Theorem 3.1
Suppose that
\(f\in C((0,1)\times (0,+ \infty )^{n-1}, \mathbb{R}_{+})\)
satisfies:
-
\((\mathrm{H}_{1})\)
:
-
There exist two functions
\(f_{1}\), \(f_{2}\in C((0,1)\times (0,+\infty )^{2(n-1)},\mathbb{R}_{+})\)
such that
$$ \begin{aligned} f(t,x_{1},x_{2},\ldots ,x_{n-1})&= f_{1}(t,x_{1},x_{2}, \ldots ,x_{n-1},x _{1},x_{2},\ldots ,x_{n-1}) \\ &\quad{} +f_{2}(t,x_{1},x_{2},\ldots ,x_{n-1},x _{1},x_{2},\ldots ,x_{n-1}). \end{aligned} $$
-
\((\mathrm{H}_{2})\)
:
-
For all
\(t\in (0,1)\)
and
\((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\), \(f_{1}(t, x _{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2}, [4] \ldots , y_{n-1})\), \(f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\)
are increasing in
\((x_{1}, x_{2}, \ldots , x_{n-1})\in (0,+\infty )^{n-1}\); for all
\(t\in (0,1)\)
and
\((x_{1}, x_{2}, \ldots , x_{n-1})\in (0,+\infty )^{n-1}\), \(f_{1}(t, x_{1}, x_{2}, \ldots , x _{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\), \(f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\)
are decreasing in
\((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\).
-
\((\mathrm{H}_{3})\)
:
-
For all
\(\mu \in (0,1)\), there exists
\(\varphi (\mu )\in (\mu , 1]\)
such that, for all
\(t\in (0,1)\)
and
\((x_{1}, x_{2}, \ldots , x_{n-1})\), \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\),
$$\begin{aligned}& \begin{aligned} &f_{1}\bigl(t, \mu x_{1}, \mu x_{2}, \ldots , \mu x_{n-1}, \mu ^{-1} y_{1}, \mu ^{-1} y_{2},\ldots , \mu ^{-1} y_{n-1}\bigr) \\ &\quad \geq \varphi (\mu ) f _{1}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}), \end{aligned} \\& \begin{aligned} &f_{2}\bigl(t, \mu x_{1}, \mu x_{2}, \ldots , \mu x_{n-1}, \mu ^{-1} y_{1}, \mu ^{-1} y_{2},\ldots , \mu ^{-1} y_{n-1}\bigr) \\ &\quad \geq \mu f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}). \end{aligned} \end{aligned}$$
-
\((\mathrm{H}_{4})\)
:
-
There exists a constant
\(\kappa >0\)
such that, for all
\(t\in (0,1)\)
and
\((x_{1}, x_{2}, \ldots , x_{n-1})\), \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\),
$$ f_{1}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}) \geq \kappa f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2}, \ldots , y_{n-1}). $$
-
\((\mathrm{H}_{5})\)
:
-
The functions
\(f_{1}\)
and
\(f_{2}\)
satisfy
$$\begin{aligned}& 0< \int _{0}^{1}f_{1}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1}, \ldots ,s^{\gamma -1}\bigr) \,ds< +\infty , \\& 0< \int _{0}^{1}f_{2}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1}, \ldots ,s^{\gamma -1}\bigr) \,ds< +\infty . \end{aligned}$$
Then BVP (1.1) has a unique solution
\(z_{\lambda }^{*}\)
in
P, and there exists a constant
\(\eta _{\lambda }\in (0, 1)\)
such that
$$ \frac{\eta _{\lambda }\varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma )}t ^{\gamma -1} \leq z_{\lambda }^{*}(t) \leq \frac{\varGamma (\gamma -\nu _{n-2})}{\eta _{\lambda }\varGamma (\gamma )}t^{\gamma -1},\quad t\in [0, 1]. $$
And at the same time, \(z_{\lambda }^{*}\)
satisfies:
-
(i)
If there exists
\(r\in (0, 1)\)
such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then
\(z_{\lambda }^{*}\)
is continuous with respect to
\(\lambda \in (0,+ \infty )\), i.e., for ∀ \(\lambda _{0}\in (0,+\infty )\),
$$ \bigl\Vert z_{\lambda }^{*}-z_{\lambda _{0}}^{*} \bigr\Vert \rightarrow 0, \quad \textit{as } \lambda \rightarrow \lambda _{0}. $$
-
(ii)
If
$$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$
then
\(0<\lambda _{1}< \lambda _{2}\)
implies
\(z_{\lambda _{1}}^{*}< z_{ \lambda _{2}}^{*}\).
