Skip to main content

Advertisement

Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters

  • 207 Accesses

Abstract

We consider a class of Caputo fractional p-Laplacian differential equations with integral boundary conditions which involve two parameters. By using the Avery–Peterson fixed point theorem, we obtain the existence of positive solutions for the boundary value problem. As an application, we present an example to illustrate our main result.

Introduction

In this paper, we investigate the following integral boundary value problem (short for BVP) of Caputo fractional differential equations with p-Laplacian operator and parameters:

$$ \left \{ \textstyle\begin{array}{l} D_{0+}^{\beta}\varphi_{p} ( D_{0+}^{\alpha}x(t) ) + f (t,x(t),D_{0+} ^{\beta}x(t) )= 0,\quad t \in(0,1),\\ ( \varphi_{p} ( D_{0+}^{\alpha}x(0) ) )^{(i)} =\varphi_{p} ( D_{0+}^{\alpha}x(1) )=0,\quad i=1,2, \ldots, m-1 , \\ x(0) + x^{{\prime}}(0) = \int_{0}^{1}g_{0}(s)x(s)\,ds + a, \\ x(1) + x^{{\prime}}(1) = \int_{0}^{1}g_{1}(s)x(s)\,ds + b, \\ x^{(j)}(0) = 0,\quad j= 2, 3, \ldots, n-1, \end{array}\displaystyle \right . $$
(1)

where \(1< n-1<\alpha<n\), \(1< m-1<\beta< m\), \(\alpha-\beta>1 \), \(D_{{0^{+}}}^{\alpha}\) and \(D_{{0^{+}}}^{\beta}\) are the Caputo fractional derivatives. \(\varphi_{p}\) is the p-Laplacian operator, \(\varphi _{p}(s)=\vert s\vert^{p-2}s\), \(p>1\), \(\varphi_{p}^{-1}=\varphi_{q}\), \(1/p+1/q=1\). \(g_{0},g_{1} \in C([0,1], [0,+\infty)) \), \(f \in C([0,1] \times[0,+\infty) \times[0,+\infty), [0,+\infty)) \) are given functions. \(a,b>0\) are disturbance parameters.

As we all know, fractional differential equation theory is becoming more and more perfect because of its extensive application, and many significant achievements have been made; see [112]. As one of many applications, turbulence problem can be well characterized by the p-Laplacian operator; see [13]. Fractional p-Laplacian equations are becoming more and more important, they can be used to describe a class of diffusion phenomena, which have been widely used in the fields of fluid mechanics, material memory, biology, plasma physics, finance and chemistry. Many important results related to the boundary value problems of fractional differential equations with p-Laplacian operator have been obtained; see [1424]. But in practical problems, disturbance is objective. As a boundary value problem with disturbance parameter can describe real problems better, many scholars turn their attention to it.

In [6], Jia et al. consider the fractional-order differential equation integral boundary value problem with disturbance parameters

$$\left \{ \textstyle\begin{array}{l} - {}^{C}{D^{\delta}}u ( t ) = f ( {t,u ( t )} ),\quad t \in J,\\ {m_{1}}u ( 0 ) - {n_{1}}u' ( 0 ) = 0,\\ {m_{2}}u ( 1 ) - {n_{2}}u' ( 1 ) = \int_{0}^{1} {g ( s )u ( s )\,ds} + a, \end{array}\displaystyle \right . $$

where \(J=[0,1]\), \(1 < \delta\leq2\), \(f \in C( [0,1]\times[0,+\infty) , [0,+\infty) )\), \(m_{i} \geq0\), \(n_{i} \geq0\), \(m^{2}_{i}+n^{2}_{i}>0\), \({i=1,2}\), \(g \in C( [0,1] , [0,+\infty))\), disturbance parameter \(a >0\), and \(^{C}{D^{\delta}}\) is the Caputo fractional derivative of order δ. By using an upper and lower solution method, the fixed point index theorem and the Schauder fixed point theorem, sufficient conditions are obtained for the problem to have at least one positive solution, two positive solutions and no solution.

