Fractional calculus [1-2] developed since 17th century. In recent years, fractional differential equations have been of great interest. Both fractional differential equations and differential equations with the $p-$Laplacian operator are widely applied in different fields. For details, see [3-10] and references therein.

Goodrich [8] considered a class of fractional boundary value problems of the form

$\left\{ \begin{aligned} & -D_{0+}^{\nu}y(t)=f(t, y(t)), 0<t<1, \\ & y^{(i)}(0)=0, \left[D_{0+}^{\alpha}y(t)\right]_{t=1}=0, \\ \end{aligned} \right.$

where $0\leq i\leq n-2, 1\leq\alpha\leq n-2, \nu>3$ satisfying $n-1<\nu\leq n, n$ is a given integer, and $D_{0+}^{\nu}, D_{0+}^{\alpha}$ is the Riemann-Liouville fractional derivative. The author obtained the Green's function of this problem and proved that the Green's function satisfied a Harnack-like inequality. By using a fixed point theorem due to Krasnoselskii, the author established the existence results for at least one positive solution of the problem.

Yang, Zhang and Liu [9] studied the fractional boundary value problem

$\left\{ \begin{aligned} & D_{0+}^{\alpha}u(t)+f\left(t, u(t), D_{0+}^{\beta}u(t)\right)=0, t\in (0, 1), \\ & u^{(i)}(0)=0, 0\leq i\leq n-2, \left[D_{0+}^{\delta}u(t)\right]_{t=1}=0, 1\leq\delta\leq n-2, \\ \end{aligned} \right.$

where $f\in C([0,1]\times R^+ \times R, R^+), 0<\beta \leq1, n-1<\alpha\leq n, n>3$ is a given integer, and $D_{0+}^{\alpha}, D_{0+}^{\beta}, D_{0+}^{\delta}$ is the Riemann-Liouville fractional derivative. By means of a fixed point theorem in a cone, the author obtained the existence results for at least one positive solution.

There are many papers [5, 6] studying eigenvalue problems for boundary value problems of integer-order differential equations. But there are few papers discussing eigenvalue problems of fractional boundary value problems with the $p-$Laplacian operator. Motivated by these works, we study the the higher-order two-point boundary value problem of fractional order differential equations with the $p-$Laplacian operator

$\left\{ \begin{aligned} & \left[\varphi_p\left(D_{0+}^{\alpha}u(t)\right)\right]'+\lambda h(t)f(u(t))=0, 0<t<1, \\ & u^{(i)}(0)=0(i=0, 1, \cdots, N-2), D_{0+}^{\beta}u(1)=0, \\ \end{aligned} \right.$

(1.1)

where $\varphi_p(s)=|s|^{p-2}s, p>1, \frac{1}{p}+\frac{1}{q}=1, \alpha>2, \lambda>0, 1\leq \beta\leq N-2, h\in C((0, 1), [0, +\infty)), f\in C([0, +\infty), $ $ [0, +\infty)), N$ is the smallest integer greater than or equal to $\alpha$, $D_{0+}^{\alpha}, D_{0+}^{\beta}$ is the Riemann-Liouville fractional derivative.

For the convenience of the reader, we list the necessary definitions from fractional calculus theory here.

Definition 2.1 [7] The Riemann-Liouville fractional integral of order $\alpha>0$ of a function $u:(0, \infty)\rightarrow R$ is given by $I_{0+}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}u(s)ds, $ provided the right-hand side is pointwise defined on $(0, \infty)$.

Definition 2.2 [7] The Riemann-Liouville fractional derivative of order $\alpha>0$ of a continuous function $u:(0, \infty)\rightarrow R$ is given by

$D_{0+}^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\left(\frac{d}{dt}\right)^n\int_0^t\frac{u(s)}{(t-s)^{\alpha-n+1}}ds,$

where $n=[\alpha]+1$, provided the right-hand side is pointwise defined on $(0, \infty)$.

Lemma 2.1 [7] Assume that $u\in C(0, 1)\bigcap L(0, 1)$ with a fractional derivative of order $\alpha>0$ that belongs to $C(0, 1)\bigcap L(0, 1)$. Then

$I_{0+}^{\alpha}D_{0+}^{\alpha}u(t)=u(t)+c_1t^{\alpha-1}+c_2t^{\alpha-2}+\cdots+c_Nt^{\alpha-N}$

for some $c_i \in R(i=1, 2, \cdots N), $ where $N$ is the smallest integer greater than or equal to $\alpha$.