-
(iii)
If there exists
\(r\in (0, \frac{1}{2})\)
such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then
$$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert z_{\lambda }^{*} \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert z_{\lambda }^{*} \bigr\Vert =+\infty . $$
Moreover, for any initial values
\(z_{0}\), \(\tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:
$$\begin{aligned}& \begin{aligned} z_{n}(t)&= I_{0^{+}}^{\nu _{n-2}} \biggl\{ \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}z _{n-1}(s),\ldots , z_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s),I_{0^{+}} ^{\nu _{n-2}-\nu _{1}}z_{n-1}(s),\ldots , z_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s)\bigr)\,ds \biggr\} , \end{aligned} \\ & \begin{aligned} \tilde{z}_{n}(t)&= I_{0^{+}}^{\nu _{n-2}} \biggl\{ \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s),I_{0^{+}}^{\nu _{n-2}- \nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}z_{n-1}(s), \ldots , z_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2}\bigl(s,I _{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}} z_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} z_{n-1}(s), \ldots , z_{n-1}(s)\bigr)\,ds \biggr\} , \\ & n=1,2,\ldots , \end{aligned} \end{aligned}$$
we have
\(z_{n}\rightarrow z_{\lambda }^{*}\)
and
\(\tilde{z}_{n}\rightarrow z_{\lambda }^{*}\)
in
E, as
\(n\rightarrow \infty \).
Proof
Let \(P_{e}=\{u\in E: u\sim e\}\), where \(e(t)=t^{ \gamma -\nu _{n-2}-1}\). Then \(P_{e}\) is a component of P. Now, we define two operators \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e} \rightarrow P\) by
$$\begin{aligned}& \begin{aligned} A_{\lambda }(u, v) (t)&=\lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds, \end{aligned} \\ & \begin{aligned} B_{\lambda }(u, v) (t)&=\lambda \int _{0}^{1}K(t,s) f_{2} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds. \end{aligned} \end{aligned}$$
Combining the definition of \(T_{\lambda }\) in (2.20) and \((H_{1})\), we have
$$ \begin{aligned}[b] (T_{\lambda }u) (t)&= \lambda \int _{0}^{1}K(t, s)f\bigl(s,I_{0+}^{\nu _{n-2}}u(s),I _{0+}^{\nu _{n-2}-\nu _{1}}u(s),\ldots ,I_{0+}^{\nu _{n-2}-\nu _{n-3}}u(s),u(s) \bigr)\,ds \\ &=\lambda \int _{0}^{1}K(t,s)f_{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}u(s),I _{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}u(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}u(s),I _{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}u(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s)\bigr)\,ds \\ &=A_{\lambda }(u, u) (t)+B_{\lambda }(u, u) (t), \quad t \in [0, 1]. \end{aligned} $$
(3.1)
Then we can conclude that u is the solution of BVP (2.1) if u satisfies \(u=A_{\lambda }(u, u)+B_{\lambda }(u, u)\).
