Theorem 3.1
Suppose that (S1)-(S4) hold. Then the BVP (1.1) has a unique positive solution
\((u_{\lambda}^{\star },v_{\lambda}^{\star})\), which satisfies
$$\begin{aligned}& \frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)}t^{\alpha-1}\leq u_{\lambda}^{\star}(t)\leq \frac{D\Gamma(\alpha-n+2)}{\Gamma (\alpha)}t^{\alpha-1}, \\ & \frac{\Gamma(\beta-m+2)\mu t^{\beta-1}}{\Gamma(\beta )} \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)} \biggr)^{\varsigma} \int_{0}^{1}\theta_{2}(s)s^{\varsigma(\alpha-1)}g(s,1,1, \ldots ,1)\,ds \\ & \quad \leq v_{\lambda}^{\star}(t) \\ & \quad \leq\frac{\Gamma(\beta-m+2)M_{2}\mu t^{\beta-1}}{\Gamma(\beta )} \biggl( \frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta _{m-2})}+1 \biggr) ^{\varsigma} \\ & \qquad {}\times\int_{0}^{1}(1-s)^{\beta-m+1}g(s,1,1,\ldots,1) \,ds, \end{aligned}$$
(3.1)
and at the same time
\(u_{\lambda}^{\star}\)
satisfies:
-
(1)
For
\(\lambda_{0}\in(0,\infty)\), \(\|u_{\lambda}^{\star }-u_{\lambda_{0}}^{\star}\|\rightarrow0\), \(\lambda\rightarrow\lambda_{0}\).
-
(2)
If
\(0<\sigma<\frac{1}{2}\), then
\(0<\lambda_{1}<\lambda_{2}\)
implies
\(u_{\lambda_{1}}^{\star}\leq u_{\lambda_{2}}^{\star}\), \(u_{\lambda_{1}}^{\star}\neq u_{\lambda_{2}}^{\star}\).
-
(3)
\(\lim_{\lambda\rightarrow0}\|u_{\lambda}^{\star}\|=0\), \(\lim_{\lambda\rightarrow+\infty}\|u_{\lambda}^{\star}\|=+\infty\).
Moreover, for any
\(u_{0}\), we construct a successive sequence
$$\begin{aligned} u_{k+1}(t) =&I_{0^{+}}^{n-2}\biggl\{ \lambda \int_{0}^{1}G(t,s) \bigl[\phi\bigl(s,u_{ k}(s), D_{0^{+}}^{\mu_{1}}u_{ k}(s), \\ &D_{0^{+}}^{\mu_{2}}u_{ k}(s), \ldots,D_{0^{+}}^{\mu_{n-2}}u_{ k}(s),Au^{(n-2)}_{k}(s) \bigr) \\ &{}+\psi\bigl(s,u_{ k}(s), D_{0^{+}}^{\mu_{1}}u_{ k}(s),D_{0^{+}}^{\mu_{2}}u_{ k}(s), \ldots,D_{0^{+}}^{\mu_{n-2}}u_{ k}(s),Au^{(n-2)}_{k}(s) \bigr) \bigr]\,ds\biggr\} , \\ &k=1,2,\ldots, \end{aligned}$$
and we have
\(\|u_{k}-u_{\lambda}^{\star}\|\rightarrow0\)
as
\(k\rightarrow \infty\), and the convergence rate
$$\bigl\Vert u_{k}-u_{\lambda}^{\star}\bigr\Vert =o \bigl(1-r^{\sigma^{k}}\bigr), $$
where
r
is a constant, \(0< r<1\), and dependent on
\(u_{0}\). Moreover,
$$v_{\lambda}^{\star}(t)=I_{0^{+}}^{m-2} \biggl(\mu \int _{0}^{1}H(t,s)g\bigl(s,u_{\lambda}^{\star}(s),D_{0^{+}}^{\eta _{1}}u_{\lambda}^{\star}(s), \ldots, D_{0^{+}}^{\eta_{m-2}}u_{\lambda}^{\star}(s)\bigr) \,ds \biggr). $$
Proof
We first consider the existence of a positive solution to problem (2.10). From the discussion in Section 2, we only need to consider the existence of a positive solution to BVP (2.11). In order to realize this purpose, let
$$Ax(t)=I_{0^{+}}^{m-2} \biggl(\mu \int_{0}^{1}H(t,s)g \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\eta_{1}}x(s), \ldots ,I_{0^{+}}^{n-2-\eta_{m-2}} x(s) \bigr)\,ds \biggr), $$
and, for any \(x,w\in Q_{e}\), we define the operator \(T_{\lambda }:Q_{e}\times Q_{e}\rightarrow P\) by
