin{aligned}[b] &\sum_{n=1}^{l} \sum_{j=1}^{l}c^{\ast}_{n}b_{nj}G_{nj} \bigl(u_{n}^{2}(t), u_{j}^{2}(t) \bigr) \\ &\quad=\sum_{Q\in\Omega}W(Q)\sum _{(n, j)\in K(C_{\Omega})} \bigl[p_{j}\bigl(u_{j}^{2}(t) \bigr)-p_{n}\bigl(u_{n}^{2}(t)\bigr) \bigr]\leq0 \end{aligned} $$
(15)
and
$$ \begin{aligned}[b] &\sum_{n=1}^{l} \sum_{j=1}^{l}c^{\ast}_{n}b^{\ast}_{nj}G^{\ast}_{nj} \bigl(v_{n}^{2}(t), v_{j}^{2}(t) \bigr) \\ &\quad=\sum_{Q\in\Omega}W(Q)\sum _{(n, j)\in K(C_{\Omega})} \bigl[p^{\ast}_{j} \bigl(v_{j}^{2}(t)\bigr)-p^{\ast}_{n} \bigl(v_{n}^{2}(t) \bigr) \bigr]\leq 0. \end{aligned} $$
(16)
Substituting (15) and (16) into (12) gives
$$ \frac{dV(t)}{dt}\leq\lambda\sum_{n=1}^{l}c_{n}^{\ast}\sum_{j=1}^{l} \bigl(-\delta \bigl[u_{n}^{2}(t)+v_{n}^{2}(t)\bigr]+N \bigr). $$
(17)
Integrating (17) from 0 to ω gives
$$ \int_{0}^{\omega}\sum_{n=1}^{l}c_{n}^{\ast}\delta \bigl[u_{n}^{2}(s)+v_{n}^{2}(s) \bigr]\,ds\leq\omega N \sum_{n=1}^{l} c_{n}^{\ast}. $$
(18)
Integrating (17) from 0 to t gives
$$ \begin{aligned}[b] V(t)\leq{}& V(0)+ \int_{0}^{t}\sum_{n=1}^{l}c_{n}^{\ast}l \bigl(\delta\bigl[u_{n}^{2}(s)+v_{n}^{2}(s) \bigr]+N \bigr)\,ds \\ \leq{}& V(0)+ \int_{0}^{\omega}\sum_{n=1}^{l}c_{n}^{\ast}l \bigl(\delta\bigl[u_{n}^{2}(s)+v_{n}^{2}(s) \bigr]+N \bigr)\,ds. \end{aligned} $$
(19)
Substituting (18) into (19) gives
$$ V(t)\leq V(0)+2l\omega N\sum_{n=1}^{l}c_{n}^{\ast}. $$
(20)
From (20) and the definitions of \(V_{n}(t)\) and \(V_{1n}(t)\), we have
$$ \begin{aligned}[b] &\sum_{n=1}^{l}c_{n}^{\ast}\bigl\{ \bigl[u_{n}(t)-c_{n}u_{n}(t-\tau _{1})\bigr]^{2}+\bigl[v_{n}(t)-c_{n}v_{n}(t- \tau_{1}\bigr]^{2} \bigr\} \\ & \quad\leq V(0)+2l\omega N\sum_{n=1}^{l}c_{n}^{\ast}. \end{aligned} $$
(21)
Letting \(|u_{n}(\xi)|=\max_{t\in[0, \omega]}|u_{n}(t)|\), \(|v_{n}(\eta)|=\max_{t\in[0, \omega]}|v_{n}(t)|\), then from (21), it follows that
$$\sum_{n=1}^{l}c_{n}^{\ast}\bigl(1- \vert c_{n} \vert \bigr)^{2} \bigl[u_{n}^{2}(\xi)+v^{2}_{n}(\eta)\bigr] \leq V(0)+2l\omega N\sum_{n=1}^{l}c_{n}^{\ast}. $$
Hence there exists a positive constant M such that \(\|(u(t), v(t))^{T}\|\leq M\). This completes the proof of Lemma 3.1. □
Theorem 3.1
Assume that
\((h_{1})\)–\((h_{4})\)
hold and
\(|c_{n}|<1\). Then system (2) has at least one
ω
periodic solution.