We prove that \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P \) are well defined at first. For any \(u, v\in P_{e}\), there exists a constant \(\omega \in (0,1)\) such that \(\omega e(t)\leq u(t)\leq \frac{1}{ \omega }e(t)\), \(\omega e(t)\leq v(t)\leq \frac{1}{\omega }e(t)\), \(t\in [0, 1]\). Moreover, by the definition of fractional integral and \(e(t)\leq 1\), for all \(t\in [0,1]\),
$$ \begin{aligned}[b] I_{0^{+}}^{\nu _{n-2}}e(t)&= \frac{1}{\varGamma (\nu _{n-2})} \int _{0}^{t}(t-s)^{ \nu _{n-2}-1}s^{\gamma -\nu _{n-2}-1} \,ds \\ &=\frac{\varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma )}t^{\gamma -1}\leq 1 \end{aligned} $$
(3.2)
and
$$ \begin{aligned}[b] I_{0^{+}}^{\nu _{n-2}-\nu _{i}}e(t)& = \frac{1}{\varGamma (\nu _{n-2}-\nu _{i})} \int _{0}^{t}(t-s)^{\nu _{n-2}-\nu _{i}-1}s^{\gamma -\nu _{n-2}-1} \,ds \\ &=\frac{\varGamma (\gamma -\nu _{n-2})}{ \varGamma (\gamma -\nu _{i})}t^{\gamma -\nu _{i}-1}\leq 1, \quad i=1,2,\ldots ,n-3. \end{aligned} $$
(3.3)
Thus, by \(\mathrm{(H_{1})}\), \(\mathrm{(H_{2})}\), \(\mathrm{(H_{3})}\), \(\mathrm{(H_{5})}\), (3.2), and (3.3), we know that, for all \(t\in [0,1]\),
$$ \begin{aligned}[b] A_{\lambda }(u, v) (t)&= \lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &\leq \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\omega ^{-1}e(s),I_{0^{+}}^{\nu _{n-2}- \nu _{1}} \omega ^{-1}e(s),\ldots ,\omega ^{-1}e(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}\omega e(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \omega e(s),\ldots ,\omega e(s) \bigr)\,ds \\ &\leq \lambda \int _{0} ^{1}K(t,s) f_{1} \biggl(s, \omega ^{-1} , \omega ^{-1},\ldots , \omega ^{-1}, \\ &\quad \frac{\varGamma (\gamma -\nu _{n-2})}{ \varGamma (\gamma )}\omega s^{\gamma -1}, \frac{\varGamma (\gamma -\nu _{n-2})}{ \varGamma (\gamma -\nu _{1})} \omega s^{\gamma -\nu _{1}-1},\ldots ,\omega s^{\gamma -\nu _{n-2}-1} \biggr)\,ds \\ &\leq \lambda \int _{0}^{1}K(t,s) f _{1} \bigl(s,(\rho \omega )^{-1}, (\rho \omega )^{-1},\ldots , ( \rho \omega )^{-1}, \\ &\quad \rho \omega s^{\gamma -1}, \rho \omega s^{\gamma -\nu _{1}-1},\ldots , \omega s^{\gamma -\nu _{n-3}-1} \bigr)\,ds \\ &\leq \lambda \frac{Q_{1}}{\varphi (\rho \omega )}e(t) \int _{0}^{1}f _{1}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{\gamma -1}\bigr) \,ds \\ &< +\infty , \end{aligned} $$
(3.4)
where
$$ \rho =\min \biggl\{ \frac{\varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma )},\frac{ \varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma -\nu _{1})}, \ldots , \frac{ \varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma -\nu _{n-3})}, 1 \biggr\} >0. $$
Similarly, for all \(t\in [0,1]\),
$$ B_{\lambda }(u, v) (t)\leq \lambda \frac{Q_{1}}{\rho \omega }e(t) \int _{0}^{1}f_{2}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1},\ldots ,s ^{\gamma -1}\bigr) \,ds< +\infty . $$
(3.5)
So, \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P\) are well defined.