$$\begin{aligned} T_{\lambda}(x,w) (t) =&\lambda \int_{0}^{1}G(t,s)\bigl[\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}x(s),Ax(s)\bigr) \\ &{}+\psi\bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu _{1}}w(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}w(s) ,Aw(s)\bigr)\bigr]\,ds. \end{aligned}$$
(3.2)
Now we prove that \(T_{\lambda}:Q_{e}\times Q_{e}\rightarrow P\) is well defined. For any \(x,w\in Q_{e}\), by (2.13), (2.16), we have
$$\begin{aligned}& \int_{0}^{1}H(s,\tau)g\bigl(\tau,I_{0^{+}}^{n-2}x( \tau ),I_{0^{+}}^{n-2-\eta_{1}}x(\tau),\ldots,I_{0^{+}}^{n-2-\eta _{m-2}}x( \tau)\bigr)\,d\tau \\& \quad \leq \int_{0}^{1}H(s,\tau)g \biggl(\tau, \frac{D\Gamma(\alpha -n+2)}{\Gamma(\alpha)}\tau^{\alpha-1} ,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{1})}\tau^{\alpha -\eta_{1}-1}, \ldots, \\& \qquad \frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha -\eta_{m-2})} \tau^{\alpha-\eta_{m-2}-1} \biggr)\,d\tau \\& \quad \leq \int_{0}^{1}H(s,\tau)g \biggl(\tau, \frac{D\Gamma(\alpha -n+2)}{\Gamma(\alpha)}+1 ,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{1})}+1,\ldots ,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{m-2})}+1 \biggr)\,d \tau \\& \quad \leq \int_{0}^{1}H(s,\tau)g \biggl(\tau, \frac{D\Gamma(\alpha -n+2)}{\Gamma(\alpha-\eta_{m-2})}+1 ,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{m-2})}+1,\ldots ,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{m-2})}+1 \biggr)\,d \tau \\& \quad \leq M_{2} \biggl(\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta _{m-2})}+1 \biggr)^{\varsigma}s^{\beta-m+1} \int_{0}^{1}(1-\tau )^{\beta-m+1}g(\tau,1,1, \ldots,1)\,d\tau,\quad s\in[0,1], \\& \int_{0}^{1}H(s,\tau)g\bigl(\tau,I_{0^{+}}^{n-2}x( \tau ),I_{0^{+}}^{n-2-\eta_{1}}x(\tau),\ldots,I_{0^{+}}^{n-2-\eta _{m-2}}x( \tau)\bigr)\,d\tau \\& \quad \geq \int_{0}^{1}H(s,\tau)g \biggl(\tau, \frac{\Gamma(\alpha -n+2)}{D\Gamma(\alpha)}\tau^{\alpha-1} ,\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha-\eta_{1})}\tau^{\alpha -\eta_{1}-1}, \ldots, \\& \qquad \frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha -\eta_{m-2})} \tau^{\alpha-\eta_{m-2}-1} \biggr)\,d\tau \\& \quad \geq \int_{0}^{1}H(s,\tau)g \biggl(\tau, \frac{\Gamma(\alpha -n+2)}{D\Gamma(\alpha)}\tau^{\alpha-1} ,\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)}\tau^{\alpha -1}, \ldots,\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)}\tau ^{\alpha-1} \biggr)\,d\tau \\& \quad \geq \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)} \biggr)^{\varsigma}s^{\beta-m+1} \int_{0}^{1}\theta_{2}(\tau)\tau ^{\varsigma(\alpha-1)}g(\tau,1,1,\ldots,1) \,d\tau, \quad s\in[0,1]. \end{aligned}$$
Hence, by (2.14), for any \(s\in[0,1]\), we have