Lemma 2.2 [8] Given $y\in C[0,1]$. The problem

$\left\{ \begin{aligned} & \left[\varphi_p\left(D_{0+}^{\alpha}u(t)\right)\right]'+y(t)=0, 0<t<1, \\ & u^{(i)}(0)=0(i=0, 1, \cdots, N-2), D_{0+}^{\beta}u(1)=0 \end{aligned} \right.$

(2.1)

is equivalent to $u(t)=\int_0^1G(t, s)\varphi_q\left(\int_0^sy(r)dr\right)ds, $ where

$G(t, s)=\left\{ \begin{aligned} & \frac{t^{\alpha-1}(1-s)^{\alpha-\beta-1}-(t-s)^{\alpha-1}} {\Gamma(\alpha)}, 0\leq s\leq t\leq1, \\ & \frac{t^{\alpha-1}(1-s)^{\alpha-\beta-1}} {\Gamma(\alpha)}, 0\leq t\leq s\leq1.\\ \end{aligned} \right.$

(2.2)

Lemma 2.3 [8] The function $G(t, s)$ in $(2.2)$ satisfies

(1)$G(t, s)>0, t, s\in(0, 1);$

(2)$\max\limits_{0\leq t\leq 1}G(t, s)\leq G(1, s)(s\in (0, 1));$

(3)$\min\limits_{\frac{1}{2}\leq t\leq 1}G(t, s)\geq \gamma_0G(1, s)(s\in (0, 1)) ,\;\text{where}\;0<\gamma_0=\min\left\{\frac{\left(\frac{1}{2}\right)^{\alpha-\beta-1}}{2^\beta-1}, \left(\frac{1}{2}\right)^{\alpha-1}\right\}\leq \frac{1}{2}.$

The following theorem is fundamental in the proofs of our main results.

Lemma 2.4 [6] Let $P$ be a cone in a Banach space $X$. Assume $\Omega_1, \Omega_2$ are open subsets of $X$ with $0\in \Omega_1\subset \overline{\Omega}_1\subset \Omega_2$. If $F:P\rightarrow P$ is completely continuous such that either

(1)$ \|Fu\|\leq \|u\|, \forall u\in P\bigcap \partial\Omega_1, \|Fu\|\geq \|u\|, \forall u\in P\bigcap \partial\Omega_2,$ or

(2)$ \|Fu\|\geq \|u\|, \forall u\in P\bigcap \partial\Omega_1, \|Fu\|\leq \|u\|, \forall u\in P\bigcap \partial\Omega_2, $

Then $F$ has a fixed point in $P\bigcap \left(\overline{\Omega}_2\backslash \Omega_1\right).$

Let $E=C([0,1], R).$ Then $E$ is a Banach space with the norm $\|u\|=\max\limits_{0\leq t\leq 1}|u(t)|.$ Define the cone $P\subseteq E$ by $P=\left\{u\in E:u(t)\geq 0( t\in [0,1]), \min\limits_{\frac{1}{2}\leq t\leq 1}u(t)\geq \gamma_0\|u\|\right\}.$ For any $u\in P, $ define $F_\lambda:P\rightarrow E, (F_\lambda u)(t)=\lambda \int_0^1G(t, s)\varphi_q\left(\int_0^sh(r)f(u(r))dr\right)ds$. For each $u\in P$, we get

$\quad \min\limits_{\frac{1}{2}\leq t\leq 1}(F_\lambda u)(t)=\min\limits_{\frac{1}{2}\leq t\leq 1}\lambda \int_0^1G(t, s)\varphi_q\left(\int_0^sh(r)f(u(r))dr\right)ds \\ \geq \lambda\gamma_0\int_0^1G(1, s)\varphi_q\left(\int_0^s h(r)f(u(r))dr\right)ds \geq\gamma_0\|F_\lambda u\|,$

$F_\lambda P\subseteq P.$ Standard arguments show that $F_\lambda :P\rightarrow P$ is completely continuous. $u$ is a positive solution of (1.1) if and only if $u\in P$ is a fixed point of $F_\lambda$.