Now, we prove that \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e} \rightarrow P_{e}\). Taking a constant \(W>1\) such that
$$ \begin{aligned}[b] W&>\max \biggl\{ \frac{\lambda Q_{1}}{\varphi (\rho \omega )} \int _{0} ^{1}f_{1}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{ \gamma -1}\bigr) \,ds, \\ &\quad \frac{\lambda Q_{1}}{\rho \omega } \int _{0} ^{1}f_{2}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{ \gamma -1}\bigr) \,ds, \\ &\quad \biggl(\lambda \varphi (\rho \omega ) \int _{0}^{1}Q _{2}(s)f_{1} \bigl(s,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{\gamma -1},1,1, \ldots ,1\bigr)\,ds \biggr)^{-1}, \\ &\quad \biggl(\lambda \rho \omega \int _{0}^{1}Q_{2}(s)f_{2} \bigl(s,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{\gamma -1},1,1, \ldots ,1\bigr)\,ds \biggr)^{-1} \biggr\} . \end{aligned} $$
(3.6)
Then from \(\mathrm{(H_{1})}\), \(\mathrm{(H_{2}),}\) and \(\mathrm{(H_{3})}\), for all \(t\in [0,1]\),
$$ \begin{aligned}[b] A_{\lambda }(u, v) (t)&= \lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &\geq \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\omega e(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \omega e(s),\ldots ,\omega e(s), \\ &\quad I_{0^{+}} ^{\nu _{n-2}}\omega ^{-1} e(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}\omega ^{-1} e(s),\ldots , \omega ^{-1} e(s) \bigr)\,ds \\ &\geq \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,\rho \omega s^{\gamma -1}, \rho \omega s^{\gamma -\nu _{1}-1},\ldots , \omega s^{\gamma -\nu -1}, \\ &\quad ( \rho \omega )^{-1}, (\rho \omega )^{-1},\ldots , (\rho \omega )^{-1} \bigr)\,ds \\ &\geq \lambda \varphi (\rho \omega )e(t) \int _{0}^{1}Q_{2}(s)f_{1} \bigl(s,s^{\gamma -1},s^{\gamma -1},\ldots ,s ^{\gamma -1},1,1, \ldots ,1\bigr)\,ds \\ &\geq W^{-1}e(t) \end{aligned} $$
(3.7)
and
$$ \begin{aligned}[b] B_{\lambda }(u, v) (t) &\geq \lambda \rho \omega e(t) \int _{0}^{1}Q_{2}(s)f _{2}\bigl(s,s^{\gamma -1},s^{\gamma -1},\ldots ,s^{\gamma -1},1,1,\ldots ,1\bigr)\,ds \\ &\geq W^{-1}e(t). \end{aligned} $$
(3.8)
On the other hand, from (3.4) and (3.5), we know, for all u, \(v\in P_{e}\), \(t\in [0, 1]\),
$$ \begin{aligned}[b] A_{\lambda }(u, v) (t) &\leq \frac{\lambda Q_{1}}{\varphi (\rho \omega )}e(t) \int _{0}^{1}f_{1}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1}, \ldots ,s^{\gamma -1}\bigr) \,ds \\ &\leq We(t) \end{aligned} $$
(3.9)
and
$$ \begin{aligned}[b] B_{\lambda }(u, v) (t)&\leq \frac{\lambda Q_{1}}{\rho \omega }e(t) \int _{0}^{1}f_{2}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1},\ldots ,s ^{\gamma -1}\bigr) \,ds \\ &\leq We(t). \end{aligned} $$
(3.10)
So, \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P_{e}\).
In fact, from \(\mathrm{(H_{2})}\), it is easy to check that \(A_{\lambda }\), \(B_{\lambda }\) are mixed monotone operators. Furthermore, it follows from \(\mathrm{(H_{3})}\) that, for all \(\mu \in (0,1)\), there exists \(\varphi (\mu )\in (\mu , 1]\) such that, for any \(u, v\in P_{e}\), \(t\in [0, 1]\),