In [25], Wang et al. consider a class of fractional differential equations with integral boundary conditions which involve two disturbance parameters. By using the Guo–Krasnoselskii fixed point theorem, new results on the existence and nonexistence of positive solutions for the boundary value problem are obtained. The problem is given by

$$\left \{ \textstyle\begin{array}{l} D_{0{+}}^{\alpha}x(t)=f (t,x(t) ), \quad t \in(0,1), \\ x(0)=x'(0)=0,\\ x(1)=\int_{0}^{1}g_{1}(s)x(s)\, {d}s+a, \\ x'(1)=\int_{0}^{1}g_{2}(s)x(s)\, {d}s-b, \end{array}\displaystyle \right . $$

where \(D_{0+}^{\alpha} \) is the standard Riemann–Liouville fractional derivative with \(3<\alpha\leq4 \), \({f:[0,1]\times[0,+\infty) \rightarrow[0,+\infty)}\) is a continuous function, \(g_{1},g_{2}\in L^{1}[0,1]\) and \(a,b\geq0\).

In [26] Hao et al. consider the existence of positive solutions of higher order fractional integral boundary value problem with a parameter

$$\left \{ \textstyle\begin{array}{l} -{D^{\eta-2}_{0+} } (u''( t ))+ \lambda f ( {t,u ( t )} ) =0,\quad t \in J,\\ u''(0)=u'''(0)= \cdots=u^{n-2}(0)=0,\\ ^{C}{D^{k-2}_{0+} } (u''( t )) \vert_{t=1}=0,\\ \alpha u(0) - \beta u'(0)= \int_{0}^{1}u(s)\,d A(s),\\ \gamma u(1) + \delta u'(1)= \int_{0}^{1}u(s)\,d B(s), \end{array}\displaystyle \right . $$

where \(D^{\eta-2}_{0+}\), \(D^{k-2}_{0+}\) are the standard Riemann–Liouville fractional derivative, \(n-1 < \eta\leq n\), \(\eta \geq4\), \(2 \leq k \leq n-2\), \(\alpha, \beta, \gamma, \delta>0 \). \(\int_{0}^{1}u(s)\,d A(s)\) and \(\int_{0}^{1}u(s)\,d B(s)\) denote the Riemann–Stieltjes integrals of u with respect to A and B. \(A(t)\), \(B(t)\) are nondecreasing on \([0, 1]\), \(f : [0, 1] \times [0, +\infty) \rightarrow[0, +\infty)\) is continuous, \(\lambda> 0\) is a parameter. By using the Guo–Krasnoselskii fixed point theorem on cones, under different conditions of nonlinearity, existence and nonexistence, results for positive solutions are derived in terms of different parameter intervals.

The purpose of this paper is to establish conditions ensuring the existence of three positive solutions of BVP (1) and give an estimate of these solutions by using the Avery–Peterson fixed point theorem. Our supposed problem is different from the problems studied before and mentioned above. Our result is new and our work extends the application of the theorem.

In this paper, a positive solution \(x=x(t)\) of BVP (1) means a solution of (1) satisfying \(x(t)>0\), \(t\in[0,1]\).

Throughout this paper, we always assume that the following condition is satisfied:

(L0):

\(0< a< b<2a<+\infty\), \(0\leq g_{0}(t) \leq g_{1}(t) \leq 2g_{0}(t)\), \(0\leq\int_{0}^{1}g_{0}(s)\,ds\), \(\int_{0}^{1}g_{1}(s)\,ds < 1\).

Preliminaries and lemmas

The basic theory of fractional-order differential equation and boundary value problem can be obtained from many places in the literature, which will not be repeated here; see [19]. Here we present some necessary basic results that will be used.

Lemma 2.1

(see [2])

The Caputo fractional derivative of order \(n-1<\alpha<n \) for \(t^{\beta}\) is given by

$$ D_{0 + }^{\alpha}{t^{\beta}} = \left \{ { \textstyle\begin{array}{l@{\quad}l} {\frac{{\varGamma(\beta + 1)}}{{\varGamma(\beta - \alpha + 1)}}{t^{\beta - \alpha}},}&{\beta \in N\textit{ and }\beta \ge n\textit{ or }\beta \notin N\textit{ and }\beta > n - 1,}\\ {0,}&{\beta \in\{ 0,1, \ldots,n - 1\} .} \end{array}\displaystyle } \right . $$

Lemma 2.2

(see [19])