For convenience, we denote

$F_0=\lim\limits_{u\rightarrow 0^+}\sup\frac{\varphi_q(f(u))}{u}, F_\infty=\lim\limits_{u\rightarrow +\infty}\sup\frac{\varphi_q(f(u))}{u}, f_0=\lim\limits_{u\rightarrow 0^+}\inf\frac{\varphi_q(f(u))}{u}, \\ f_\infty=\lim\limits_{u\rightarrow +\infty}\inf \frac{\varphi_q(f(u))}{u}, C_1=\int_0^1 G(1, s)\varphi_q\left(\int_0^sh(r)dr\right) ds, C_2\\ \quad\quad=\gamma_0^2\int_\frac{1}{2}^1 G(1, s)\varphi_q\left(\int_0^sh(r)dr\right) ds. $

Theorem 3.1 If $f_\infty C_2>F_0C_1$ holds, then for each $\lambda \in \left(\frac{1}{f_\infty C_2}, \frac{1}{F_0 C_1}\right)$, $(1.1)$ has at least one positive solution. Here we impose $\frac{1}{f_\infty C_2}=0$ if $f_\infty =+\infty$ and $\frac{1}{F_0 C_1}=+\infty$ if $F_0=0$.

Proof For $\lambda \in \left(\frac{1}{f_\infty C_2}, \frac{1}{F_0 C_1}\right)$, let $\varepsilon>0$ be such that $\frac{1}{(f_\infty-\varepsilon) C_2}\leq \lambda \leq \frac{1}{(F_0+\varepsilon) C_1}$. There exists $r_1>0$ such that $f(u)\leq \varphi_p\left[\left(F_0+\varepsilon\right)u\right]$, for $0<u\leq r_1$. If $u\in P$ with $\|u\|=r_1$,

$\quad \|F_\lambda u\| \leq \lambda \int_0^1G(1, s)\varphi_q\left\{\int_0^sh(r)\varphi_p\left[\left(F_0+\varepsilon\right)u(r)\right]dr\right\}ds \\ \leq \lambda r_1\left(F_0+\varepsilon\right)\int_0^1G(1, s)\varphi_q\left(\int_0^sh(r)dr\right)ds = \lambda r_1\left(F_0+\varepsilon\right)C_1\leq r_1=\|u\|.$

If we choose $\Omega_1=\left\{u\in E:\|u\|<r_1\right\}$, then $\|F_\lambda u\|\leq \|u\|$, for $u\in P\bigcap \partial \Omega_1$. Let $r_3>0$ be such that $f(u)\geq \varphi_p\left[\left(f_\infty-\varepsilon\right)u\right]$, for $u\geq r_3$. If $u\in P$ with $\|u\|=r_2=\max\left\{2r_1, \frac{r_3}{\gamma_0}\right\}$,

$\quad\|F_\lambda u\|\geq \min\limits_{\frac{1}{2}\leq t\leq 1}\lambda \int_{\frac{1}{2}}^{1}G(t, s)\varphi_q\left(\int_0^sh(r)f(u(r))dr\right)ds \\ \geq \lambda\gamma_0\int_{\frac{1}{2}}^{1}G(1, s)\varphi_q\left\{\int_0^sh(r)\varphi_p\left[\left(f_\infty-\varepsilon\right)u(r)\right]dr\right\}ds \\ \geq \lambda \left(f_\infty-\varepsilon\right)\gamma_0^2\|u\|\int_{\frac{1}{2}}^{1}G(1, s)\varphi_q\left(\int_0^sh(r)dr\right) ds = \lambda \left(f_\infty-\varepsilon\right)C_2\|u\|\geq \|u\|.$

If we choose $\Omega_2=\left\{u\in E:\|u\|<r_2\right\}$, then $\|F_\lambda u\|\geq \|u\|$, for $u\in P\bigcap \partial \Omega_2$. By Lemma 2.4, $F_\lambda$ has a fixed point $u\in P\bigcap \left(\overline{\Omega}_2 \backslash \Omega_1\right)$ with $r_1\leq\|u\|\leq r_2$. The proof is completed.

Theorem 3.2 If $f_0 C_2>F_\infty C_1$ holds, then for each $\lambda \in \left(\frac{1}{f_0 C_2}, \frac{1}{F_\infty C_1}\right)$, $(1.1)$ has at least one positive solution. Here we impose $\frac{1}{f_0 C_2}=0$ if $f_0 =+\infty$ and $\frac{1}{F_\infty C_1}=+\infty$ if $F_\infty=0$.

