Let \(\alpha _{i}, \beta _{i}, \gamma _{i}\geqslant 0\) (\(i=1,2\)) with \(\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2} \neq 0\), \(\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2} \neq 0\), and \(r^{-1}(L_{\alpha _{i},\beta _{i},\gamma _{i}})= \lambda _{\alpha _{i},\beta _{i},\gamma _{i}}\) for \(i=1,2\). Now, we list our assumptions for f as follows:
-
(H1)
\(f\in C([0,1]\times \mathbb{R}^{3},\mathbb{R})\).
-
(H2)
There exist two nonnegative functions \(b(t),c(t)\in C[0,1]\) with \(c(t)\not \equiv 0\) and a function \(K(x_{1},x_{2},x_{3})\in C[ \mathbb{R}^{3}, \mathbb{R}^{+}]\) such that
$$ f(t,x_{1},x_{2},x_{3})\geqslant -b(t)-c(t)K(x_{1},x_{2},x_{3}), \quad \forall x_{i}\in \mathbb{R}, t\in [0,1], i=1,2,3. $$
-
(H3)
\(\lim_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\to +\infty } \frac{K(x_{1},x_{2},x_{3})}{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+ \gamma _{1}|x_{3}|}=0\).
-
(H4)
\(\liminf_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\to + \infty }\frac{f(t,x_{1},x_{2},x_{3})}{\alpha _{1}|x_{1}|+\beta _{1}|x _{2}|+\gamma _{1}|x_{3}|}>\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}\), uniformly for \(t\in [0,1]\).
-
(H5)
\(\limsup_{\alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x_{3}|\to 0}\frac{|f(t,x _{1},x_{2},x_{3})|}{\alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x _{3}|}<\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}\), uniformly for \(t\in [0,1]\).
We now present our main result.
Theorem 3.1
Suppose that (H0)–(H5) hold. Then (1.1) has at least one nontrivial solution.
Proof
From (H4) there exist \(\varepsilon _{0}>0\) and \(X_{0}>0\) such that
$$\begin{aligned}& f(t,x_{1},x_{2},x_{3})\geqslant ( \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0} ) \bigl(\alpha _{1} \vert x_{1} \vert +\beta _{1}|x_{2}|+ \gamma _{1}|x_{3}|\bigr), \\& \quad \forall t\in [0,1], \alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|>X_{0}. \end{aligned}$$
(3.1)
For any given ε with \(\varepsilon _{0} - \|c\|\varepsilon >0\), and from (H3) there exists \(X_{1}>X_{0}\) such that
$$ K(x_{1},x_{2},x_{3}) \leqslant \varepsilon \bigl(\alpha _{1} \vert x_{1} \vert +\beta _{1}|x_{2}|+\gamma _{1}|x_{3}| \bigr), \quad \forall \alpha _{1}|x_{1}|+\beta _{1}|x _{2}|+\gamma _{1}|x_{3}|>X_{1}. $$
(3.2)
It follows from (H2), (3.1), (3.2) that
$$\begin{aligned} f(t,x_{1},x_{2},x_{3}) &\geqslant (\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0} ) \bigl(\alpha _{1} \vert x_{1} \vert +\beta _{1} \vert x_{2} \vert + \gamma _{1} \vert x_{3} \vert \bigr)-b(t)-c(t)K(x_{1},x_{2},x_{3}) \\ &\geqslant (\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0} ) \bigl(\alpha _{1} \vert x _{1} \vert +\beta _{1} \vert x_{2} \vert +\gamma _{1} \vert x_{3} \vert \bigr) \\ &\quad {}-b(t)-\varepsilon c(t) \bigl(\alpha _{1} \vert x_{1} \vert +\beta _{1} \vert x_{2} \vert +\gamma _{1} \vert x_{3} \vert \bigr) \\ &\geqslant \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}-\varepsilon c(t) \bigr) \bigl( \alpha _{1} \vert x_{1} \vert +\beta _{1} \vert x_{2} \vert +\gamma _{1} \vert x_{3} \vert \bigr)-b(t) \\ &\geqslant \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \bigl(\alpha _{1} \vert x_{1} \vert +\beta _{1} \vert x_{2} \vert + \gamma _{1} \vert x_{3} \vert \bigr) \\ &\quad {}-\|b\|,\quad \forall \alpha _{1} \vert x_{1} \vert +\beta _{1} \vert x_{2} \vert +\gamma _{1} \vert x_{3} \vert >X_{1}. \end{aligned}$$
(3.3)
Let \(C_{X_{1}}= (\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}-\varepsilon \|c\| )X_{1}+\max_{0\leqslant t\leqslant 1, \alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\leqslant X_{1}}|f(t,x_{1},x_{2},x_{3})|\), \(K^{*}=\max_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\leqslant X_{1}}K(x_{1},x_{2},x_{3})\). Then it easy to see that
$$\begin{aligned} f(t,x_{1},x_{2},x_{3}) \geqslant& \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}-\varepsilon \Vert c \Vert \bigr) \bigl(\alpha _{1} \vert x_{1} \vert + \beta _{1}|x_{2}|+\gamma _{1}|x_{3}| \bigr) \\ &{}-b(t)-C_{X_{1}},\quad \forall (t,x_{1},x _{2},x_{3})\in [0,1]\times \mathbb{R}^{3}. \end{aligned}$$
(3.4)
Note that ε can be chosen arbitrarily small, and we let
$$\begin{aligned} R >& \max \biggl\{ \frac{(\|b\|+\|c\|K^{*}+C_{X_{1}})\int _{0}^{1} \phi _{h}(s) \,ds}{1-\varepsilon M_{\alpha _{1},\beta _{1},\gamma _{1}} \|c\| }, \\ &\frac{( \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+2\varepsilon _{0}-2\|c\| \varepsilon )(\|b\|+\|c\|K^{*}+C_{X_{1}})\int _{0}^{1} \phi _{h}(s) \,ds}{ \varepsilon _{0}-\|c\|\varepsilon -\varepsilon \|c\|M_{\alpha _{1},\beta _{1},\gamma _{1}}(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+2 \varepsilon _{0}-2\|c\|\varepsilon )} \biggr\} , \end{aligned}$$
where \(M_{\alpha _{1},\beta _{1},\gamma _{1}}=\int _{0}^{1} \phi _{h}(s) (\alpha _{1}\int _{0}^{1}G_{1}(s,\tau )\,d\tau +\beta _{1}\int _{0} ^{1}G_{2}(s,\tau )\,d\tau +\gamma _{1} ) \,ds\), and \(\phi _{h}(s)=\frac{(1-s)^{ \beta -2}}{\varGamma (\beta )}+\frac{\int _{0}^{1} h(t)\widetilde{H}_{2}(t,s) \,dt}{1-\int _{0}^{1} h(t) t^{\beta -2} \,dt}\), \(s \in [0,1]\).
Now we prove that
$$ v-Av\neq \mu \varphi _{\alpha _{1},\beta _{1},\gamma _{1}},\quad \forall v \in \partial B_{R}, \mu \geqslant 0, $$
(3.5)
where \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\) is the positive eigenfunction of \(L_{\alpha _{1},\beta _{1},\gamma _{1}}\) corresponding to the eigenvalue \(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}\), and then \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}= \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\) and \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\in P_{0}\) by (2.11).