$$ \begin{aligned}[b] A_{\lambda }\bigl(\mu u, \mu ^{-1}v\bigr) (t)&= \lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\mu u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \mu u(s),\ldots , \mu u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}\mu ^{-1}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}\mu ^{-1}v(s),\ldots , \mu ^{-1}v(s) \bigr)\,ds \\ &=\lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s, \mu I_{0^{+}}^{\nu _{n-2}} u(s), \mu I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , \mu u(s), \\ &\quad \mu ^{-1}I_{0^{+}}^{\nu _{n-2}}v(s), \mu ^{-1}I_{0^{+}} ^{\nu _{n-2}-\nu _{1}}v(s),\ldots , \mu ^{-1}v(s) \bigr)\,ds \\ &\geq \lambda \varphi (\mu ) \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}}^{ \nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &= \varphi (\mu ) A_{\lambda }(u, v) (t) \end{aligned} $$
(3.11)
and
$$ \begin{aligned}[b] B_{\lambda }\bigl(\mu u, \mu ^{-1}v\bigr) (t)&= \lambda \int _{0}^{1}K(t,s) f_{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\mu u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \mu u(s),\ldots , \mu u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}\mu ^{-1}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}\mu ^{-1}v(s),\ldots , \mu ^{-1}v(s) \bigr)\,ds \\ &\geq \lambda \mu \int _{0}^{1}K(t,s) f_{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I _{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &=\mu B _{\lambda }(u, v) (t). \end{aligned} $$
(3.12)
By \(\mathrm{(H_{4})}\), we infer that there exists \(\kappa >0\) such that, for all u, \(v\in P_{e}\), \(t\in [0, 1]\),
$$ \begin{aligned}[b] A_{\lambda }(u, v) (t)&= \lambda \int _{0}^{1}K(t,s) f_{1} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &\geq \kappa \lambda \int _{0}^{1}K(t,s)f _{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}u(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}z(s), \ldots , u(s), \\ &\quad I_{0^{+}}^{\nu _{n-2}}v(s), I _{0^{+}}^{\nu _{n-2}-\nu _{1}}v(s), \ldots , v(s) \bigr)\,ds \\ &=\kappa B _{\lambda }(u, v) (t). \end{aligned} $$
(3.13)
Combining (3.11)–(3.13) and using Lemma 2.2, we infer that there exists a unique fixed point \(u^{*}_{\lambda }\in P_{e}\) such that
$$ A_{\lambda }\bigl(u^{*}_{\lambda }, u^{*}_{\lambda }\bigr)+B_{\lambda } \bigl(u^{*} _{\lambda }, u^{*}_{\lambda } \bigr)=u^{*}_{\lambda }. $$
That is, BVP (2.1) has a unique solution \(u^{*}_{\lambda }\in P_{e}\). Since \(u^{*}_{\lambda }\in P_{e}\), there exists a constant \(\eta _{\lambda }\in (0, 1)\) such that
$$ \eta _{\lambda } t^{\gamma -\nu _{n-2}-1} \leq u^{*}_{\lambda }(t) \leq \frac{1}{\eta _{\lambda } }t^{\gamma -\nu _{n-2}-1},\quad t\in [0, 1]. $$
(3.14)
Moreover, for any \(\lambda >0\), let \(\tilde{A}= (\lambda ^{-1} A_{ \lambda })\) and \(\tilde{B}= (\lambda ^{-1} B_{\lambda })\). Obviously, Ã and B̃ satisfy all the conditions of Lemma 2.3. With the preceding proof, we can infer that \(u^{*}_{\lambda }\) is the unique positive solution of the following equation:
$$ \lambda \tilde{A}(u, u)+\lambda \tilde{B}(u, u)=A(u, u)+ B(u, u)= u. $$
By means of Lemma 2.3, we know that \(u^{*}_{\lambda }\) satisfies:
-
(1)
If there exists \(r\in (0, 1)\) such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then \(u^{*}_{\lambda }\) is continuous with respect to \(\lambda \in (0, +\infty )\). That is, for ∀ \(\lambda _{0}\in (0,+\infty )\),
$$ \bigl\Vert u^{*}_{\lambda }-u^{*}_{\lambda _{0}} \bigr\Vert \rightarrow 0, \quad \text{as } \lambda \rightarrow \lambda _{0}. $$
-
(2)
If
$$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$
then \(0<\lambda _{1}< \lambda _{2}\) implies \(u^{*}_{\lambda _{1}}< u^{*} _{\lambda _{2}}\).