$$\begin{aligned}& Ax(s) =I_{0^{+}}^{m-2} \biggl(\mu \int_{0}^{1}H(s,\tau)g\bigl(\tau ,I_{0^{+}}^{n-2}x(\tau),I_{0^{+}}^{n-2-\eta_{1}}x(\tau), \ldots ,I_{0^{+}}^{n-2-\eta_{m-2}} x(\tau)\bigr)\,d\tau \biggr) \\& \hphantom{Ax(s)}\leq \frac{\Gamma(\beta-m+2)M_{2}\mu s^{\beta-1}}{\Gamma(\beta )} \biggl(\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta _{m-2})}+1 \biggr) ^{\varsigma} \\& \hphantom{Ax(s) ={}}{}\times\int_{0}^{1}(1-\tau)^{\beta-m+1}g(\tau,1,1, \ldots ,1)\,d\tau, \\& Ax(s) =I_{0^{+}}^{m-2} \biggl(\mu \int_{0}^{1}H(s,\tau)g\bigl(\tau ,I_{0^{+}}^{n-2}x(\tau),I_{0^{+}}^{n-2-\eta_{1}}x(\tau), \ldots ,I_{0^{+}}^{n-2-\eta_{m-2}} x(\tau)\bigr)\,d\tau \biggr) \\& \hphantom{Ax(s)}\geq\frac{\mu\Gamma(\beta-m+2)s^{\beta-1}}{\Gamma(\beta)} \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)} \biggr)^{\varsigma} \int_{0}^{1}\theta_{2}(\tau) \tau^{\varsigma(\alpha-1)}g(\tau ,1,1,\ldots,1)\,d\tau. \end{aligned}$$
(3.3)
By (3.3), (2.13), (2.15), (S1), and Remark 2.1, we have
$$\begin{aligned}& \phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots ,I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr) \\& \quad \leq \phi\biggl(s,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha )}s^{\alpha-1}+1,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\mu_{1})} s^{\alpha-\mu_{1}-1}+1,\ldots, \\& \qquad \frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\mu _{n-2})}s^{\alpha-\mu_{n-2}-1}+1,\frac{\Gamma(\beta-m+2)M_{2}\mu }{\Gamma(\beta)} \biggl( \frac{D\Gamma(\alpha-n+2)}{ \Gamma(\alpha-\eta_{m-2})}+1 \biggr)^{\varsigma} \\& \qquad {} \times s^{\beta-1} \int_{0}^{1}(1-\tau)^{\beta-m+1}g(\tau ,1,1, \ldots,1)\,d\tau+1\biggr) \\& \quad \leq \phi(s,Db+1,Db+1,\ldots,Db+1) \\& \quad \leq (Db+1)^{\sigma}\phi(s,1,1,\ldots,1) \\& \quad \leq 2^{\sigma}b^{\sigma}D^{\sigma}\phi(s,1,1,\ldots,1), \quad s\in(0,1), \end{aligned}$$
(3.4)
where
$$\begin{aligned}& D> \max\biggl\{ \biggl[2^{\sigma}b^{\sigma}M_{1} \lambda \int _{0}^{1}(1-s)^{\alpha-n+1}\phi(s,1,1, \ldots,1)\,ds \\& \hphantom{D>{}}{}+M_{1}c^{-\sigma }\lambda \int_{0}^{1}s^{-\sigma(\alpha-1)}(1-s)^{\alpha-n+1} \psi(s,1,1,\ldots,1)\,ds \biggr]^{\frac{1}{1-\sigma}}, 1, 2c, b^{-1}, \\& \hphantom{D>{}}{}\biggl[c^{\sigma}\lambda \int_{0}^{1}\theta_{1}(s)s^{\sigma (\alpha-1)} \phi(s,1,1,\ldots,1)\,ds+2^{-\sigma}b^{-\sigma} \lambda \int_{0}^{1}\theta_{1}(s)\psi(s,1,1, \ldots,1)\,ds \biggr]^{-\frac{1}{1-\sigma}} \biggr\} , \\& b=\max\biggl\{ \frac{\Gamma(\alpha-n+2)}{ \Gamma(\alpha-\mu_{n-2})},1,\frac{\Gamma(\beta-m+2)}{\Gamma (\beta)(M_{2}\mu)^{-1}}\biggl(\frac{\Gamma(\alpha-n+2)}{\Gamma (\alpha-\eta_{m-2})} +1\biggr) ^{\varsigma} \\& \hphantom{b={}}{}\times\int_{0}^{1}(1-\tau)^{\beta-m+1}g(\tau,1,1, \ldots ,1)\,d\tau\biggr\} . \end{aligned}$$
By (3.3), (2.13), (2.15), (S1), and (S2), we also have