Proof For $\lambda \in \left(\frac{1}{f_0 C_2}, \frac{1}{F_\infty C_1}\right)$, let $\varepsilon>0$ be such that $\frac{1}{(f_0-\varepsilon) C_2}\leq \lambda \leq \frac{1}{(F_\infty+\varepsilon) C_1}$. There exists $r_1>0$ such that $f(u)\geq \varphi_p\left[\left(f_0-\varepsilon\right)u\right]$ for $0<u\leq r_1$. If $u\in P$ with $\|u\|=r_1$, then similar to the proof of Theorem 3.1, we can obtain that $\|F_\lambda u\|\geq \|u\|$.

If we choose $\Omega_1=\left\{u\in E:\|u\|<r_1\right\}$, then $\|F_\lambda u\|\geq \|u\|$, for $u\in P\bigcap \partial \Omega_1$. Let $r_3>0$ be such that $f(u)\leq \varphi_p\left[\left(F_\infty+\varepsilon\right)u\right]$, for $u\geq r_3$. We consider two cases:

Case 1 If $f$ is bounded, there exists $M>0$ such that $f(u)\leq \varphi_p(M)(u\in (0, +\infty))$. Let $r_4=\max\{2r_1, \lambda M C_1\}.$ For $u\in P$ with $\|u\|=r_4$, $\|F_\lambda u\|\leq \lambda M \int_0^1G(1, s)\varphi_q\left(\int_0^sh(r)dr\right) ds \leq \lambda MC_1\leq r_4=\|u\|.$ Thus $\|F_\lambda u\|\leq \|u\|$, for $u\in \partial P_{r_4}$.

Case 2 If $f$ is unbounded, there exists $r_5>\max\{2r_1, r_3\}$ such that $f(u)\leq f(r_5)$ for $0<u\leq r_5$. For $u\in P$ with $\|u\|=r_5$, $\|F_\lambda u\|\leq \lambda r_5 \left(F_\infty+\varepsilon\right) \int_0^1G(1, s)\varphi_q\left(\int_0^sh(r)dr\right)ds \leq r_5=\|u\|.$ Thus $\|F_\lambda u\|\leq \|u\|$, for $u\in \partial P_{r_5}$.

In both Cases 1 and 2, if we set $\Omega_2=\{u\in E: \|u\|<r_2=max\{r_4, r_5\}\}$, then $\|F_\lambda u\|\leq \|u\|$, for $u\in P\bigcap \partial \Omega_2$. By Lemma 2.4, $F_\lambda$ has a fixed point $u\in P\bigcap \left(\overline{\Omega}_2 \backslash \Omega_1\right)$ with $r_1\leq\|u\|\leq r_2$. The proof is completed.

Theorem 3.3 Suppose there exist $r_2>r_1>0$ or $\gamma_0r_1>r_2>0$ such that

$\max\limits_{0\leq u\leq r_2}f(u)\leq \varphi_p\left(\frac{r_2}{\lambda C_1}\right), \min\limits_{\gamma_0r_1\leq u\leq r_1}f(u)\geq \varphi_p\left(\frac{r_1}{\lambda C_2}\right).$

Then $(1.1)$ has at least one positive solution $u\in P$.

The proof of Theorem 3.3 is similar to that of Theorem 3.1, we omit it here.

For the reminder of the paper, we will need the condition $(H_1) \sup\limits_{r>0}\min\limits_{\gamma_0r\leq u\leq r}f(u)>0.$ Denote

$\lambda_1=\sup\limits_{r>0}\frac{r}{C_1\max\limits_{0\leq u\leq r}\varphi_q(f(u)) }, \lambda_2=\inf\limits_{r>0}\frac{r}{C_2\min\limits_{\gamma_0r\leq u\leq r}\varphi_q(f(u)) }.$

In view of the continuity of $f(u)$ and $(H_1)$, we have $0<\lambda_1\leq +\infty, 0\leq\lambda_2< +\infty$.

Theorem 3.4 Assume $(H_1)$ holds. If $f_0=+\infty$ and $f_\infty=+\infty$, then $(1.1)$ has at least two positive solutions for each $\lambda \in (0, \lambda_1)$.

