Denote
$$\begin{aligned} &f^{s}_{0}=\limsup_{u+v\rightarrow0}\max _{t\in[0,1]}\frac{f(t,u,v)}{\varphi _{p_{1}}(u+v)}, \qquad g^{s}_{0}=\limsup _{u+v\rightarrow0}\max_{t\in[0,1]}\frac{g(t,u,v)}{\varphi _{p_{2}}(u+v)}, \\ &f^{i}_{0}=\liminf_{u+v\rightarrow0}\min _{t\in[\theta_{1},\theta_{2}]}\frac {f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{i}_{0}=\liminf _{u+v\rightarrow0}\min_{t\in[\theta_{1},\theta_{2}]}\frac {g(t,u,v)}{\varphi_{p_{2}}(u+v)}, \\ &f^{s}_{\infty}=\limsup_{u+v\rightarrow\infty}\max _{t\in[0,1]}\frac {f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{s}_{\infty}=\limsup _{u+v\rightarrow\infty}\max_{t\in[0,1]}\frac {g(t,u,v)}{\varphi_{p_{2}}(u+v)}, \\ &f^{i}_{\infty}=\liminf_{u+v\rightarrow\infty}\min _{t\in[\theta_{1},\theta _{2}]}\frac{f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{i}_{\infty}=\liminf _{u+v\rightarrow\infty}\min_{t\in[\theta_{1},\theta _{2}]}\frac{g(t,u,v)}{\varphi_{p_{2}}(u+v)}. \\ &A= \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\,d\tau \biggr) \,ds,\qquad B= \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)\,d\tau \biggr) \,ds, \\ &C= \int_{\theta_{1}}^{\theta_{2}} l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{1}(s,\tau)\,d\tau \biggr) \,ds,\qquad D= \int_{\theta _{1}}^{\theta_{2}} l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau)\,d\tau \biggr) \,ds. \end{aligned}$$
For \(f^{s}_{0}, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty)\), we define the symbols \(L_{1}, L_{2}, L_{3}\) and \(L_{4}\) as follows:
$$\begin{aligned} &L_{1}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma _{1}\gamma} \biggr) \frac{1}{f_{\infty}^{i}},\qquad L_{2}=\varphi_{p_{1}} \biggl( \frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr)\frac{1}{f_{0}^{s}}, \\ &L_{3}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma _{2}\gamma} \biggr) \frac{1}{g_{\infty}^{i}},\qquad L_{4}=\varphi_{p_{2}} \biggl( \frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{0}^{s}}. \end{aligned}$$
Theorem 3.1
(1) If
\(f^{s}_{0}, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{1} < L_{2}, L_{3} < L_{4}\), then for each
\(\lambda\in(L_{1},L_{2})\)
and
\(\mu\in (L_{3},L_{4})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(2) If
\(f^{s}_{0} = 0, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{3} < L_{4}\), then for each
\(\lambda\in(L_{1},\infty)\)
and
\(\mu\in(L_{3}, L_{4})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(3) If
\(g^{s}_{0} = 0, f^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{1} < L_{2}\), then for each
\(\lambda\in(L_{1}, L_{2})\)
and
\(\mu\in(L_{3}, \infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(4) If
\(f^{s}_{0} = g^{s}_{0} = 0, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty)\), then for each
\(\lambda\in(L_{1},\infty)\)
and
\(\mu\in(L_{3},\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(5) If
\(\{f^{s}_{0}, g^{s}_{0} \in(0,\infty), f^{i}_{\infty}= \infty\}\)
or
\(\{ f^{s}_{0}, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty\}\), then for each
\(\lambda\in(0,L_{2})\)
and
\(\mu\in(0,L_{4})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(6) If
\(\{f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty\}\)
or
\(\{ f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), f^{i}_{\infty}=\infty\}\), then for each
\(\lambda\in(0,\infty)\)
and
\(\mu\in(0,L_{4})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(7) If
\(\{f^{s}_{0}\in(0,\infty), g^{s}_{0} = 0, g^{i}_{\infty}= \infty\}\)
or
\(\{ f^{s}_{0}\in(0,\infty), g^{s}_{0} = 0, f^{i}_{\infty}= \infty\}\), then for each
\(\lambda\in(0,L_{2})\)
and
\(\mu\in(0,\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(8) If
\(\{f^{s}_{0} = g^{s}_{0} = 0, g^{i}_{\infty}= \infty\}\)
or
\(\{f^{s}_{0} = g^{s}_{0} = 0, f^{i}_{\infty}= \infty\}\), then for each
\(\lambda\in(0,\infty )\)
and
\(\mu\in(0,\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
Proof
Because the proofs of the above cases are similar, in what follows we will prove two of them, namely cases (1) and (6).