$$\begin{aligned}& \psi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots ,I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr) \\& \quad \leq \psi \biggl(s,\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha )}s^{\alpha-1},\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha-\mu_{1})} s^{\alpha-\mu_{1}-1},\ldots,\frac{\Gamma(\alpha-n+2)}{D\Gamma (\alpha-\mu_{n-2})}s^{\alpha-\mu_{n-2}-1}, \\& \qquad \frac{\mu\Gamma(\beta-m+2)s^{\beta -1}}{\Gamma(\beta)} \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma (\alpha)} \biggr)^{\varsigma} \int_{0}^{1}\theta_{2}(\tau) \tau^{\varsigma (\alpha-1)}g(\tau,1,1,\ldots,1)\,d\tau \biggr) \\& \quad \leq \psi \biggl(s,\frac{c}{D}s^{\alpha-1}, \frac{c}{D}s^{\alpha -\mu_{1}-1},\ldots,\frac{c}{D}s^{\alpha-\mu_{n-2}-1} , \frac{c}{D^{\varsigma}}s^{\beta-1} \biggr) \\& \quad \leq \psi \biggl(s,\frac{c}{D}s^{\alpha-1}, \frac{c}{D}s^{\alpha -1},\ldots,\frac{c}{D}s^{\alpha-1} \biggr) \\& \quad \leq \biggl(\frac{c}{D}s^{\alpha-1} \biggr)^{-\sigma}\psi (s,1,1,\ldots,1) \\& \quad = c^{-\sigma}D^{\sigma}s^{-\sigma(\alpha-1)}\psi(s,1,1,\ldots,1), \quad s\in(0,1), \end{aligned}$$
(3.5)
where
$$\begin{aligned} c =&\min \biggl\{ \frac{\Gamma(\alpha-n+2)}{\Gamma(\alpha )},1,\frac{\Gamma(\beta-m+2)\mu}{\Gamma(\beta)} \biggl(\frac {\Gamma(\alpha-n+2)}{ \Gamma(\alpha)} \biggr)^{\varsigma} \\ &{}\times\int_{0}^{1}\theta_{2}(\tau) \tau^{\varsigma(\alpha-1)}g(\tau ,1,\ldots,1)\,d\tau \biggr\} . \end{aligned}$$
Noting \(\frac{c}{D}s^{\alpha-1}<1\), by (3.3), (2.13), (2.15), (S1), and (S2), we have
$$\begin{aligned}& \phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots ,I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr) \\& \quad \geq \phi \biggl(s,\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha )}s^{\alpha-1},\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha-\mu_{1})} s^{\alpha-\mu_{1}-1},\ldots,\frac{\Gamma(\alpha-n+2)}{D\Gamma (\alpha-\mu_{n-2})}s^{\alpha-\mu_{n-2}-1}, \\& \qquad \frac{\mu\Gamma(\beta-m+2)}{\Gamma (\beta)} \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)} \biggr)^{\varsigma} s^{\beta-1} \int_{0}^{1}\theta_{2}(\tau)\tau ^{\varsigma(\alpha-1)}g(\tau,1,1,\ldots,1)\,d\tau \biggr) \\& \quad \geq \phi \biggl(s,\frac{c}{D}s^{\alpha-1}, \frac{c}{D}s^{\alpha -\mu_{1}-1},\ldots,\frac{c}{D} s^{\alpha-\mu_{n-2}-1}, \frac{c}{D}s^{\beta-1} \biggr) \\& \quad \geq \phi \biggl(s,\frac{c}{D}s^{\alpha-1}, \frac{c}{D}s^{\alpha -1},\ldots,\frac{c}{D}s^{\alpha-1} \biggr) \\& \quad \geq \biggl(\frac{c}{D}s^{\alpha-1} \biggr)^{\sigma}\phi (s,1,1,\ldots,1) \\& \quad = c^{\sigma}D^{-\sigma}s^{\sigma(\alpha-1)}\phi(s,1,1,\ldots,1), \quad s\in(0,1), \end{aligned}$$
(3.6)
and by (3.3), (2.13), (2.15), (S1), and Remark 2.1, we also get