Suppose (3.5) is not true. Then there exists \(v_{0}\in \partial B_{R}\) and \(\mu _{0}>0\) such that
$$ v_{0}-Av_{0}=\mu _{0}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}. $$
(3.6)
Let
$$\begin{aligned} \widetilde{v}(t) =& \int _{0}^{1} H_{1}(t,s) \biggl[b(s)+c(s)K \biggl( \int _{0}^{1} G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v _{0}(\tau )\,d\tau ,v_{0}(s) \biggr) \\ &{}+C_{X_{1}} \biggr]\,ds,\quad \forall t\in [0,1]. \end{aligned}$$
(3.7)
Then we have
$$\begin{aligned} \widetilde{v}(t) \leqslant& \int _{0}^{1}H_{1}(t,s) \biggl[b(s)+c(s) \biggl[\varepsilon \biggl(\alpha _{1} \biggl\vert \int _{0}^{1}G_{1}(s,\tau ) v_{0}(\tau )\,d\tau \biggr\vert \\ &{}+\beta _{1} \biggl\vert \int _{0}^{1}G_{2}(s,\tau )v _{0}(\tau )\,d\tau \biggr\vert +\gamma _{1} \bigl\vert v_{0}(s) \bigr\vert \biggr)+K^{*} \biggr]+C _{X_{1}} \biggr]\,ds \\ \leqslant& \int _{0}^{1}t^{\beta -1} \phi _{h}(s) \biggl[b(s)+c(s) \biggl[\varepsilon \biggl(\alpha _{1} \int _{0}^{1}G_{1}(s, \tau ) \bigl\vert v_{0}(\tau ) \bigr\vert \,d\tau \\ &{}+\beta _{1} \int _{0}^{1}G_{2}(s,\tau ) \bigl\vert v_{0}( \tau ) \bigr\vert \,d\tau +\gamma _{1} \bigl\vert v_{0}(s) \bigr\vert \biggr)+K^{*} \biggr]+C_{X_{1}} \biggr]\,ds \\ \leqslant& t^{\beta -1} \int _{0}^{1} \phi _{h}(s) \biggl[\|b\|+ \|c\| \biggl[\varepsilon \biggl(\alpha _{1} \int _{0}^{1}G_{1}(s,\tau )\,d\tau \\ &{}+ \beta _{1} \int _{0}^{1}G_{2}(s,\tau )\,d\tau + \gamma _{1} \biggr)\|v_{0}\|+K ^{*} \biggr]+C_{X_{1}} \biggr]\,ds \\ \leqslant& t^{\beta -1}\bigl( \Vert b \Vert +\|c \|K^{*}+C_{X_{1}} \bigr) \int _{0}^{1} \phi _{h}(s) \,ds \\ &{}+ t^{\beta -1} \varepsilon \|c\| \int _{0}^{1} \phi _{h}(s) \biggl(\alpha _{1} \int _{0}^{1}G_{1}(s, \tau )\,d\tau +\beta _{1} \int _{0}^{1}G_{2}(s,\tau )\,d\tau + \gamma _{1} \biggr) \|v_{0}\| \,ds \\ \le& \bigl( \Vert b \Vert +\|c\|K^{*}+C_{X_{1}}\bigr) \int _{0}^{1} \phi _{h}(s) \,ds \\ &{}+ \varepsilon \|c\| R \int _{0}^{1} \phi _{h}(s) \biggl(\alpha _{1} \int _{0}^{1}G_{1}(s,\tau )\,d\tau +\beta _{1} \int _{0}^{1}G_{2}(s, \tau )\,d\tau + \gamma _{1} \biggr) \,ds. \end{aligned}$$
(3.8)
Consequently, we have
$$\begin{aligned} \|\widetilde{v}\| &\leqslant \bigl( \Vert b \Vert +\|c\|K^{*}+C_{X_{1}} \bigr) \int _{0}^{1} \phi _{h}(s) \,ds \\ &\quad {}+ \varepsilon \|c\| R \int _{0}^{1} \phi _{h}(s) \biggl(\alpha _{1} \int _{0}^{1}G_{1}(s,\tau )\,d\tau +\beta _{1} \int _{0}^{1}G_{2}(s, \tau )\,d\tau + \gamma _{1} \biggr) \,ds \\ & < R. \end{aligned}$$
(3.9)
Note that \(\widetilde{v}\in P_{0}\), and then from (2.9), \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\in P_{0}\), and
$$\begin{aligned} v_{0}(t)+\widetilde{v}(t) =&Av_{0}(t)+\mu _{0} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}(t)+\widetilde{v}(t) \\ =& \int _{0}^{1}H_{1}(t,s) \biggl[f \biggl(s, \int _{0}^{1}G_{1}(s,\tau )v_{0}( \tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v_{0}(\tau )\,d\tau ,v_{0}(s) \biggr) \\ &{}+b(s)+c(s)K \biggl( \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v_{0}(\tau )\,d\tau ,v_{0}(s) \biggr)+C_{X _{1}} \biggr]\,ds \\ &{}+\mu _{0}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}(t), \end{aligned}$$
we have
$$ v_{0}+\widetilde{v}\in P_{0}, $$
(3.10)
using the fact that
$$\begin{aligned}& f \biggl(s, \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G _{2}(s,\tau )v_{0}(\tau )\,d\tau ,v_{0}(s) \biggr) +b(s) \\& \quad {}+c(s)K \biggl( \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v _{0}(\tau )\,d\tau ,v_{0}(s) \biggr)+C_{X_{1}}\in P. \end{aligned}$$