Let \(h\in C[0,1] \)and \(1< m-1<\beta<m\). Then the BVP

$$ \left \{ \textstyle\begin{array}{l} D_{{0^{+} }}^{\beta}u(t)=h(t),\quad 0< t< 1, \\ u(1) = u^{(i)}(0) = 0, \quad i=1,2,\ldots,m-1, \end{array}\displaystyle \right . $$

has an unique solution

$$ u(t)=- \int_{0}^{1}H(t,s)h(s)\,ds, $$

where

$$ H(t,s)=\frac{1}{\varGamma(\beta)}\left \{ \textstyle\begin{array}{l@{\quad}l} (1-s)^{\beta-1}-(t-s)^{\beta-1}, & 0\leq s\leq t\leq1, \\ (1-s)^{\beta-1},&0\leq t\leq s\leq1. \end{array}\displaystyle \right . $$
(2)

Denote

$$ \begin{gathered} M_{1}= \int_{0}^{1}g_{0}(s)\,ds,\qquad M_{2}= \int_{0}^{1}sg_{0}(s)\,ds,\\ N_{1}= \int_{0}^{1}g_{1}(s)\,ds,\qquad N_{2}= \int_{0}^{1}sg_{1}(s)\,ds,\\ \delta^{-1}=1+M_{2}+N_{1}-M_{2}N_{1}+M_{1}(N_{2}-2)-N_{2},\\ \omega(t) =\delta \bigl( b(M_{2}-1+t-M_{1}t)+a(2-N_{2}-t+N_{1}t) \bigr).\end{gathered} $$
(3)

From (L0), we know, for \(t \in(0,1) \),

$$ 1>N_{1}>N_{2}>M_{2}>0,\qquad 1>N_{1}>M_{1}>M_{2}>0,\qquad 2M_{1}>N_{1},2M_{2}>N_{2}. $$
(4)

Thus

$$\begin{aligned} \delta^{-1} =& 1+M_{2}+N_{1}-M_{2}N_{1}+M_{1}(N_{2}-2)-N_{2} \\ =& 1+M_{2}+N_{1}-M_{2}N_{1}+M_{1}N_{2}-2M_{1}-N_{2} \\ =& (1+M_{1}N_{2}-2M_{1})+M_{2}+N_{1}-M_{2}N_{1}-N_{2} \\ =& (1-M_{1})+(M_{1}N_{2}-N_{2})+(M_{2}-M_{2}N_{1})+(N_{1}-M_{1}) \\ =& (1-M_{1})+N_{2}(M_{1}-1)+M_{2}(1-N_{1})+(N_{1}-M_{1}) \\ =& (1-M_{1}) (1-N_{2})+M_{2}(1-N_{1})+(N_{1}-M_{1}) \\ >& 0. \end{aligned}$$

Thus, the following lemma holds.

Lemma 2.3

Let (L0) hold, \(y\in C[0,1] \)and \(1< n-1<\alpha<n\), \(1< m-1<\beta <m\), then the following boundary value problem:

$$ \left \{ \textstyle\begin{array}{l} D_{{0^{+} }}^{\alpha}x(t)=y(t),\quad0< t< 1, \\ x^{(j)}(0) = 0, \quad j= 2, 3, \ldots, n-1,\\ x(0) + x^{{\prime}}(0) = \int_{0}^{1}g_{0}(s)x(s)\,ds + a, \\ x(1) + x^{{\prime}}(1) = \int_{0}^{1}g_{1}(s)x(s)\,ds + b, \end{array}\displaystyle \right . $$
(5)

has an unique solution

$$ x(t)= \int_{0}^{1}G(t,s)y(s)\,ds+\omega(t) $$
(6)

and

$$ D_{0+}^{\beta}x(t)=\frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{t}(t-s)^{\alpha-\beta-1}y(s)\,ds, $$
(7)

where

$$\begin{aligned}& G(t,s) = G_{1}(t,s)+G_{2}(t,s), \end{aligned}$$
(8)
$$\begin{aligned}& G_{1}(t,s) = \frac{1}{\varGamma(\alpha)}\left \{ \textstyle\begin{array}{l@{\quad}l} (t-s)^{\alpha-1}+(1-t)(1-s)^{\alpha-2}(\alpha -s), & 0\leq s\leq t\leq1, \\ (1-t)(1-s)^{\alpha-2}(\alpha-s),&0\leq t\leq s\leq1, \end{array}\displaystyle \right . \end{aligned}$$
(9)
$$\begin{aligned}& G_{2}(t,s) = \delta \int_{0}^{1} \bigl((M_{2}-1+t-M_{1}t)g_{1}( \tau)+(2-N_{2}-t+N_{1}t)g_{0}(\tau) \bigr) G_{1}(\tau,s)\,d\tau. \end{aligned}$$
(10)