-
(3)
If there exists \(r\in (0, \frac{1}{2})\) such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then
$$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert u^{*}_{\lambda } \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert u^{*}_{\lambda } \bigr\Vert =+\infty . $$
Furthermore, by Lemma 2.2, we can infer that, for any initial values \(u_{0}, v_{0}\in P_{e}\), by constructing successively the sequences as follows:
$$\begin{aligned}& \begin{aligned} u_{n}(t)&= \lambda \int _{0}^{1}K(t,s)f_{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}u _{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u_{n-1}(s), \ldots , u_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}v_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v_{n-1}(s), \ldots , v_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}u_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u_{n-1}(s), \ldots , u_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}v_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v_{n-1}(s), \ldots , v_{n-1}(s)\bigr)\,ds, \end{aligned} \\& \begin{aligned} v_{n}(t)&= \lambda \int _{0}^{1}K(t,s)f_{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}v _{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v_{n-1}(s), \ldots , v_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}u_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u_{n-1}(s), \ldots , u_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2} \bigl(s,I_{0^{+}} ^{\nu _{n-2}}v_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}v_{n-1}(s), \ldots , v_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}u_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}u_{n-1}(s), \ldots , u_{n-1}(s)\bigr)\,ds, \\ &\quad t\in [0, 1],\quad n=1,2,\ldots , \end{aligned} \end{aligned}$$
we have
$$ u_{n}\rightarrow u^{*}_{\lambda },\qquad v_{n}\rightarrow u^{*}_{ \lambda }, \quad \text{in } E, \text{ as }n\rightarrow \infty . $$
Finally, by what we have proved in Lemma 2.9, we know \(z^{*}_{\lambda }=I^{\nu _{n-2}}u^{*}_{\lambda }\) is the unique positive solution of BVP (1.1). From (3.14), we know that \(z^{*}\) satisfies
$$ \frac{\eta _{\lambda }\varGamma (\gamma -\nu _{n-2})}{\varGamma (\gamma )}t ^{\gamma -1} \leq z^{*}_{\lambda }(t) \leq \frac{\varGamma (\gamma -\nu _{n-2})}{\eta _{\lambda }\varGamma (\gamma )}t^{\gamma -1},\quad t\in [0, 1]. $$
(3.15)
And from the monotonicity and continuity of fractional integral, we get:
-
(i)
If there exists \(r\in (0, 1)\) such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then for ∀ \(\lambda _{0}\in (0,+\infty )\),
$$ \bigl\Vert z^{*}_{\lambda }-z^{*}_{\lambda _{0}} \bigr\Vert \rightarrow 0, \quad \text{as } \lambda \rightarrow \lambda _{0}. $$
-
(ii)
If
$$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$
then \(0<\lambda _{1}< \lambda _{2}\) implies \(z^{*}_{\lambda _{1}}< z^{*} _{\lambda _{2}}\).
-
(iii)
If there exists \(r\in (0, \frac{1}{2})\) such that
$$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$
then
$$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert z^{*}_{\lambda } \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert z^{*}_{\lambda } \bigr\Vert =+\infty . $$
Furthermore, for any initial values \(z_{0}, \tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:
$$\begin{aligned}& \begin{aligned} z_{n}(t)&= I_{0^{+}}^{\nu _{n-2}} \biggl\{ \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}}z _{n-1}(s),\ldots , z_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2} \bigl(s,I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s),I_{0^{+}} ^{\nu _{n-2}-\nu _{1}}z_{n-1}(s),\ldots , z_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s)\bigr)\,ds \biggr\} , \end{aligned} \\& \begin{aligned} \tilde{z}_{n}(t)&= I_{0^{+}}^{\nu _{n-2}} \biggl\{ \lambda \int _{0}^{1}K(t,s)f _{1} \bigl(s,I_{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s),I_{0^{+}}^{\nu _{n-2}- \nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}}z_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}}z_{n-1}(s), \ldots , z_{n-1}(s)\bigr)\,ds \\ &\quad{} +\lambda \int _{0}^{1}K(t,s)f_{2}\bigl(s,I _{0^{+}}^{\nu _{n-2}}\tilde{z}_{n-1}(s),I_{0^{+}}^{\nu _{n-2}-\nu _{1}} \tilde{z}_{n-1}(s),\ldots , \tilde{z}_{n-1}(s), \\ & \quad I_{0^{+}}^{\nu _{n-2}} z_{n-1}(s), I_{0^{+}}^{\nu _{n-2}-\nu _{1}} z_{n-1}(s), \ldots , z_{n-1}(s)\bigr)\,ds \biggr\} , \\ & \quad t\in [0, 1],\quad n=1,2,\ldots , \end{aligned} \end{aligned}$$
we have
$$ z_{n}\rightarrow z^{*}_{\lambda },\qquad \tilde{z}_{n}\rightarrow z ^{*}_{\lambda }, \quad \text{in } E, \text{ as } n\rightarrow \infty . $$
The proof of Theorem 3.1 is completed. □