$$\begin{aligned}& \psi\bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots ,I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s)\bigr) \\& \quad \geq \psi\biggl(s,\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha )}s^{\alpha-1},\frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\mu_{1})} s^{\alpha-\mu_{1}-1},\ldots,\frac{D\Gamma(\alpha-n+2)}{\Gamma (\alpha-\mu_{n-2})}s^{\alpha-\mu_{n-2}-1}, \\& \qquad \frac{\Gamma(\beta-m+2)M_{2}\mu}{\Gamma(\beta)} \biggl(\frac{D\Gamma(\alpha-n+2)}{ \Gamma(\alpha)}+1\biggr)^{\varsigma}s^{\beta-m+1} \int _{0}^{1}(1-\tau)^{\beta-m+1}g(\tau,1, \ldots,1)\,d\tau\biggr) \\& \quad \geq \psi\bigl(s,Dbs^{\alpha-1},Dbs^{\alpha-\mu_{1}-1},\ldots ,Dbs^{\alpha-\mu_{n-2}-1}, D^{\varsigma}bs^{\beta-m+1}\bigr) \\& \quad \geq \psi(s,Db+1,Db+1,\ldots,Db+1) \\& \quad \geq (Db+1)^{-\sigma}\psi(s,1,1,\ldots,1) \\& \quad \geq 2^{-\sigma}b^{-\sigma}D^{-\sigma}\psi(s,1,1, \ldots,1),\quad s\in(0,1). \end{aligned}$$
(3.7)
For \(x,w\in Q_{e}\), it follows from (3.4), (3.5) that
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr)\,ds \\& \quad \leq M_{1}t^{\alpha-n+1}\lambda \int_{0}^{1}(1-s)^{\alpha-n+1}\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr)\,ds \\& \quad \leq 2^{\sigma}b^{\sigma}D^{\sigma}M_{1} \lambda t^{\alpha -n+1} \int_{0}^{1}(1-s)^{\alpha-n+1}\phi(s,1,1, \ldots,1)\,ds< +\infty,\quad t\in[0,1], \end{aligned}$$
(3.8)
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}w(s) ,Aw(s) \bigr)\,ds \\& \quad \leq M_{1}t^{\alpha-n+1}\lambda \int_{0}^{1}\psi \bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}w(s) ,Aw(s) \bigr)\,ds \\& \quad \leq c^{-\sigma}D^{\sigma}M_{1}\lambda t^{\alpha-n+1} \int _{0}^{1}(1-s)^{\alpha-n+1}s^{-\sigma(\alpha-1)} \psi(s,1,1,\ldots ,1)\,ds< +\infty,\quad t\in[0,1]. \end{aligned}$$
(3.9)
By (H4), (3.8), and (3.9), we see that \(T_{\lambda }:Q_{e}\times Q_{e}\rightarrow P\) is well defined.
Next, we will prove that \(T_{\lambda}:Q_{e}\times Q_{e}\rightarrow Q_{e}\). It follows from (3.8), (3.9) that
$$ T_{\lambda}(x,w) (t)\leq Dt^{\alpha-n+1}=De(t),\quad t \in[0,1]. $$
(3.10)
At the same time, by (3.6), (3.7), we have
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr)\,ds \\& \quad \geq t^{\alpha-n+1}\lambda \int_{0}^{1}\theta_{1}(s)\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr)\,ds \\& \quad \geq c^{\sigma}D^{-\sigma}\lambda t^{\alpha-n+1} \int _{0}^{1}\theta_{1}(s)s^{\sigma(\alpha-1)} \phi(s,1,1,\ldots,1)\,ds,\quad t\in[0,1], \end{aligned}$$
(3.11)
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}w(s) ,Aw(s) \bigr)\,ds \\& \quad \geq t^{\alpha-n+1}\lambda \int_{0}^{1}\theta_{1}(s)\psi \bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}w(s) ,Aw(s) \bigr)\,ds \\& \quad \geq 2^{-\sigma}D^{-\sigma}b^{-\sigma}\lambda t^{\alpha-n+1} \int _{0}^{1}\theta_{1}(s)\psi(s,1,1, \ldots,1)\,ds,\quad t\in[0,1]. \end{aligned}$$
(3.12)
Equations (3.11) and (3.12) imply that
$$ T_{\lambda}(x,w) (t)\geq\frac{1}{D}t^{\alpha-n+1}= \frac{1}{D}e(t),\quad t\in[0,1]. $$
(3.13)
Hence, \(T_{\lambda}:Q_{e}\times Q_{e}\rightarrow Q_{e}\) is well defined.