Proof

Consider BVP (5), we have

$$ x(t)=\frac{1}{\varGamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s) \,ds+C_{0}+C_{1}t+C_{2}t^{2}+ \cdots+C_{n-1}t^{n-1}. $$

In view of \(x^{(j)}(0)=0\) (\(j=2, 3,\ldots,n-1\)), we know \(C_{2}=C_{3}=\cdots=C_{n-1}=0\), and

$$ \begin{gathered} x(0)=C_{0},\qquad x^{{\prime}}(0)=C_{1}, \qquad x(1)= \frac{1}{\varGamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-1}y(s) \,ds+C_{0}+C_{1}, \\ x^{{\prime}}(1)=\frac{1}{\varGamma(\alpha-1)} \int_{0}^{1}(1-s)^{\alpha-2}y(s) \,ds+C_{1},\end{gathered} $$

so that

$$ x(t)=\frac{1}{\varGamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s) \,ds+C_{0}+C_{1}t . $$
(11)

From the boundary condition of BVP (5), by methods similar to Lemma 2.4 in [19], through traditional analytical calculation and integration techniques, we have

$$\begin{aligned} x(t) =& \int_{0}^{1}G_{1}(t,s)y(s)\,ds+ \int_{0}^{1}G_{2}(t,s)y(s)\,ds+\omega(t) \\ =& \int_{0}^{1}G(t,s)y(s)\,ds+\omega(t), \end{aligned}$$

where \(\omega(t)\), \(G_{1}(t,s)\) and \(G_{2}(t,s)\) are given by (3), (9) and (10).

On the other hand, in view of (11), because \(1< m-1<\beta<\alpha -1<n-1\), by Lemma 2.1, we have

$$\begin{aligned} D_{0+}^{\beta}x(t) =& D_{0+}^{\beta} \biggl(\frac{1}{\varGamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s) \,ds+C_{0}+C_{1}t \biggr) \\ =& D_{0+}^{\beta} \bigl(I_{0+}^{\alpha}y(t)+C_{0}+C_{1}t \bigr) \\ =& D_{0+}^{\beta}I_{0+}^{\alpha}y(t)+D_{0+}^{\beta }(C_{0})+D_{0+}^{\beta}(C_{1}t) \\ =& D_{0+}^{\beta}I_{0+}^{\alpha}y(t) \\ =& I_{0+}^{\alpha-\beta}y(t) \\ =& \frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{t}(t-s)^{\alpha-\beta-1}y(s)\,ds. \end{aligned}$$
(12)

 □

Lemma 2.4

The BVP (1) is equivalent to the following integral equation:

$$ x(t)= \int_{0}^{1}G(t,s)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds+\omega(t) $$
(13)

and

$$ D_{0+}^{\beta}x(t)=\frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{t}(t-s)^{\alpha-\beta-1} \varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds, $$
(14)

where \(H(t,s)\), \(\omega(t) \)and \(G(t,s)\)are given by (2), (3) and (8).

Proof

From Lemma 2.2 and Lemma 2.3, let \(y(t)=\varphi_{q}(u(t))\), \(h(t)=-f (t,x(t),D_{0+}^{\beta}x(t) )\), we have

$$\begin{aligned} y(t) =& \varphi_{q} \bigl(u(t) \bigr)= \varphi_{q} \biggl( \int_{0}^{1}H(t,s)f \bigl(s,x(s),D_{0+}^{\beta}x(s) \bigr)\,ds \biggr) \\ =& \varphi_{q} \biggl( \int_{0}^{1}H(t,s)f \bigl(s,x(s),D_{0+}^{\beta}x(s) \bigr)\,ds \biggr). \end{aligned}$$

Immediately we obtain

$$\begin{aligned} x(t)&= \int_{0}^{1}G(t,s)y(s)\,ds+\omega(t) \\ &= \int_{0}^{1}G(t,s)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x(\tau ),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds+\omega(t). \end{aligned} $$

From (12), (14) holds.