Proof
We will prove the existence of periodic solutions of system (2) by means of using Lemma 2.1. We are concerned with the Banach spaces: \(X^{\ast}=Z^{\ast}=\{(u(t), v(t))^{T}\in C(R, R^{2l}): u(t+\omega)=u(t), v(t+\omega)=v(t)\} \) with the norm \(\|(u(t), v(t))^{T}\|=\sum_{n=1}^{l}\max_{t\in[0, \omega]}(|u_{n}(t)|+|v_{n}(t)|)\). Set \(L^{\ast}: \operatorname{Dom}L^{\ast}\subset X^{\ast}\rightarrow X^{\ast}\), \(L^{\ast}(u(t), v(t))= (\frac{d[K_{1}u_{1}(t)]}{dt}, \frac{d[K_{2}u_{2}(t)]}{dt},\ldots, \frac{d[K_{l}u_{l}(t])}{dt}, \frac{d[K_{1}v_{1}(t)]}{dt}, \frac {d[K_{2}v_{2}(t)]}{dt},\ldots, \frac{d[K_{l}v_{1}(t)]}{dt} )^{T}\) and
$$N^{\ast}\bigl(u(t),v(t)\bigr)= \bigl(f_{1}(t), f_{2}(t),\ldots, f_{l}(t), f^{\ast}_{1}(t), f^{\ast}_{2}(t),\ldots, f^{\ast}_{l}(t) \bigr), $$
where, for \(n\in\textbf{L}\),
$$ \begin{gathered} \begin{aligned} f_{n}(t)={}&{-}d_{n}(t)u_{n}(t)+ \sum_{j=1}^{l}b^{R}_{nj}(t)F_{j}^{R} \bigl(u_{j}(t),v_{j}(t)\bigr)-\sum _{j=1}^{l}b_{nj}^{I}(t)F_{j}^{I} \bigl(u_{j}(t), v_{j}(t)\bigr) \\ &+\sum_{j=1}^{l}e_{nj}^{R}(t) G_{j}^{R}\bigl(u_{j}^{t}, v_{j}^{t}\bigr)-\sum_{j=1}^{l}e_{nj}^{I}(t)G_{j}^{I} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P^{R}_{n}(t), \end{aligned} \\ \begin{aligned}f^{\ast}_{n}(t)={}&{-}d_{n}(t)v_{n}(t)+ \sum_{j=1}^{l}b^{R}_{nj}(t)F_{j}^{I} \bigl(u_{j}(t),v_{j}(t)\bigr)+\sum _{j=1}^{l}b_{nj}^{I}(t) F_{j}^{R}\bigl(u_{j}(t), v_{j}(t) \bigr) \\ & +\sum_{j=1}^{l}e_{nj}^{R}(t) G_{j}^{I}\bigl(u_{j}^{t}, v_{j}^{t}\bigr)+\sum_{j=1}^{l}e_{nj}^{I}(t)G_{j}^{R} \bigl(u_{j}^{t}, v_{j}^{t} \bigr)+P^{I}_{n}(t). \end{aligned} \end{gathered} $$
Thus \(\operatorname{Ker}L^{\ast}=\{u=(u(t),v(t))^{T}\in X^{\ast}: u\in R^{2l}\}\), \(\operatorname{Im} L^{\ast}=\{w\in Z^{\ast}: \int_{0}^{\omega}w(t)\,dt=0\}\) is closed in \(Z^{\ast}\) and \(\operatorname{Dim} \operatorname{Ker} L^{\ast}=2l=\operatorname{Codim} \operatorname{Im}L^{\ast}\). Hence, the operator \(L^{\ast}\) is a Fredholm mapping of index 0. We construct the projectors \(P^{\ast}: X^{\ast}\cap \operatorname{Dom} L^{\ast}\rightarrow \operatorname{Ker}L^{\ast}\) and \(Q^{\ast}: Z^{\ast}\rightarrow Z^{\ast}\) as
$$\begin{gathered} P^{\ast}u=\frac{1}{\omega} \int_{0}^{\omega}u(t)\,dt,\quad u\in X^{\ast}; \\ Q^{\ast}w=\frac{1}{\omega} \int_{0}^{\omega}w(t)\,dt,\quad w\in Z^{\ast}.\end{gathered} $$