(1) For any \(\lambda\in(L_{1},L_{2})\) and \(\mu\in(L_{3},L_{4})\), there exists \(0<\varepsilon< \min\{f^{i}_{\infty},g^{i}_{\infty}\}\) such that
$$\begin{aligned} &\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\frac{1}{f_{\infty}^{i}-\varepsilon} \leq \lambda\leq\varphi _{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac {1}{f_{0}^{s}+\varepsilon}, \\ &\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma_{2}\gamma } \biggr)\frac{1}{g_{\infty}^{i}-\varepsilon} \leq \mu\leq\varphi _{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr) \frac {1}{g_{0}^{s}+\varepsilon}. \end{aligned}$$
By the definitions of \(f^{s}_{0}\) and \(g^{s}_{0}\), there exists \(R_{1} > 0\) such that
$$\begin{aligned} &f(t,u,v) < \bigl(f^{s}_{0}+\varepsilon\bigr) \varphi_{p_{1}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{1}, \\ &g(t,u,v) < \bigl(g^{s}_{0}+\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{1}. \end{aligned}$$
Denote \(\Omega_{1} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{1}\}\), for any \((u,v)\in P\cap\partial\Omega_{1}\) and \(t\in[0,1]\), we have \(0\leq u(t)+v(t)\leq \Vert u \Vert + \Vert v \Vert = \Vert (u,v) \Vert _{Y} = R_{1}\), then
$$\begin{aligned} T_{1}(u,v) (t)={}&\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(t,s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \leq{}&\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac {M_{1}}{\Gamma(\beta_{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{0}+\varepsilon\bigr)\varphi_{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \leq{}&\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda)\varphi_{q_{1}} \bigl(f^{s}_{0}+\varepsilon\bigr) \int_{0}^{1}l_{1}(s)\varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau) \varphi_{p_{1}}\bigl( \Vert u \Vert + \Vert v \Vert \bigr)\,d\tau \biggr)\,ds \\ ={}&\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{0}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\\ \leq{}& \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
so \(\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}\). In a similar manner, we deduce
$$\begin{aligned} T_{2}(u,v) (t)&=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(t,s) \varphi_{q_{2}} \biggl( \int _{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac {M_{2}}{\Gamma(\beta_{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau ) \bigl(g^{s}_{0}+\varepsilon\bigr)\varphi_{p_{2}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{0}+ \varepsilon\bigr)\bigr) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau )\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{0}+ \varepsilon\bigr)\bigr)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
(3.1)
then \(\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}\). Hence
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{1}. \end{aligned}$$
(3.2)
On the other hand, by the definitions of \(f^{i}_{\infty}\) and \(g^{i}_{\infty}\), there exists \(\overline{R_{2}} > 0\) such that
$$\begin{aligned} &f(t,u,v) \geq\bigl(f^{i}_{\infty}-\varepsilon\bigr) \varphi_{p_{1}}(u+v),\quad t\in [\theta_{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{2}}, \\ &g(t,u,v) \geq\bigl(g^{i}_{\infty}-\varepsilon\bigr) \varphi_{p_{2}}(u+v), \quad t\in [\theta_{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{2}}. \end{aligned}$$
Denote \(R_{2}=\max\{2R_{1},\frac{\overline{R_{2}}}{\gamma}\}\) and \(\Omega_{2} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{2}\}\). For any \((u,v)\in P\cap\partial\Omega _{2}\), we have \(\min_{t\in[\theta_{1},\theta_{2}]}(u(t)+v(t))\geq\gamma \Vert (u,v) \Vert _{Y} = \gamma R_{2}\geq\overline{R_{2}}\), then
$$\begin{aligned} &T_{1}(u,v) (\theta_{1})\\ &\quad=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(\theta_{1},s) \varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{1}}(\lambda) \int_{0}^{1}\frac{\beta _{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \bigl(f^{i}_{\infty}-\varepsilon \bigr)\varphi_{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}\bigl(\lambda \bigl(f^{i}_{\infty}-\varepsilon\bigr)\bigr) \int_{\theta_{1}}^{\theta _{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau ) \varphi_{p_{1}}\bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y} \bigr)\,d\tau \biggr)\,ds \\ &\quad=\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}}\bigl( \lambda\bigl(f^{i}_{\infty}-\varepsilon\bigr)\bigr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
and \(\Vert T_{1}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}\). Similarly, we have
$$\begin{aligned} &T_{2}(u,v) (\theta_{2})\\ &\quad=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{2}}(\mu) \int_{0}^{1}\frac{\beta _{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{\infty}-\varepsilon\bigr)\varphi_{p_{2}} \bigl(u(\tau )+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{\infty}-\varepsilon\bigr)\varphi _{p_{2}} \bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr) \,ds \\ &\quad=\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}}\bigl(\mu \bigl(g^{i}_{\infty}-\varepsilon\bigr)\bigr)D \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
then \(\Vert T_{2}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}\). So,
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap\partial\Omega_{2}. \end{aligned}$$
(3.3)
Therefore, by (3.2), (3.3) and Lemma 2.8, we conclude that Q has at least one fixed point \((u,v)\in P\cap(\overline{\Omega_{2}}\backslash \Omega_{1})\) with \(R_{1}\leq \Vert (u,v) \Vert _{Y}\leq R_{2}\).