On the other hand, if \(x(t)\) satisfies (13), we can easily prove that \(x(t)\) satisfies BVP (1). □

Lemma 2.5

Assume (L0) hold, then the function \(H(t,s)\)defined by (2), the function \(G(t,s)\)defined by (8), and then the function \(\omega(t)\)defined by (3) satisfies

  1. (1)

    \(H(t,s)\geq0\)is continuous for all \(t,s\in[0,1]\);

  2. (2)

    \(H(t,s)\leq H(s,s)\)for all \(t,s\in[0,1]\);

  3. (3)

    \(\int_{0}^{1}H(t,s)\,ds=\frac{1-t^{\beta}}{\varGamma(\beta +1)} \leq\frac{1}{\varGamma(\beta+1)}\)for all \(t\in[0,1]\);

  4. (4)

    \(G(t,s)\geq0\)is continuous for all \(t,s\in[0,1]\);

  5. (5)

    \(\omega(t)>0\)for all \(t\in[0,1]\).

Proof

(1) and (2) are proved in [19], we omit the proofs.

(3) For \(t \in[0,1]\), by a simple integral operation, we can obtain

$$\begin{aligned} \int_{0}^{1}H(t,s)\,ds&= \int_{0}^{t} \bigl((1-s)^{\beta-1}-(t-s)^{\beta-1} \bigr)\,ds + \int_{t}^{1} (1-s)^{\beta-1}\,ds\\ &= \frac{1-t^{\beta}}{\varGamma(\beta +1)} \leq\frac{1}{\varGamma(\beta+1)}.\end{aligned} $$

(4) From (9), we know \(G_{1}(t,s)\geq0\), \(t,s\in[0,1]\), and \(G_{1}(t,s)>0\), \(t,s\in(0,1)\). Combined with (L0), for \(t\in[0,1]\), we have

$$\begin{aligned} \frac{\partial G_{2}(t,s)}{\partial t} =& \delta \int_{0}^{1} \bigl((1-M_{1})g_{1}( \tau)+(-1+N_{1})g_{0}(\tau) \bigr)G_{1}(\tau,s)\,d \tau \\ =& \delta \int_{0}^{1} \bigl(g_{1}( \tau)-M_{1}g_{1}(\tau)-g_{0}(\tau)+N_{1}g_{0}( \tau) \bigr)G_{1}(\tau,s)\,d\tau \\ =& \delta \int_{0}^{1} \bigl(g_{1}( \tau)-g_{0}(\tau)-M_{1}g_{1}(\tau)+N_{1}g_{0}( \tau) \bigr)G_{1}(\tau,s)\,d\tau \\ >& \delta \int_{0}^{1} \bigl(g_{1}( \tau)-g_{0}(\tau)-M_{1}g_{1}(\tau)+M_{1}g_{0}( \tau) \bigr)G_{1}(\tau,s)\,d\tau \\ =& \delta \int_{0}^{1} \bigl(g_{1}( \tau)-g_{0}(\tau)-M_{1} \bigl(g_{1}( \tau)-g_{0}(\tau) \bigr) \bigr)G_{1}(\tau,s)\,d\tau \\ =& \delta \int_{0}^{1} \bigl(g_{1}( \tau)-g_{0}(\tau) \bigr) (1-M_{1}) G_{1}(\tau,s)\,d \tau \\ >&0, \end{aligned}$$

so that \(G_{2}(t,s)\) monotonically increase with respect to t.

As a consequence, from (10), we get

$$\begin{aligned} G_{2}(t,s) \geq& G_{2}(0,s)=\delta \int_{0}^{1} \bigl((M_{2}-1)g_{1}( \tau)+(2-N_{2})g_{0}(\tau) \bigr)G_{1}(\tau,s)\,d \tau \\ \geq&\delta \int_{0}^{1} \biggl((M_{2}-1)g_{1}( \tau)+\frac{1}{2}(2-N_{2})g_{1}(\tau) \biggr)G_{1}(\tau,s)\,d\tau \\ =&\delta \int_{0}^{1} \biggl(M_{2}- \frac{1}{2}N_{2} \biggr)g_{1}(\tau)G_{1}( \tau,s)\,d\tau \\ \geq&0. \end{aligned}$$

Hence, \(G(t,s)\geq0\).