Therefore, \(\operatorname{Lm}P^{\ast}=\operatorname{Ker}L^{\ast}\), \(\operatorname{Lm}L^{\ast}=\operatorname{Ker}Q^{\ast}=\operatorname {Im}(I-Q^{\ast})\). Moreover, the generalized inverse \(K_{p} \) of \(L^{\ast}\) is given as \(K_{p}=(L^{\ast})^{-1} (\int_{0}^{t}w(s)\,ds )\). Since \(|c_{n}|<1\), from Lemma 2.3, it is not difficult to show that \(N^{\ast}\) is \(L^{\ast}\)-compact on Ω̅. The concrete form of the operator equation \(L^{\ast}(u, v)=\lambda N^{\ast}(u, v)\), \((u, v)^{T}\in X^{\ast}\), \(\lambda\in(0, 1) \) is system (2). From Lemma 3.1, for every periodic solution \((u(t), v(t))^{T}=(u_{1}(t), u_{2}(t),\ldots, u_{l}(t), v_{1}(t),v_{2}(t),\ldots, v_{l}(t))^{T} \) of system (2), there exists a positive constant M such that \(\|(u(t), v(t))^{T}\|< M\). We set \(\Omega=\{(u(t), v(t))^{T}\in X^{\ast}: \|(u(t), v(t))^{T}\|< M\}\), \(M>\sqrt{\frac{2Nl\sum_{n=1}^{l}c_{n}^{\ast}}{\min_{1\leq n\leq l}\{c^{\ast}_{n}\}}}\). Then, for each \((u(t), v(t))^{T}\in \partial\Omega\cap \operatorname{Dom} L^{\ast}\), \(L^{\ast}(u(t), v(t))\neq\lambda N^{\ast}(u(t), v(t))\), \(\lambda\in(0, 1)\). Hence, condition (1) in Lemma 2.2 is satisfied. Secondly, we will show that when \((u(t), v(t))^{T}\in\partial\Omega\cap\operatorname{Ker}L^{\ast}\), \(Q^{\ast}N^{\ast}(u(t), v(t))\neq0\). Since \((u, v)^{T}\in\partial\Omega\cap\operatorname{Ker}L^{\ast}\), \((u, v)^{T} \) is a constant vector with \(\|(u, v)^{T}\|=M\), then when \((u, v)^{T}\in\partial\Omega\cap \operatorname{Ker}L^{\ast}\), \(Q^{\ast}N^{\ast}(u, v)= ( f_{1}(\xi_{1}), f_{2}(\xi_{2}),\ldots, f_{l}(\xi_{l}), f_{1}^{\ast}(\xi_{1}), f_{2}^{\ast}(\xi_{2}),\ldots, f_{l}^{\ast}(\xi_{l}) )\), where \(\xi_{i}\ (i=1, 2,\ldots,l)\in[0, \omega]\). When \((u, v)^{T}\in\partial\Omega\cap\operatorname{Ker}L^{\ast}\), we have
$$ \begin{aligned}[b] &[u_{n}-c_{n}u_{n}, v_{n}-c_{n}v_{n}] \bigl[Q^{\ast}N^{\ast}(u, v)_{n}\bigr]^{T} \\ &\quad=[u_{n}-c_{n}u_{n}, v_{n}-c_{n}v_{n}] \bigl(f_{n}(\xi_{n}), f_{n}^{\ast}( \xi_{n})\bigr)^{T} \\ &\quad =(u_{n}-c_{n}u_{n}) \Biggl[-d_{n}( \xi_{n})u_{n}+\sum_{j=1}^{l}b_{nj}^{R}( \xi _{n})F_{j}^{R}(u_{j},v_{j}) \\ &\qquad{} -\sum_{j=1}^{l}b_{nj}^{I}( \xi_{n}) F_{j}^{I}(u_{j},v_{j})+ \sum_{j=1}^{l}e^{R}_{nj}( \xi_{n}) G_{j}^{R}(u_{j},v_{j})- \sum_{j=1}^{l}e_{nj}^{I}( \xi_{n})G_{j}^{I}(u_{j}, v_{j})+P^{R}_{n}(\xi_{n}) \Biggr] \\ &\qquad{}+(v_{n}-c_{n}v_{n}) \Biggl[-d_{n}(\xi_{n})v_{n}+ \sum _{j=1}^{l}b_{nj}^{R}( \xi_{n})F_{j}^{I}(u_{j},v_{j}) \\ &\qquad{}-\sum_{j=1}^{l}b_{nj}^{I}( \xi_{n}) F_{j}^{R}(u_{j},v_{j})+ \sum_{j=1}^{l}e^{R}_{nj}( \xi_{n}) G_{j}^{I}(u_{j},v_{j})- \sum_{j=1}^{l}e_{nj}^{I}( \xi_{n})G_{j}^{R}(u_{j}, v_{j})+P^{I}_{n}(\xi_{n}) \Biggr] \\ &\qquad{}+0. \end{aligned} $$
(22)
It is obvious that
$$ \begin{aligned}[b] 0={}& \vert c_{n} \vert \Biggl(\sum_{j=1}^{l}B_{nj} \delta+\delta^{2}\overline{P^{R}_{n}}\Biggr) \bigl(u_{n}^{2}-u_{n}^{2}\bigr)+ \frac{1+ \vert c_{n} \vert }{1-\sigma}\sum_{j=1}^{l}E_{nj} \bigl(u_{j}^{2}-u_{j}^{2}\bigr) \\ &+ \vert c_{n} \vert \Biggl(\sum_{j=1}^{l}B_{nj\delta}^{\ast}+ \delta^{2}\overline{ P^{I}_{n}}\Biggr) \bigl(v_{n}^{2}-v_{n}^{2}\bigr)+ \frac{1+ \vert c_{n} \vert }{1-\sigma}\sum_{j=1}^{l}f_{nj} \bigl(v_{j}^{2}-v_{j}^{2}\bigr). \end{aligned} $$
(23)
Substituting (23) into (22) gives
$$\begin{aligned} & [u_{n}-c_{n}u_{n}, v_{n}-c_{n}v_{n}] \bigl[QN(u, v)_{n} \bigr]^{T} \\ &\quad=[u_{n}-c_{n}u_{n}, v_{n}-c_{n}v_{n}] \bigl(f_{n}(\xi_{n}), f_{n}^{\ast}( \xi_{n})\bigr)^{T} \\ &\quad\leq(u_{n}-c_{n}u_{n}) \Biggl[-d_{n}( \xi_{n})u_{n}+\sum_{j=1}^{l}b_{nj}^{R}( \xi _{n})F_{j}^{R}(u_{j},v_{j})-\sum_{j=1}^{l}b_{nj}^{I}( \xi_{n}) F_{j}^{I}(u_{j},v_{j}) \\ &\qquad{}+ \sum_{j=1}^{l} e^{R}_{nj}( \xi_{n}) G_{j}^{R}(x_{j},y_{j})-\sum_{j=1}^{l}e_{nj}^{I}( \xi_{n})G_{j}^{I}(u_{j}, v_{j})+P^{R}_{n}(\xi_{n}) \Biggr] \\ & \qquad{}+(v_{n}-c_{n}v_{n}) \Biggl[-d_{n}( \xi_{n})v_{n}+\sum_{j=1}^{l}b_{nj}^{R}( \xi_{n})F_{j}^{I}(u_{j},v_{j})+ \sum_{j=1}^{l}b_{nj}^{I}( \xi_{n}) F_{j}^{R}(u_{j},v_{j}) \\ &\qquad{}+ \sum_{j=1}^{l}e^{R}_{nj}( \xi_{n}) G_{j}^{I}(u_{j},v_{j})+\sum_{j=1}^{l}e_{nj}^{I}( \xi_{n}) G_{j}^{R}(u_{j}, v_{j})+P^{I}_{n}(\xi_{n}) \Biggr] \\ &\qquad{}+ \vert c_{n} \vert \Biggl(\sum_{j=1}^{l}B_{nj} \delta+\delta^{2}\overline{P^{R}_{n}}\Biggr) \bigl(u_{n}^{2}-u_{n}^{2}\bigr)+ \frac{1+ \vert c_{n} \vert }{1-\sigma}\sum_{j=1}^{l}E_{nj} \bigl(u_{j}^{2}-u_{j}^{2}\bigr) \\ &\qquad{}+ \vert c_{n} \vert (\sum_{j=1}^{l}B_{nj\delta}^{\ast}+ \delta^{2}\overline{ P^{I}_{n}}\bigl(v_{n}^{2}-v_{n}^{2} \bigr)+\frac{1+ \vert c_{n} \vert }{1-\sigma}\sum_{j=1}^{l}f_{nj} \bigl(v_{j}^{2}-v_{j}^{2}\bigr). \end{aligned}$$