(6) Suppose \(f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty\), then for any \(\lambda\in(0,\infty)\) and \(\mu\in(0,L_{4})\), there exists \(\varepsilon> 0\) such that
$$0 < \lambda< \varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac{1}{\varepsilon},\qquad \varphi_{p_{2}} \biggl(\frac{\Gamma(\beta _{2})}{(\beta_{2}-2)D\gamma_{2}\gamma} \biggr) \varepsilon< \mu< \varphi _{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr) \frac {1}{g_{0}^{s}+\varepsilon}. $$
By the definitions of \(f^{s}_{0}\) and \(g^{s}_{0}\), there exists \(R_{3} > 0\) such that
$$\begin{aligned} &f(t,u,v) < \varepsilon\varphi_{p_{1}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{3}, \\ &g(t,u,v) < \bigl(g^{s}_{0}+\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{3}. \end{aligned}$$
Denote \(\Omega_{3} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{3}\}\). For any \((u,v)\in P\cap\partial\Omega_{3}\) and \(t\in[0,1]\), we have
$$\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\varepsilon \varphi _{p_{1}}\bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda\varepsilon) \int_{0}^{1}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda\varepsilon)A \bigl\Vert (u,v) \bigr\Vert _{Y}< \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
then \(\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{3}\).
Similar to (3.1) of (1), we get \(\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial\Omega_{3}\), then
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap\partial\Omega_{3}. \end{aligned}$$
(3.4)
On the other hand, by \(g^{i}_{\infty}=\infty\), there exists \(\overline {R_{4}} > 0\) such that
$$g(t,u,v) \geq\frac{1}{\varepsilon}\varphi_{p_{2}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{4}}. $$
Let \(R_{4}=\max\{2R_{3},\frac{\overline{R_{4}}}{\gamma}\}\) and \(\Omega_{4} = \{ (u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{4}\}\). For any \((u,v)\in P\cap\partial\Omega _{4}\), we have \(\min_{t\in[\theta_{1},\theta_{2}]}(u(t)+v(t))\geq\gamma \Vert (u,v) \Vert _{Y} = \gamma R_{4}\geq\overline{R_{4}}\), then
$$\begin{aligned} T_{2}(u,v) (\theta_{2})&=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\varphi_{q_{2}}(\mu) \int_{0}^{1}\frac{\beta _{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau) \frac{1}{\varepsilon}\varphi_{p_{2}}\bigl(u(\tau )+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \frac{1}{\varepsilon}\varphi _{p_{2}}\bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}} \biggl( \frac{\mu}{\varepsilon} \biggr)D \bigl\Vert (u,v) \bigr\Vert _{Y}> \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}$$
Therefore
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert T_{2}(u,v) \bigr\Vert \geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap \partial\Omega_{4}. \end{aligned}$$
(3.5)
By (3.4), (3.5) and Lemma 2.8, we conclude that Q has at least one fixed point \((u,v)\in P\cap(\overline{\Omega_{4}}\backslash\Omega_{3})\) with \(R_{3}\leq \Vert (u,v) \Vert _{Y}\leq R_{4}\). This completes the proof. □