(5) From (3), for \(t\in[0,1]\), we know

$$\begin{aligned} \omega'(t) =& b(1-M_{1})+a(N_{1}-1) \\ >& a(1-M_{1})+a(N_{1}-1) \\ >& a(1-N_{1})+a(N_{1}-1) \\ =& 0, \end{aligned}$$

so that

$$\begin{aligned} \omega(t) \geq& \omega(0)=b(M_{2}-1)+a(2-N_{2}) \\ >& b(M_{2}-1)+a(2-2M_{2}) \\ =& (1-M_{2}) (2a-b) \\ >& 0,\quad t\in[0,1]. \end{aligned}$$

 □

Lemma 2.6

Let \(\eta\in(0,\frac{1}{2})\), then

$$\begin{gathered} \max_{t\in[0,1] }G(t,s) \leq G_{1}(0,s)+G_{2}(1,s) , \\ \min_{t\in[0,\eta]}G(t,s) \geq \rho \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr),\end{gathered} $$

where

$$\begin{aligned} \rho=\frac{M_{2}-\frac{1}{2}N_{2}}{1-M_{1}+M_{2}+N_{1}-N_{2}}. \end{aligned}$$
(15)

Proof

Step 1: We prove

$$\begin{aligned} \min_{t\in[0,\eta]}G_{1}(t,s)\geq (1-\eta)G_{1}(0,s)> \frac{1}{2} \max_{t\in[0,1] }G_{1}(t,s). \end{aligned}$$
(16)

For \(0\leq s < t\leq1\) and \(t\in[0,\eta]\),

$$\varGamma(\alpha)\frac{\partial G_{1}(t,s)}{\partial t}=(\alpha -1) (t-s)^{\alpha-2}-(\alpha-s) (1-s)^{\alpha-2}\leq(\alpha -s) (t-s)^{\alpha-2}-(\alpha-s) (1-s)^{\alpha-2}< 0, $$

so that

$$ G_{1}(t,s)\leq G_{1}(s,s)=(\alpha-s) (1-s)^{\alpha-1}< ( \alpha-s) (1-s)^{\alpha-2}=G_{1}(0,s) $$

and

$$\begin{aligned} \frac{G_{1}(t,s)}{G_{1}(0,s)} =& \frac{(t-s)^{\alpha-1}+(\alpha -s)(1-t)(1-s)^{\alpha-2}}{(\alpha-s)(1-s)^{\alpha-2}} \\ =&\frac{(t-s)^{\alpha-1}}{(\alpha-s)(1-s)^{\alpha-2}}+1-t \\ \geq& 1-t \\ \geq& 1-\eta> \frac{1}{2}. \end{aligned}$$

For \(s\geq t\) and \(t \in[0,\eta]\),

$$ \varGamma(\alpha)\frac{\partial G_{1}(t,s)}{\partial t}=-(\alpha-s) (1-s)^{\alpha-2}< 0 $$

so that

$$ G_{1}(t,s)\leq G_{1}(0,s)=(\alpha-s) (1-s)^{\alpha-2} $$

and

$$\begin{aligned} \frac{G_{1}(t,s)}{G_{1}(0,s)} =& \frac{(\alpha-s)(1-t)(1-s)^{\alpha -2}}{(\alpha-s)(1-s)^{\alpha-2}}=1-t \geq1-\eta > \frac{1}{2}. \end{aligned}$$

Therefore, (16) holds.

Step 2: We prove

$$\begin{aligned} \min_{t\in[0,\eta]}G_{2}(t,s)\geq \rho G_{2}(1,s)=\rho\max_{t\in[0,1] }G_{2}(t,s). \end{aligned}$$
(17)

From Lemma 2.5, we know that \(G_{2}(t,s)\) is a monotone increasing function with respect to \(t \in[0,1]\), so that

$$\min_{t\in[0,\eta]}G_{2}(t,s)=G_{2}(0,s),\qquad \max _{t\in[0,1] }G_{2}(t,s)=G_{2}(1,s). $$