(24)
From (24), the same proofs as those of (7)–(17) give
$$ \begin{aligned}[b] &\sum_{n=1}^{l}c^{\ast}_{n}[u_{n}-c_{n}y_{n}, v_{n}-c_{n}v_{n}] \bigl[Q^{\ast}N^{\ast}(u, v)_{n}\bigr]^{T} \\ &\quad=\sum_{n=1}^{l}c^{\ast}_{n}[u_{n}-c_{n}u_{n}, v_{n}-c_{n}v_{n}]\bigl(f_{n}( \xi_{n}), f_{n}^{\ast}(\xi_{m}) \bigr)^{T} \\ &\quad\leq\sum_{n=1}^{l}c^{\ast}_{n} \sum_{j=1}^{l} \biggl\{ \biggl[ 2 \frac{\underline{d_{j}}}{l}- \vert c_{j} \vert \frac{\overline{d_{j}}}{l}-A_{jn\delta}- \vert c_{j} \vert \bigl(B_{jn\delta }+l\delta^{2}\overline{P^{R}_{j}} \bigr)-\delta \biggr]u_{j}^{2} \\ & \qquad{}- \biggl[ 2 \frac{\underline{d_{n}}}{l}- \vert c_{n} \vert \frac{\overline{d_{n}}}{l}-A_{nj\delta}- \vert c_{n} \vert \bigl(B_{nj\delta}+l \delta^{2}\overline{P^{R}_{n}}\bigr)-\delta \biggr]u_{n}^{2} \biggr\} \\ & \qquad{} +\sum_{j=1}^{l} \biggl\{ \biggl[ 2\frac{\underline{d_{j}}}{l}- \vert c_{j} \vert \frac{\overline{d_{j}}}{l}-A^{\ast}_{jn\delta}- \vert c_{j} \vert \bigl(B^{\ast}_{jn\delta}+l \delta^{2}\overline{P^{I}_{j}}\bigr)-\delta \biggr] v_{j}^{2} \\ &\qquad{}- \biggl[ 2\frac{\underline{d_{n}}}{l}- \vert c_{n} \vert \frac{\overline{d_{n}}}{l}-A^{\ast}_{nj\delta}- \vert c_{n} \vert \bigl(B^{\ast}_{nj\delta} +l\delta^{2}\overline{P^{I}_{n}}\bigr) - \delta \biggr]v_{n}^{2}(t) \biggr\} \\ &\qquad{}-\delta\bigl(u_{n}^{2}+v_{n}^{2} \bigr)+N \} \\ &\quad\leq\sum_{n=1}^{l}\sum _{j=1}^{l}c_{n}^{\ast}\bigl[-\delta \bigl(u_{n}^{2}+v_{n}^{2}\bigr)+N\bigr]. \end{aligned} $$
(25)
Since \(\sum_{n=1}^{l}(|u_{n}|+|v_{n}|)=M\), then
$$ M^{2}\leq l\sum_{n=1}^{l} \bigl(u_{n}^{2}+v_{n}^{2}+2 \vert u_{n} \vert \vert v_{n} \vert \bigr)\leq 2l\sum _{n=1}^{l}\bigl(u_{n}^{2}+v_{n}^{2} \bigr). $$
Namely,
$$ \sum_{n=1}^{l} \bigl(u_{n}^{2}+v_{n}^{2}\bigr)\geq \frac{M^{2}}{2l}. $$
(26)
Substituting (26) into (25) gives
$$ \begin{aligned}[b] &\sum_{n=1}^{l}c^{\ast}_{n}[u_{n}-c_{n}y_{n}, v_{n}-c_{n}v_{n}] \bigl[Q^{\ast}N^{\ast}(u, v)_{n}\bigr]^{T} \\ &\quad\leq-\min_{1\leq n\leq l}\bigl\{ c_{n}^{\ast}\bigr\} \frac{M^{2}}{2}+Nl\sum_{n=l}^{l} \bigl\{ c_{n}^{\ast}\bigr\} < 0. \end{aligned} $$
(27)
Thus when \((u,v)\in\partial\Omega\cap\operatorname{Ker}L^{\ast}\), \(Q^{\ast}N^{\ast}(u, v)\neq0\). Thus, condition (b) in Lemma 2.1 is satisfied.