For \(f^{i}_{0}, g^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty)\), we define the symbols \(\widetilde{L_{1}}, \widetilde{L_{2}}, \widetilde{L_{3}}, \widetilde{L_{4}}\) as follows:
$$\begin{aligned} &\widetilde{L_{1}}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta _{1}-2)C\gamma_{1}\gamma} \biggr)\frac{1}{f_{0}^{i}},\qquad \widetilde{L_{2}}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr)\frac{1}{f_{\infty}^{s}}, \\ &\widetilde{L_{3}}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta _{2}-2)D\gamma_{2}\gamma} \biggr)\frac{1}{g_{0}^{i}}, \qquad \widetilde{L_{4}}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{\infty}^{s}}. \end{aligned}$$
Theorem 3.2
(1) If
\(f^{s}_{\infty}, g^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty)\), and
\(\widetilde{L_{1}} < \widetilde{L_{2}}, \widetilde{L_{3}} < \widetilde {L_{4}}\), then for each
\(\lambda\in(\widetilde{L_{1}}, \widetilde{L_{2}})\)
and
\(\mu\in(\widetilde{L_{3}}, \widetilde{L_{4}})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(2) If
\(f^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty), g^{s}_{\infty}= 0\), and
\(\widetilde{L_{1}} <\widetilde{L_{2}}\), then for each
\(\lambda\in(\widetilde {L_{1}},\widetilde{L_{2}})\)
and
\(\mu\in(\widetilde{L_{3}},\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(3) If
\(g^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty), f^{s}_{\infty}= 0, \widetilde{L_{3}} < \widetilde{L_{4}}\), then for each
\(\lambda\in(\widetilde {L_{1}},\infty)\)
and
\(\mu\in(\widetilde{L_{3}}, \widetilde{L_{4}})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(4) If
\(f^{i}_{0}, g^{i}_{0}\in(0,\infty), f^{s}_{\infty}= g^{s}_{\infty}= 0\), then for each
\(\lambda\in(\widetilde{L_{1}},\infty)\)
and
\(\mu\in (\widetilde{L_{3}},\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(5) If
\(\{f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty), f^{i}_{0} = \infty\}\)
or
\(\{f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty), g^{i}_{0} = \infty\}\), then for each
\(\lambda\in(0,\widetilde{L_{2}})\)
and
\(\mu\in(0,\widetilde{L_{4}})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(6) If
\(\{f^{i}_{0} = \infty, f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0\}\)
or
\(\{f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0, g^{i}_{0} = \infty\}\), then for each
\(\lambda\in(0,\widetilde{L_{2}})\)
and
\(\mu\in(0,\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(7) If
\(\{f^{i}_{0}=\infty, g^{s}_{\infty}\in(0,\infty), f^{s}_{\infty}= 0\}\)
or
\(\{g^{s}_{\infty}\in(0,\infty), g^{i}_{0} = \infty, {f^{s}_{\infty}= 0 }\}\), then for each
\(\lambda\in(0,\infty)\)
and
\(\mu\in (0,\widetilde{L_{4}})\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
(8) If
\(\{f^{s}_{\infty}= g^{s}_{\infty}= 0, f^{i}_{0} = \infty\}\)
or
\(\{ f^{s}_{\infty}= g^{s}_{\infty}= 0, g^{i}_{0} = \infty\}\), then for each
\(\lambda \in(0,\infty)\)
and
\(\mu\in(0,\infty)\), system (1.1) has at least one positive solution
\((u(t),v(t)), t\in(0,1)\).
Proof
Because the proofs of the above cases are similar, in what follows we will prove two of them, namely cases (1) and (6).