Therefore, (2.10), (3.4) and (3.7) enable us to obtain
$$\begin{aligned}& Av_{0}(t)+ \widetilde{v}(t) \\& \quad = \int _{0}^{1}H_{1}(t,s)f \biggl(s, \int _{0} ^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v_{0}( \tau )\,d\tau ,v_{0}(s) \biggr)\,ds \\& \qquad {}+ \int _{0}^{1}H_{1}(t,s) \biggl[b(s)+c(s)K \biggl( \int _{0}^{1}G_{1}(s, \tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v_{0}(\tau ) \,d\tau ,v_{0}(s) \biggr) \\& \qquad {}+C_{X_{1}} \biggr]\,ds \\& \quad \geqslant \bigl(\lambda _{ \alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \int _{0}^{1}H_{1}(t,s) \biggl[\alpha _{1} \biggl\vert \int _{0}^{1}G_{1}(s, \tau )v_{0}(\tau )\,d\tau \biggr\vert \\& \qquad {}+\beta _{1} \biggl\vert \int _{0}^{1}G_{2}(s, \tau )v_{0}(\tau )\,d\tau \biggr\vert +\gamma _{1} \bigl\vert v_{0}(s) \bigr\vert \biggr]\,ds \\& \qquad {}- \int _{0}^{1}H_{1}(t,s) \bigl[b(s)+C_{X_{1}}\bigr]\,ds \\& \qquad {}+ \int _{0}^{1}H_{1}(t,s) \biggl[b(s)+c(s)K \biggl( \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau , \int _{0}^{1}G_{2}(s,\tau )v_{0}(\tau )\,d\tau ,v_{0}(s) \biggr) \\& \qquad {}+C_{X _{1}} \biggr]\,ds \\& \quad \geqslant \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+ \varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \int _{0}^{1}H_{1}(t,s) \biggl[\alpha _{1} \biggl\vert \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau \biggr\vert \\& \qquad {}+ \beta _{1} \biggl\vert \int _{0}^{1}G_{2}(s,\tau )v_{0}(\tau )\,d\tau \biggr\vert + \gamma _{1} \bigl\vert v_{0}(s) \bigr\vert \biggr]\,ds \\& \quad \geqslant \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \biggl\vert \alpha _{1} \int _{0}^{1}H_{1}(t,s) \int _{0}^{1}G_{1}(s,\tau )v_{0}(\tau )\,d\tau \,ds \\& \qquad {}+\beta _{1} \int _{0}^{1}H_{1}(t,s) \int _{0}^{1}G_{2}(s,\tau )v _{0}(\tau )\,d\tau \,ds+\gamma _{1} \int _{0}^{1}H_{1}(t,s)v_{0}(s) \,ds \biggr\vert \\& \quad \geqslant \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+ \varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \biggl\vert \alpha _{1} \int _{0}^{1}H _{3}(t,\tau )v_{0}(\tau )\,d\tau +\beta _{1} \int _{0}^{1}H_{2}(t,\tau )v _{0}(\tau )\,d\tau \\& \qquad {}+\gamma _{1} \int _{0}^{1}H_{1}(t,s)v_{0}(s) \,ds \biggr\vert \\& \quad = \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \biggl\vert \int _{0}^{1}H_{\alpha _{1},\beta _{1}, \gamma _{1}}(t,s)v_{0}(s) \,ds \biggr\vert \\& \quad \geqslant \bigl(\lambda _{\alpha _{1}, \beta _{1},\gamma _{1}}+\varepsilon _{0}- \varepsilon \Vert c \Vert \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s)v_{0}(s) \,ds. \end{aligned}$$
(3.11)
From the definition of operator \(L_{\alpha _{1},\beta _{1},\gamma _{1}}\), we get
$$\begin{aligned} & \bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s)v _{0}(s)\,ds \\ &\quad =\lambda _{\alpha _{1},\beta _{1},\gamma _{1}} \int _{0}^{1}H _{\alpha _{1},\beta _{1},\gamma _{1}}(t,s) \bigl(v_{0}(s)+\widetilde{v}(s)\bigr)\,ds+\bigl( \varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1}, \gamma _{1}}(t,s)v_{0}(s) \,ds \\ &\qquad {}-\lambda _{\alpha _{1},\beta _{1},\gamma _{1}} \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s)\widetilde{v}(s) \,ds \\ &\quad =\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}(v_{0}+\widetilde{v}) (t)+\bigl(\varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s) \bigl(v_{0}(s)+ \widetilde{v}(s)\bigr)\,ds \\ &\qquad {}-\bigl(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+ \varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1}, \gamma _{1}}(t,s)\widetilde{v}(s) \,ds. \end{aligned}$$