Next, we shall prove that \(T_{\lambda}:Q_{e}\times Q_{e}\rightarrow Q_{e}\) is a mixed monotone operator. In fact, for any \(x_{1},x_{2}\in Q_{e}\) and \(x_{1}\leq x_{2}\), by the monotonicity of \(I_{0^{+}}^{n-2-\mu_{i}}\), A and ϕ, for any \(t\in[0, 1]\), we have
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}x_{1}(s),I_{0^{+}}^{n-2-\mu_{1}}x_{1}(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}x_{1}(s) ,Ax_{1}(s) \bigr)\,ds \\& \quad \leq \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}x_{2}(s),I_{0^{+}}^{n-2-\mu_{1}}x_{2}(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}x_{2}(s) ,Ax_{2}(s) \bigr)\,ds. \end{aligned}$$
(3.14)
Hence, by (3.14), we have
$$ T_{\lambda}(x_{1},w)\leq T_{\lambda}(x_{2},w), \quad w\in Q_{e}, $$
(3.15)
that is, \(T_{\lambda}(x,w)\) is nondecreasing on x for any \(w\in Q_{e}\). Similarly, if \(w_{1}\geq w_{2}\), \(w_{1}, w_{2}\in Q_{e}\), from (S1), for any \(t\in[0, 1]\), we have
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}w_{1}(s),I_{0^{+}}^{n-2-\mu_{1}}w_{1}(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}w_{1}(s) ,Aw_{1}(s) \bigr)\,ds \\& \quad \leq \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}w_{2}(s),I_{0^{+}}^{n-2-\mu_{1}}w_{2}(s), \ldots, I_{0^{+}}^{n-2-\mu_{n-2}}w_{2}(s) ,Aw_{2}(s) \bigr)\,ds. \end{aligned}$$
(3.16)
Hence, by (3.16), we have
$$ T_{\lambda}(x,w_{1})\leq T_{\lambda}(x,w_{2}), \quad x\in Q_{e}, $$
(3.17)
i.e., \(T_{\lambda}(x,w)\) is nonincreasing on w for any \(x\in Q_{e}\). Hence, by (3.15) and (3.17), we see that \(T_{\lambda }:Q_{e}\times Q_{e}\rightarrow Q_{e}\) is a mixed monotone operator.
Finally, we show that the operator \(T_{\lambda}\) satisfies (2.12). For any \(x,w\in Q_{e}\) and \(l\in(0,1)\), \(t\in[0, 1]\), by (S2) and (S3), we have
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}lx(s),I_{0^{+}}^{n-2-\mu_{1}}lx(s), \ldots,I_{0^{+}} ^{n-2-\mu_{n-2}}lx(s) ,Alx(s) \bigr)\,ds \\& \quad \geq \lambda \int_{0}^{1}G(t,s)\phi \bigl(s,lI_{0^{+}}^{n-2}x(s),lI_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots, lI_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Alx(s) \bigr)\,ds \\& \quad \geq \lambda l^{\sigma} \int_{0}^{1}G(t,s)\phi \bigl(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-2-\mu_{1}}x(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}x(s) ,Ax(s) \bigr)\,ds, \end{aligned}$$
(3.18)
$$\begin{aligned}& \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}l^{-1}w(s),I_{0^{+}}^{n-2-\mu_{1}}l^{-1}w(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}l^{-1}w(s), Al^{-1}w(s) \bigr)\,ds \\& \quad \geq \lambda \int_{0}^{1}G(t,s)\psi \bigl(s,l^{-1}I_{0^{+}}^{n-2}w(s),l^{-1}I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots,l^{-1}I_{0^{+}}^{n-2-\mu_{n-2}}w(s), l^{-1}Aw(s) \bigr)\,ds \\& \quad \geq \lambda l^{\sigma} \int_{0}^{1}G(t,s)\psi \bigl(s,I_{0^{+}}^{n-2}w(s),I_{0^{+}}^{n-2-\mu_{1}}w(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}w(s), Aw(s) \bigr)\,ds. \end{aligned}$$
(3.19)
Equations (3.18), (3.19) imply that
$$ T_{\lambda} \biggl(lx,\frac{1}{l}w \biggr)\geq l^{\sigma}T_{\lambda }(x,w), \quad x,w\in Q_{e}. $$
(3.20)
Hence, as regards Lemma 2.6 assume that there exists a unique positive solution \(x_{\lambda}^{\star}\in Q_{e}\) such that \(T_{\lambda }(x_{\lambda}^{\star},x_{\lambda}^{\star})=x_{\lambda}^{\star}\). It is easy to check that \(x_{\lambda}^{\star}\) is a unique positive solution of (2.8) for any given \(\lambda>0\). Moreover, by Lemma 2.7 we have the following conclusions:
-
(1)
For any \(\lambda_{0}\in(0,+\infty)\), \(\|x_{\lambda}^{\star }-x_{\lambda_{0}}^{\star}\|\rightarrow0\), \(\lambda\rightarrow\lambda_{0}\).