(1) For any \(\lambda\in(\widetilde{L_{1}},\widetilde{L_{2}})\) and \(\mu\in (\widetilde{L_{3}},\widetilde{L_{4}})\), there exists \(0<\varepsilon< \min\{ f^{i}_{0},g^{i}_{0}\}\) such that
$$\begin{aligned} &\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\frac{1}{f_{0}^{i}-\varepsilon} \leq \lambda\leq\varphi _{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac{1}{f_{\infty }^{s}+\varepsilon}, \\ &\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma_{2}\gamma } \biggr)\frac{1}{g_{0}^{i}-\varepsilon} \leq \mu\leq\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{\infty}^{s}+\varepsilon}. \end{aligned}$$
By the definitions of \(f^{i}_{0}\) and \(g^{i}_{0}\), there exists \(R_{1} > 0\) such that
$$\begin{aligned} &f(t,u,v) \geq\bigl(f^{i}_{0}-\varepsilon\bigr) \varphi_{p_{1}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\leq R_{1}, \\ &g(t,u,v) \geq\bigl(g^{i}_{0}-\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\leq R_{1}. \end{aligned}$$
Denote \(\Omega_{1} = \{(u,v)\in Y: \Vert (u,v) \Vert < R_{1}\}\), for any \((u,v)\in P\cap\partial\Omega_{1}\), we have
$$\begin{aligned} &T_{1}(u,v) (\theta_{1})\\ &\quad=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(\theta_{1},s) \varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{1}}(\lambda) \int_{\theta _{1}}^{\theta_{2}}\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi _{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \bigl(f^{i}_{0}-\varepsilon\bigr)\varphi _{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}}\bigl( \lambda\bigl(f^{i}_{0}-\varepsilon\bigr)\bigr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
then \(\Vert T_{1}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}\). Similarly, we have
$$\begin{aligned} &T_{2}(u,v) (\theta_{2})\\ &\quad=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{2}}(\mu) \int_{\theta _{1}}^{\theta_{2}}\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi _{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{0}-\varepsilon\bigr)\varphi _{p_{2}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}}\bigl(\mu \bigl(g^{i}_{0}-\varepsilon\bigr)\bigr)D \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
then \(\Vert T_{2}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}\). Therefore
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap\partial\Omega_{1}. \end{aligned}$$
(3.6)
On the other hand, we define \(f^{*},g^{*}:[0,1]\times[0,+\infty)\rightarrow [0,+\infty)\) as follows:
$$f^{*}(t,x)=\max_{0\leq u + v \leq x}f(t,u,v),\qquad g^{*}(t,x)=\max _{0\leq u + v \leq x}g(t,u,v), $$
then
$$\begin{aligned} &f(t,u,v)\leq f^{*}(t,x),\quad t\in[0,1], u,v\geq0, u + v \leq x, \\ &g(t,u,v)\leq g^{*}(t,x),\quad t\in[0,1], u,v\geq0, u + v \leq x. \end{aligned}$$
Clearly, \(f^{*}(t,x)\) and \(g^{*}(t,x)\) are nondecreasing on x, by the proof of [35], we have
$$\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]}\frac{f^{*}(t,x)}{\varphi _{p_{1}}(x)} \leq f^{s}_{\infty}, \qquad \limsup_{x\rightarrow+\infty}\max _{t\in [0,1]}\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}\leq g^{s}_{\infty}. $$
From the above inequalities, there exists \(\overline{R_{2}} > 0\) such that
$$\begin{aligned} &\frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)} \leq\limsup_{x\rightarrow+\infty }\max_{t\in[0,1]} \frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)}+\varepsilon\leq f^{s}_{\infty}+\varepsilon,\quad t \in[0,1], x \geq\overline{R_{2}}, \\ &\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}\leq\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]} \frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}+\varepsilon\leq g^{s}_{\infty}+\varepsilon,\quad t \in[0,1], x \geq\overline{R_{2}}. \end{aligned}$$
Then \(f^{*}(t,x)\leq(f^{s}_{\infty}+\varepsilon)\varphi_{p_{1}}(x), g^{*}(t,x)\leq (g^{s}_{\infty}+\varepsilon)\varphi_{p_{2}}(x), t\in[0,1], x \geq\overline{R_{2}}\).
Denote \(R_{2} = \max\{2R_{1},\overline{R_{2}}\}\), \(\Omega_{2} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y} < {R_{2} }\}\). For any \((u,v)\in P\cap\partial \Omega_{2}\), by the definitions of \(f^{*}\) and \(g^{*}\), we have
$$f\bigl(t,u(t),v(t)\bigr)\leq f^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr), \qquad g\bigl(t,u(t),v(t)\bigr)\leq g^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr),\quad t\in[0,1], $$
so
$$\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{1}}(\lambda)\frac{M_{1}}{\Gamma (\beta_{1})} \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{\infty}+\varepsilon\bigr)\varphi_{p_{1}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{\infty}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
and \(\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}\). Similarly, we have
$$\begin{aligned} T_{2}(u,v) (t)&\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac{M_{2}}{\Gamma(\beta _{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu)\frac{M_{2}}{\Gamma(\beta _{2})} \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau) \bigl(g^{s}_{\infty }+\varepsilon\bigr)\varphi_{p_{2}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{\infty}+ \varepsilon\bigr)\bigr)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
so \(\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}\). Therefore
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{2}. \end{aligned}$$
(3.7)
From (3.6), (3.7) and Lemma 2.8, we get that Q has at least one fixed point \((u,v)\in {P\cap(\overline{\Omega_{2}}\backslash\Omega _{1})}\) with \(R_{1}\leq \Vert (u,v) \Vert _{Y}\leq R_{2}\).