(3.12)
From (3.10), we get \(v_{0}(t)+\widetilde{v}(t)\geqslant t^{ \beta -1}\|v_{0}+\widetilde{v}\|\geqslant t^{\beta -1}(\|v_{0}\|-\| \widetilde{v}\|)\), \(t\in [0,1]\), and hence from (3.8) we have
$$\begin{aligned}& \bigl(\varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s) \bigl(v_{0}(s)+\widetilde{v}(s)\bigr)\,ds \\& \qquad {}-\bigl( \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \int _{0}^{1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s)\widetilde{v}(s) \,ds \\& \quad \geqslant \bigl(\varepsilon _{0}- \Vert c \Vert \varepsilon \bigr) \bigl(R- \Vert \widetilde{v} \Vert \bigr) \int _{0}^{1}s^{\beta -1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s) \,ds-\bigl( \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}- \Vert c \Vert \varepsilon\bigr) \\& \qquad {}\times \biggl[\bigl( \Vert b \Vert + \Vert c \Vert K^{*}+C_{X_{1}}\bigr) \int _{0}^{1} \phi _{h}(s) \,ds+ \varepsilon \Vert c \Vert R M_{\alpha _{1},\beta _{1},\gamma _{1}} \biggr] \int _{0} ^{1}s^{\beta -1}H_{\alpha _{1},\beta _{1},\gamma _{1}}(t,s) \,ds \\& \quad \geqslant 0. \end{aligned}$$
(3.13)
Combining (3.11), (3.12) and (3.13), and we obtain
$$ Av_{0}(t)+\widetilde{v}(t)\geqslant \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}(v_{0}+\widetilde{v}) (t), \quad t\in [0,1]. $$
(3.14)
Therefore, using (3.6) and (3.14) we have
$$ v_{0}+\widetilde{v}=Av_{0}+\mu _{0} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}+\widetilde{v}\geqslant \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}(v_{0}+ \widetilde{v})+\mu _{0} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}} \geqslant \mu _{0} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}. $$
Define
$$ \mu ^{*}=\sup \{\mu >0:v_{0}+\widetilde{v}\geqslant \mu \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\}. $$
It is easy to see that \(\mu ^{*}\geqslant \mu _{0}\) and \(v_{0}+ \widetilde{v}\geqslant \mu ^{*}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\). From \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}= \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\), we obtain
$$ \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}(v_{0}+\widetilde{v})\geqslant \lambda _{\alpha _{1},\beta _{1},\gamma _{1}} L_{\alpha _{1},\beta _{1}, \gamma _{1}}\mu ^{*}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}=\mu ^{*} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}. $$
Hence
$$ v_{0}+\widetilde{v}\geqslant \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L _{\alpha _{1},\beta _{1},\gamma _{1}}(v_{0}+\widetilde{v})+\mu _{0} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\geqslant \bigl(\mu _{0}+\mu ^{*}\bigr) \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}, $$
which contradicts the definition of \(\mu ^{*}\). Therefore, (3.5) holds, and from Lemma 2.8 we obtain
$$ \deg (I-A,B_{R},0)=0. $$
(3.15)
From (H5) there exist \(0<\varepsilon _{1}< \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}\) and \(0< r< R\) such that
$$ \bigl\vert f(t,x_{1},x_{2},x_{3}) \bigr\vert \leqslant ( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1}) \bigl(\alpha _{2} \vert x _{1} \vert +\beta _{2}|x_{2}|+ \gamma _{2}|x_{3}|\bigr), $$