-
(2)
If \(0<\sigma<\frac{1}{2}\), then \(0<\lambda_{1}<\lambda_{2}\) implies \(x_{\lambda_{1}}^{\star}\leq x_{\lambda_{2}}^{\star}\), \(x_{\lambda_{1}}^{\star}\neq x_{\lambda_{2}}^{\star}\).
-
(3)
\(\lim_{\lambda\rightarrow0}\|x_{\lambda}^{\star}\|=0\), \(\lim_{\lambda\rightarrow+\infty}\|x_{\lambda}^{\star}\|=+\infty\).
By Lemma 2.5, we have
$$ \left \{ \textstyle\begin{array}{l} u_{\lambda}^{\star}(t)=I_{0^{+}}^{n-2}x_{\lambda }^{\star}(t), \\ v_{\lambda}^{\star}(t)=I_{0^{+}}^{m-2}y_{\lambda}^{\star}(t), \quad t\in[0,1]. \end{array}\displaystyle \right . $$
(3.21)
Hence, by (3.21) and the monotonicity and continuity of \(I_{0^{+}}^{n-2}\), we get:
-
(1)
\(\|u_{\lambda}^{\star}-u_{\lambda_{0}}^{\star}\|\rightarrow 0\), \(\lambda\rightarrow\lambda_{0}\).
-
(2)
If \(0<\sigma<\frac{1}{2}\), then \(0<\lambda_{1}<\lambda_{2}\) implies \(u_{\lambda_{1}}^{\star}\leq u_{\lambda_{2}}^{\star}\), \(u_{\lambda_{1}}^{\star}\neq u_{\lambda_{2}}^{\star}\).
-
(3)
\(\lim_{\lambda\rightarrow0}\|u_{\lambda}^{\star}\|=0\), \(\lim_{\lambda\rightarrow+\infty}\|u_{\lambda}^{\star}\|=+\infty\).
Moreover, for any \(u_{0}(t)=I_{0^{+}}^{n-2}x_{0}\in Q_{e}\), by Lemma 2.6, constructing a successive sequence
$$\begin{aligned} x_{k+1}(t) =&\lambda \int_{0}^{1}G(t,s)\bigl[\phi \bigl(s,I_{0^{+}}^{n-2}x_{k}(s), I_{0^{+}}^{n-2-\mu_{1}}x_{ k}(s), \ldots,I_{0^{+}}^{n-2-\mu_{n-2}}x_{ k}(s),Ax_{ k}(s) \bigr) \\ &{}+\psi\bigl(s,I_{0^{+}}^{n-2}x_{ k}(s), I_{0^{+}}^{n-2-\mu_{1}}x_{ k}(s),\ldots, I_{0^{+}}^{n-2-\mu_{n-2}}x_{ k}(s) ,Ax_{ k}(s)\bigr)\bigr]\,ds, \quad k=1,2,\ldots, \end{aligned}$$
by \(u_{k+1}(t)=I_{0^{+}}^{n-2}x_{k+1}(t)\), then
$$\begin{aligned} u_{k+1}(t) =&I_{0^{+}}^{n-2}\biggl\{ \lambda \int_{0}^{1}G(t,s) \bigl[\phi\bigl(s,u_{ k}(s), D_{0^{+}}^{\mu_{1}}u_{ k}(s),D_{0^{+}}^{\mu_{2}}u_{ k}(s), \ldots,D_{0^{+}}^{\mu_{n-2}}u_{ k}(s),Au^{(n-2)}_{k}(s) \bigr) \\ &{}+\psi\bigl(s,u_{ k}(s), D_{0^{+}}^{\mu_{1}}u_{ k}(s),D_{0^{+}}^{\mu_{2}}u_{ k}(s), \ldots,D_{0^{+}}^{\mu_{n-2}}u_{ k}(s),Au^{(n-2)}_{k}(s) \bigr) \bigr]\,ds\biggr\} , \\ &k=1,2,\ldots, \end{aligned}$$