Thirdly, we show that when \((u, v)^{T}\in\partial\Omega\cap\operatorname{Ker}L^{\ast}\), \(\operatorname{deg}\{J^{\ast}Q^{\ast}N^{\ast}, \Omega\cap\operatorname {Ker}L^{\ast}, 0\}\neq0\). We construct a mapping \(H(u, v, \mu^{\ast}) \) by setting
$$ \begin{aligned} H\bigl(u, v, \mu^{\ast}\bigr) ={}&{-} \mu^{\ast}(\underline{d_{1}}u_{1}, \underline{d_{2}}u_{2},\ldots, \underline{d_{l}}v_{l}, \underline{d_{1}}v_{1}, \underline{d_{2}}v_{2}, \ldots, \underline{d_{l}}v_{l}) \\ &+\bigl(1-\mu^{\ast}\bigr) \bigl(f_{1}(\xi_{1}), f_{2}(\xi_{2}),\ldots, f_{l}(\xi_{l}), f_{1}^{\ast}(\xi_{1}), f_{2}^{\ast}( \xi_{2}),\ldots, f_{l}^{\ast}(\xi_{l}) \bigr), \end{aligned} $$
where \(\forall(u, v, \mu^{\ast})\in\partial\Omega\cap \operatorname{Ker}L^{\ast}\times[0, 1]\). If when \((u, v, \mu^{\ast})\in\partial\Omega\cap\operatorname{Ker}L^{\ast}=R^{2l}\cap \operatorname{Ker} L^{\ast}\), \(H(u, v,\mu^{\ast})=0\), then for \(n\in\textbf{L}\),
$$ 0=-\mu^{\ast}\underline{d_{n}}u_{n}+ \bigl(1-\mu^{\ast}\bigr)f_{n}(\xi_{n}) $$
(28)
and
$$ 0=-\mu^{\ast}\underline{d_{n}}u_{n}+ \bigl(1-\mu^{\ast}\bigr)f^{\ast}_{n}(\xi_{n}). $$
(29)
From (28) and (29), we have
$$\begin{aligned} 0 ={}&(u_{n}-c_{n}u_{n}) \bigl[-\mu^{\ast}\underline{d_{n}}u_{n}+\bigl(1- \mu^{\ast}\bigr)f_{n}(\xi_{n})\bigr] \\ &+(v_{n}-c_{n}v_{n})\bigl[-\mu^{\ast}\underline{d_{n}}v_{n}+\bigl(1-\mu^{\ast}\bigr)f_{n}^{\ast}(\xi_{n})\bigr] \\ \leq{}&{-}(1-c_{n})\mu^{\ast}\underline{d_{n}}u_{n}^{2}-(1-c_{n}) \bigl(1-\mu^{\ast}\bigr)\underline{d_{n}}u^{2}_{n}+u_{n} \Biggl\{ \sum_{j=1}^{l}b_{nj}^{R}( \xi_{n})F_{j}^{R}(u_{j}, v_{j}) \\ & -\sum_{j=1}^{l}b_{nj}^{I}( \xi_{n}) F_{j}^{I}(u_{j}, v_{j})+\sum_{j=1}^{l}e_{nj}^{R}( \xi_{n})G_{j}^{R}(u_{j}, v_{j})-\sum_{j=1}^{l} e_{nj}^{I}(\xi_{n})G_{j}^{I}(u_{j}, v_{j})+P^{R}_{n}(\xi_{n})] \Biggr\} \\ &-(1-c_{n}) \mu^{\ast}\underline{d_{n}}v_{n}^{2} -(1-c_{n}) \bigl(1-\mu^{\ast}\bigr)\overline{d_{n}}v^{2}_{n}+v_{n} \Biggl\{ \sum_{j=1}^{l}b_{nj}^{R}( \xi_{n})F_{j}^{I}(u_{j}, v_{j}) \\ &+\sum_{j=1}^{l}b_{nj}^{I}( \xi_{n})F_{j}^{R}(u_{j}, v_{j}) +\sum_{j=1}^{l}e_{nj}^{R}( \xi_{n}) G_{j}^{I}(u_{j}, v_{j})+\sum_{j=1}^{l} e_{nj}^{I}(\xi_{n})G_{j}^{R}(u_{j}, v_{j})+P^{I}_{n}(\xi_{n}) \Biggr\} \\ \leq{}& -(1-c_{n})\underline{d_{n}}u_{n}^{2}+(1-c_{n}) \bigl(1-\mu^{\ast}\bigr) \vert u_{n} \vert \Biggl\{ \sum _{j=1}^{l}\overline{b_{nj}^{R}} \bigl\vert F_{j}^{R}(u_{j}, v_{j}) \bigr\vert \\ &+\sum_{j=1}^{l}\overline{b_{nj}^{I}} \bigl\vert F_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l} \overline{e_{nj}^{R}} \bigl\vert