for all \(x_{i}\in \mathbb{R}\), \(i=1, 2,3\), \(t\in [0,1]\) with \(0\leqslant \alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x_{3}|< r\). Consequently, we obtain
$$\begin{aligned} \bigl\vert (Av) (t) \bigr\vert &\leqslant (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}- \varepsilon _{1}) \int _{0}^{1}H_{1}(t,s) \biggl(\alpha _{2} \biggl\vert \int _{0}^{1}G_{1}(s,\tau )v(\tau )\,d\tau \biggr\vert \\ &\quad {}+\beta _{2} \biggl\vert \int _{0}^{1}G_{2}(s,\tau )v(\tau )\,d \tau \biggr\vert +\gamma _{2} \bigl\vert v(s) \bigr\vert \biggr) \,ds \\ & \le (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1}) \int _{0}^{1}H_{1}(t,s) \biggl(\alpha _{2} \int _{0}^{1}G_{1}(s,\tau ) \bigl\vert v(\tau ) \bigr\vert \,d\tau \\ &\quad {}+\beta _{2} \int _{0}^{1}G_{2}(s,\tau ) \bigl\vert v(\tau ) \bigr\vert \,d\tau +\gamma _{2} \bigl\vert v(s) \bigr\vert \biggr)\,ds \\ &=( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1}) \int _{0} ^{1}H_{\alpha _{2},\beta _{2},\gamma _{2}}(t,s) \bigl\vert v(s) \bigr\vert \,ds \\ &=( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1}) \bigl(L_{\alpha _{2},\beta _{2},\gamma _{2}} \vert v \vert \bigr) (t),\quad \forall t\in [0,1], v\in E, \|v \|\leqslant r. \end{aligned}$$
Now for this r, we prove that
$$ Av\neq \lambda v, \quad \forall v\in \partial B_{r}, \lambda \geqslant 1. $$
(3.16)
Assume the contrary. Then there exist \(v_{0}\in \partial B_{r}\) and \(\lambda _{0}\geqslant 1\) such that \(Av_{0}=\lambda _{0}v_{0}\). Let \(\omega (t)=|v_{0}(t)|\). Then \(\omega \in \partial B_{r}\cap P\) and
$$ \omega \leqslant \frac{1}{\lambda _{0}}( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1})L_{\alpha _{2},\beta _{2},\gamma _{2}}\omega \leqslant ( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}- \varepsilon _{1})L_{\alpha _{2},\beta _{2},\gamma _{2}}\omega . $$
By induction, we have \(\omega \leqslant ( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1})^{n}L^{n} _{\alpha _{2},\beta _{2},\gamma _{2}}\omega \), for \(n=1,2,\ldots \) . As a result, we have
$$ \|\omega \|\leqslant (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}- \varepsilon _{1})^{n} \bigl\Vert L^{n}_{\alpha _{2},\beta _{2},\gamma _{2}} \bigr\Vert \| \omega \|, $$
and thus
$$ 1\leqslant (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}- \varepsilon _{1} )^{n} \bigl\Vert L^{n}_{\alpha _{2},\beta _{2},\gamma _{2}} \bigr\Vert . $$
Therefore, by Gelfand’s theorem, we have
$$ (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1})r(L_{\alpha _{2},\beta _{2},\gamma _{2}})= ( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}- \varepsilon _{1} ) \lim_{n\to \infty } \sqrt[n]{ \bigl\Vert L^{n}_{\alpha _{2},\beta _{2},\gamma _{2}} \bigr\Vert }\geqslant 1. $$
This contradicts
$$ (\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1})r(L_{\alpha _{2},\beta _{2},\gamma _{2}})=1- \varepsilon _{1}r(L_{\alpha _{2},\beta _{2},\gamma _{2}})< 1. $$
Thus (3.16) holds and from Lemma 2.9 we have
$$ \deg (I-A,B_{r},0)=1. $$
(3.17)
Now (3.15) and (3.17) imply that
$$ \deg (I-A,B_{R}\setminus \overline{B}_{r},0)=\deg (I-A,B_{R},0)- \deg (I-A,B_{r},0)=-1. $$
Therefore the operator A has at least one fixed point in \(B_{R} \setminus \overline{B}_{r}\). Equivalently, (1.1) has at least one nontrivial solution. This completes the proof. □