By (L0) and (4), we have

$$\begin{aligned} \frac{G_{2}(0,s)}{G_{2}(1,s)} =& \frac{\delta\int_{0}^{1} ((M_{2}-1)g_{1}(\tau)+(2-N_{2})g_{0}(\tau) ) G_{1}(\tau,s)\,d\tau }{\delta\int_{0}^{1} ((M_{2}-1+1-M_{1})g_{1}(\tau )+(2-N_{2}-1+N_{1})g_{0}(\tau) ) G_{1}(\tau,s)\,d\tau} \\ \geq& \frac{ \int_{0}^{1} ((M_{2}-1)g_{1}(\tau)+\frac {1}{2}(2-N_{2})g_{1}(\tau) ) G_{1}(\tau,s)\,d\tau}{ \int_{0}^{1} ((M_{2}-M_{1})g_{1}(\tau)+(1-N_{2}+N_{1})g_{1}(\tau) ) G_{1}(\tau,s)\, d\tau} \\ =& \frac{ \int_{0}^{1}(M_{2}-1+1-\frac{1}{2}N_{2})g_{1}(\tau)G_{1}(\tau ,s)\,d\tau}{ \int_{0}^{1}(M_{2}-M_{1}+1-N_{2}+N_{1})g_{1}(\tau) G_{1}(\tau,s)\,d\tau} \\ =& \frac{ (M_{2}-\frac{1}{2}N_{2}) \int_{0}^{1}g_{1}(\tau)G_{1}(\tau ,s)\,d\tau}{ (1-M_{1}+M_{2}+N_{1}-N_{2})\int_{0}^{1}g_{1}(\tau) G_{1}(\tau,s)\,d\tau} \\ =& \frac{ M_{2}-\frac{1}{2}N_{2}}{ 1-M_{1}+M_{2}+N_{1}-N_{2}} \\ =&\rho. \end{aligned}$$

Obviously, \(\rho>0 \) and

$$\rho- \frac{1}{2}=\frac{ M_{2}-\frac{1}{2}N_{2}}{ 1-M_{1}+M_{2}+N_{1}-N_{2}}- \frac{1}{2}= \frac{ M_{2}- 1+M_{1}-N_{1}}{ 2 (1-M_{1}+M_{2}+N_{1}-N_{2} ) }< 0, $$

so that \(0<\rho<\frac{1}{2}\) and (17) hold.

Finally, from (16) and (17), we can easily show that the following results hold:

$$\begin{aligned} \max_{t\in[0,1] }G(t,s) =&\max_{t\in[0,1] } \bigl( G_{1}(t,s)+G_{2}(t,s) \bigr) \\ \leq& \max_{t\in[0,1] }G_{1}(t,s)+\max _{t\in[0,1] }G_{2}(t,s) \\ = & G_{1}(0,s)+G_{2}(1,s) \end{aligned}$$

and

$$\begin{aligned} \min_{t\in[0,\eta]}G(t,s) \geq&\min _{t\in[0,\eta]}G_{1}(t,s)+\min_{t\in[0,\eta]}G_{2}(t,s) \\ \geq& \frac{1}{2}G_{1}(0,s)+\rho G_{2}(1,s) \\ >&\rho G_{1}(0,s)+\rho G_{2}(1,s) \\ =&\rho \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr) \\ \geq&\rho\max_{t\in[0,1] }G(t,s). \end{aligned}$$

 □

Lemma 2.7

Assume (L0) hold, then the function \(\omega(t)\)satisfies the following properties:

  1. (1)

    \(\omega(t)\leq\omega(1)=\max_{t\in[0,1]}\omega(t)\);

  2. (2)

    \(\min_{t\in[0,\eta]}\omega(t)\geq\rho\max_{t\in [0,1]}\omega(t)\), whereρis given by (15).

Proof

From Lemma 2.5 and (3), we have

$$ \begin{gathered}\min_{t\in[0,\eta]}\omega(t)=\omega(0)= \delta \bigl(b(M_{2}-1)+a(2-N_{2}) \bigr), \\ \max_{t\in[0,1]}\omega(t)=\omega(1)=\delta \bigl(b(M_{2}-M_{1})+a(1-N_{2}+N_{1}) \bigr), \end{gathered} $$

and

$$\begin{aligned} \frac{\omega(0)}{\omega(1)} =& \frac{\delta (b(M_{2}-1)+a(2-N_{2}))}{\delta(b(M_{2}-M_{1})+a(1-N_{2}+N_{1}))} \\ \geq& \frac{b(M_{2}-1)+\frac {1}{2}b(2-N_{2})}{b(M_{2}-M_{1})+b(2-N_{2}-1+N_{1})} \\ =& \frac{ M_{2}-\frac{1}{2}N_{2}}{ 1-M_{1}+M_{2}+N_{1}-N_{2}} \\ =& \rho. \end{aligned}$$

Hence,

$$ \omega(t)\leq\omega(1)=\max_{t\in[0,1]}\omega(t),\qquad \min _{t\in[0,\eta]}\omega(t)\geq\rho\max_{t\in[0,1]}\omega(t). $$

 □

To finish this section, we present the well-known Avery–Peterson fixed point theorem as follows.