(6) Suppose \(f^{i}_{0} = \infty, f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0\), for any \(\lambda\in(0,\widetilde{L_{2}})\) and \(\mu\in(0,\infty)\), there exists \(\varepsilon> 0\) such that
$$\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\varepsilon< \lambda< \varphi_{p_{1}} \biggl(\frac{\Gamma(\beta _{1})}{2AM_{1}} \biggr)\frac{1}{f_{\infty}^{0}+\varepsilon},\quad 0 < \mu< \varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac {1}{\varepsilon}. $$
By \(f^{i}_{0}=\infty\), there exists \(R_{3} > 0\) such that
$$f(t,u,v) \geq\frac{1}{\varepsilon}\varphi_{p_{1}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u,v\geq0, 0 \leq u + v\leq R_{3}. $$
Choose \(\Omega_{3} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{3}\}\), then for any \((u,v)\in P\cap\partial\Omega_{3}\), we have
$$\begin{aligned} T_{1}(u,v) (\theta_{1})&\geq\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta _{2}}\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi _{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \frac{1}{\varepsilon }\varphi_{p_{1}}\bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr) \,ds \\ &\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}} \biggl( \frac{\lambda}{\varepsilon} \biggr) \int _{\theta_{1}}^{\theta_{2}}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta _{2}} H_{1}(s,\tau)\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}} \biggl( \frac{\lambda}{\varepsilon} \biggr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}$$
Thus,
$$\begin{aligned} {\bigl\Vert Q(u,v) \bigr\Vert _{Y} }\geq \bigl\Vert T_{1}(u,v) \bigr\Vert \geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap\partial\Omega_{3}. \end{aligned}$$
(3.8)
On the other hand, we define \(f^{*},g^{*}:[0,1]\times[0,+\infty)\rightarrow [0,+\infty)\) as follows:
$$f^{*}(t,x)=\max_{0\leq u + v \leq x}f(t,u,v), \qquad g^{*}(t,x)=\max _{0\leq u + v \leq x}g(t,u,v). $$
By the proof of [35], we have
$$\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]}\frac{f^{*}(t,x)}{\varphi _{p_{1}}(x)} \leq f^{s}_{\infty},\qquad \lim_{x\rightarrow+\infty}\max _{t\in [0,1]}\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}=0. $$
For above \(\varepsilon>0\), there exists \(\overline{R_{4}}>0\) such that, for any \(t\in[0,1], x\geq\overline{R_{4}}\), we have
$$\begin{aligned} &\frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)} \leq\limsup_{x\rightarrow+\infty }\max_{t\in[0,1]} \frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)}+\varepsilon\leq f^{s}_{\infty}+\varepsilon, \\ &\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)} \leq\lim_{x\rightarrow+\infty}\max_{t\in[0,1]} \frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}+\varepsilon=\varepsilon, \end{aligned}$$
hence \(f^{*}(t,x)\leq(f^{s}_{\infty}+\varepsilon)\varphi_{p_{1}}(x), g^{*}(t,x) \leq\varepsilon\varphi_{p_{2}}(x)\).
Let \(R_{4} = \max\{2R_{3},\overline{R_{4}}\}\) and \({\Omega_{4} } = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y} < R_{4}\}\). For any \((u,v)\in P\cap\partial \Omega_{4}\) and \(t\in[0,1]\), we have
$$f\bigl(t,u(t),v(t)\bigr)\leq f^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr),\qquad g\bigl(t,u(t),v(t)\bigr)\leq g^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr), $$
then
$$\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{1}}(\lambda)\frac{M_{1}}{\Gamma (\beta_{1})} \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{\infty}+\varepsilon\bigr)\varphi_{p_{1}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{\infty}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
so \(\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{4}\). In a similar manner, we deduce
$$\begin{aligned} T_{2}(u,v) (t)&\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac{M_{2}}{\Gamma(\beta _{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu)\frac{M_{2}}{\Gamma(\beta _{2})} \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)\varepsilon \varphi_{p_{2}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr) \,d\tau \biggr)\,ds \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}(\mu \varepsilon)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq\frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}$$
so \(\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{4}\). Therefore
$$\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{4}. \end{aligned}$$
(3.9)
From (3.8), (3.9) and Lemma 2.8, we conclude that Q has at least one fixed point \((u,v)\in P\cap(\overline{\Omega_{4}}\backslash\Omega_{3})\) with \(R_{3}\leq \Vert (u,v) \Vert _{Y}\leq R_{4}\). This completes the proof. □