G_{j}^{R}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l}\overline{e_{nj}^{I}} \bigl\vert G_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\overline{P^{R}_{n}} \Biggr\} \\ &-(1-c_{n}) \underline{d_{n}}v_{n}^{2}+(1-c_{n}) \bigl(1-\mu^{\ast}\bigr) \vert v_{n} \vert \Biggl\{ \sum_{j=1}^{l} \overline{b_{nj}^{R}} \bigl\vert G_{j}^{I}(u_{j}, v_{j}) \bigr\vert \\ &+\sum_{j=1}^{l} \overline{b_{nj}^{I}} \bigl\vert F_{j}^{R}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l}\overline{e_{nj}^{R}} \bigl\vert G_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l} \overline{e_{nj}^{I}} \bigl\vert G_{j}^{R}(u_{j}, v_{j}) \bigr\vert +\overline{P^{I}_{n}} \Biggr\} . \end{aligned}$$
(30)
By the same proofs as those in (7)–(17), from (30), we obtain
$$ \begin{aligned}[b] & {-}(1-c_{n}) \underline{d_{n}}u_{n}^{2}+(1-c_{n}) \bigl(1-\mu^{\ast}\bigr) \vert u_{n} \vert \Biggl\{ \sum _{j=1}^{l}\overline{b_{nj}^{R}} \bigl\vert F_{j}^{R}(u_{j}, v_{j}) \bigr\vert \\ &\qquad{} +\sum_{j=1}^{l} \overline{b_{nj}^{I}} \bigl\vert F_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l} \overline{e_{nj}^{R}} \bigl\vert G_{j}^{R}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l} \overline{e_{nj}^{I}} \bigl\vert G_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\overline{P^{R}_{n}} \Biggr\} \\ &\qquad{}-(1-c_{n})\underline{d_{n}}v_{n}^{2}+(1-c_{n}) \bigl(1-\mu^{\ast}\bigr) \vert v_{n} \vert \{\sum_{j=1}^{l} \overline {b_{nj}^{R}} \bigl\vert F_{j}^{I}(u_{j}, v_{j}) \bigr\vert \\ &\qquad{}+ \sum_{j=1}^{l} \overline{b_{nj}^{I}}|F_{j}^{R}(u_{j}, v_{j}) +\sum_{j=1}^{l} \overline{e_{nj}^{R}} \bigl\vert G_{j}^{I}(u_{j}, v_{j}) \bigr\vert +\sum_{j=1}^{l} \overline{e_{nj}^{I}} \bigl\vert G_{j}^{R}(u_{j}, v_{j}) \bigr\vert \\ &\quad< 0. \end{aligned} $$
(31)
Equation (31) contradicts (30), hence \(H(u, v, \mu^{\ast})\neq0 \) when \((u, v,\mu^{\ast})\in\partial\Omega\cap R^{2l}\cap\operatorname{Ker} L^{\ast}\). Hence, \(L^{\ast}(u, v, \mu^{\ast})\) is a homotopic mapping. Thus, we have
$$ \begin{gathered} \operatorname{deg} \bigl(J^{\ast}Q^{\ast}N^{\ast}\bigl(u, v, \mu^{\ast}\bigr), \partial \Omega\cap \operatorname{Ker}L^{\ast}, (0, 0,\ldots, 0) \bigr) \\ \quad=\operatorname{deg} \bigl(H(u, v, 0), \partial\Omega\cap \operatorname{Ker}L^{\ast}, (0, 0,\ldots, 0) \bigr) \\ \quad=\operatorname{deg} \bigl(H(u, v, 1), \partial\Omega\cap \operatorname{Ker}L^{\ast}, (0, 0,\ldots, 0) \bigr) \\ \quad\neq0. \end{gathered} $$
Thus condition (c) in Lemma 2.1 is satisfied. Hence the proof of Theorem 3.1 is complete. □