and we have \(\|u_{k}-u_{\lambda}^{\star}\|=\| I_{0^{+}}^{n-2}x_{k}-I_{0^{+}}^{n-2}x_{\lambda}^{\star}\|\rightarrow 0\) as \(k\rightarrow\infty\), the convergence rate is
$$\bigl\Vert u_{k}-u_{\lambda}^{\star}\bigr\Vert = \bigl\Vert I_{0^{+}}^{n-2}x_{k}-I_{0^{+}}^{n-2}x_{\lambda}^{\star} \bigr\Vert =o\bigl(1-r^{\sigma^{k}}\bigr), $$
r is a constant, \(0< r<1\), and dependent on \(u_{0}\), where \(u_{\lambda }^{\star}(t)=I_{0^{+}}^{n-2}x_{\lambda}^{\star}(t)\), and we easily get
$$v_{\lambda}^{\star}(t) =I_{0^{+}}^{m-2} \biggl(\mu \int _{0}^{1}H(t,s)g\bigl(s,I_{0^{+}}^{n-2}x_{\lambda}^{\star}(s), I_{0^{+}}^{n-2-\eta_{1}}x_{\lambda}^{\star}(s),\ldots ,I_{0^{+}}^{n-2-\eta_{m-2}}x_{\lambda}^{\star}(s)\bigr)\,ds \biggr), $$
so by \(u_{\lambda}^{\star}(t)=I_{0^{+}}^{n-2}x_{\lambda}^{\star }(t)\), we have
$$ v_{\lambda}^{\star}(t)=I_{0^{+}}^{m-2} \biggl(\mu \int _{0}^{1}H(t,s)g\bigl(s,u_{\lambda}^{\star}(s),D_{0^{+}}^{\eta _{1}}u_{\lambda}^{\star}(s), \ldots, D_{0^{+}}^{\eta_{m-2}}u_{\lambda}^{\star}(s)\bigr) \,ds \biggr). $$
(3.22)
By (3.3), (3.22) , for any \(t\in[0,1]\), we easily get
$$\begin{aligned}& \frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)}t^{\alpha-1}\leq u_{\lambda}^{\star}(t)=I_{0^{+}}^{n-2}x_{\lambda}^{\star}(t) \leq \frac{D\Gamma(\alpha-n+2)}{\Gamma(\alpha)}t^{\alpha-1}, \\& \frac{\Gamma(\beta-m+2)\mu}{\Gamma(\beta)} \biggl(\frac{\Gamma(\alpha-n+2)}{D\Gamma(\alpha)}\biggr)^{\varsigma} t^{\beta-1} \int_{0}^{1}\theta_{2}(s)s^{\varsigma(\alpha -1)}g(s,1,1, \ldots,1)\,ds \\& \quad \leq v_{\lambda}^{\star }(t)=I_{0^{+}}^{m-2}y_{\lambda}^{\star}(t) \\& \quad \leq\frac{\Gamma(\beta-m+2)M_{2}\mu}{\Gamma(\beta)}\biggl(\frac {D\Gamma(\alpha-n+2)}{\Gamma(\alpha-\eta_{m-2})}+1\biggr) ^{\varsigma}t^{\beta-1} \\& \qquad {}\times\int_{0}^{1}(1-s)^{\beta-m+1}g(s,1,1,\ldots,1) \,ds. \end{aligned}$$
Therefore, the proof of Theorem 3.1 is completed. □
Remark 3.1
Compared with previous work [23, 25], the fractional orders are involved not only in the nonlinearity f but also in the nonlinearity g and the uniqueness positive solution of equation (1.1) is dependent on eigenvalue λ. Moreover, compared with [25], an iterative sequence and the convergence rate are also given.