Proof Define $a(r)=\frac{r}{C_1\max\limits_{0\leq u\leq r}\varphi_q(f(u)) }$. $a(r):(0, +\infty)\rightarrow (0, +\infty)$ is continuous and $\lim\limits_{r\rightarrow 0}a(r)=\lim\limits_{r\rightarrow +\infty}a(r)=0$. There exists $r_0\in (0, +\infty)$ such that $a(r_0)=\sup\limits_{r>0}a(r)=\lambda_1$. For $\lambda \in (0, \lambda_1)$, there exist $c_1, c_2(0<c_1<r_0<c_2<+\infty)$ with $a(c_1)=a(c_2)=\lambda.$

$f(u)\leq \varphi_p\left(\frac{c_1}{\lambda C_1}\right)(u\in [0, c_1]), f(u)\leq \varphi_p\left(\frac{c_2}{\lambda C_1}\right)(u\in [0, c_2]).$

On the other hand, for $f_0=+\infty$ and $f_\infty=+\infty$, there exist $d_1, d_2(0<d_1<c_1<r_0<c_2<\gamma_0d_2<+\infty)$ satisfying $\frac{\varphi_q(f(u))}{u}\geq \frac{1}{\gamma_0\lambda C_2}$, for $u\in (0, d_1]\bigcup \left[\gamma_0d_2, +\infty\right)$. Thus

$\min\limits_{\gamma_0d_1\leq u\leq d_1}f(u)\geq \varphi_p\left(\frac{d_1}{\lambda C_2}\right), \min\limits_{\gamma_0d_2\leq u\leq d_2}f(u)\geq \varphi_p\left(\frac{d_2}{\lambda C_2}\right).$

By Theorem 3.3, (1.1) has at least two positive solutions for each $\lambda \in (0, \lambda_1)$. The proof is completed.

Corollary 3.1 Assume $(H_1)$ holds. If $f_0=+\infty$ or $f_\infty=+\infty$, then (1.1) has at least one positive solution for each $\lambda \in (0, \lambda_1)$.

Theorem 3.5 Assume $(H_1)$ holds. If $F_0=0$ and $F_\infty=0$, then $(1.1)$ has at least two positive solutions for each $\lambda \in (\lambda_2, +\infty)$.

Proof Define $b(r)=\frac{r}{C_2\min\limits_{\gamma_0r\leq u\leq r}\varphi_q(f(u)) }$. $b(r):(0, +\infty)\rightarrow (0, +\infty)$ is continuous and $\lim\limits_{r\rightarrow 0}b(r)=\lim\limits_{r\rightarrow +\infty}b(r)=+\infty$. There exists $r_0\in (0, +\infty)$ such that $b(r_0)=\inf\limits_{r>0}b(r)=\lambda_2$. For $\lambda \in (\lambda_2, +\infty)$, there exist $d_1, d_2(0<d_1<r_0<d_2<+\infty)$ satisfying $b(d_1)=b(d_2)=\lambda.$ Thus,

$f(u)\geq \varphi_p\left(\frac{d_1}{\lambda C_2}\right)\left(u\in \left[\gamma_0d_1, d_1\right]\right), f(u)\geq \varphi_p\left(\frac{d_2}{\lambda C_2}\right)\left(u\in \left[\gamma_0d_2, d_2\right]\right).$

On the other hand, applying the condition $F_0=0$, there exist $c_1\left(0<c_1<\gamma_0d_1\right)$ satisfying $\frac{\varphi_q(f(u))}{u}\leq \frac{1}{\lambda C_1}$, for $u\in (0, c_1]$. Thus $\max\limits_{0\leq u \leq c_1}f(u)\leq \varphi_p\left(\frac{c_1}{\lambda C_1}\right)$. For $F_\infty=0$, there exists $c_3(c_3>d_2)$ satisfying $\frac{\varphi_q(f(u))}{u}\leq \frac{1}{\lambda C_1}$, for $u\in (c_3, +\infty)$. Let

$M=\max\limits_{0 \leq u\leq c_3}f(u), c_2=\max\left\{2c_3, \lambda C_1\varphi_q(M)\right\}.$

Thus $\max\limits_{0 \leq u\leq c_2}f(u)\leq \varphi_p\left(\frac{c_2}{\lambda C_1}\right)$. By Theorem 3.3, (1.1) has at least two positive solutions for each $\lambda \in (\lambda_2, +\infty)$. The proof is completed.

Corollary 3.2 Assume $(\mathrm{H}_1)$ holds. If $F_0=0$ or $F_\infty=0$, then (1.1) has at least one positive solution for each $\lambda \in (\lambda_2, +\infty)$.