Let γ and θ be nonnegative continuous convex functionals on P, φ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. For \(A,B,C,D>0\), we define the following convex set:

$$\begin{gathered} P(\gamma; D)= \bigl\{ x\in P: \gamma(x)< D \bigr\} , \\ P(\gamma, \varphi; B, D) = \bigl\{ x\in P: B\leq\varphi(x),\gamma (x)\leq D \bigr\} , \\ P(\gamma, \theta, \varphi; B, C, D) = \bigl\{ x\in P: B\leq\varphi (x),\theta(x) \leq C, \gamma(x)\leq D \bigr\} ,\end{gathered} $$

and a closed set

$$ P(\gamma, \psi; A,D) = \bigl\{ x\in P: A\leq\psi(x),\gamma(x)\leq D \bigr\} . $$

Lemma 2.8

(see [27])

LetPbe a cone in a real Banach spaceE. Letγandθbe nonnegative continuous convex functionals onP, φbe a nonnegative continuous concave functional onP, andψbe a nonnegative continuous functional onPsatisfying \(\psi(\lambda x)\leq\lambda\psi(x)\)for \(0\leq\lambda\leq1\), such that, for some positive numbersMandD, \(\varphi(x)\leq\psi(x)\), \(\Vert x \Vert\leq M\gamma(x)\)for all \(x\in\overline{P(\gamma;D)}\). Suppose

$$T:\overline{P(\gamma;D)}\rightarrow\overline{P(\gamma;D)} $$

is completely continuous and there exist positive numbersA, B, andCwith \(A < B\)such that

\((H1)\):

\(\{x\in P(\gamma,\theta,\varphi;B,C,D):\varphi(x)>B\} \neq{\O}\), and \(\varphi(x)>B \)for \(x\in P(\gamma,\theta,\varphi;B,C,D)\);

\((H2)\):

\(\varphi(Tx)>B\)for \(x\in P(\gamma,\varphi;B,D)\)with \(\theta(Tx)>A\);

\((H3)\):

\(0\notin P(\gamma,\psi;A,D)\)and \(\psi(Tx)< A\)for \(x\in P(\gamma,\psi;A,D)\)with \(\psi(x)=A\).

Then T has at least three fixed point \(x_{1},x_{2},x_{3}\in\overline {P(\gamma;D)}\) such that

$$ \gamma(x_{i})\leq D,\quad i=1,2,3; \qquad\varphi(x_{1})>B, \qquad A< \varphi(x_{2}),\qquad \psi(x_{2})< B; \qquad \psi(x_{3})< A. $$

Main results

In this section, we prove the existence of positive solution of BVP (1) by applying the following Avery–Peterson fixed point theorem.

We consider the Banach space \(E=\{x\in C[0,1]:D_{0+}^{\beta}x\in C[0,1]\}\) with the norm

$$ \Vert x \Vert =\max \Bigl\{ \max_{t\in[0,1]} \bigl\vert x(t) \bigr\vert ,\max_{t\in[0,1]} \bigl\vert D_{0+}^{\beta}x(t) \bigr\vert \Bigr\} . $$

Let

$$ P= \Bigl\{ x\in E: x(t)\geq0, D_{0+}^{\beta}x(t) \geq0, \min _{t\in[0,\eta]}x(t) \geq\rho\max_{t\in[0,1]} x(t) \Bigr\} , $$

then P is a cone in E.

Define the operator \(T:P \rightarrow E\) by

$$ Tx(t):= \int_{0}^{1}G(t,s)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds+\omega(t). $$

Lemma 3.1

Assume (L0) hold, then \(T:P \rightarrow P\)is a completely continuous operator.

Proof

For \(x \in P\), it is easy to see that T is continuous operator and \(Tx(t)\geq0\). By (14), we have

$$D_{0+}^{\beta}Tx(t)=\frac{1}{\varGamma(\alpha-\beta)} \int _{0}^{t}(t-s)^{\alpha-\beta-1} \varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau ) \bigr)\,d\tau\biggr)\,ds \geq0. $$

From Lemma 2.5 and Lemma 2.6 and Lemma 2.7, similar to Lemma 3.1 in [19], we can easily prove that T is a completely continuous operator. □

Define continuous nonnegative convex functionals as

$$ \gamma(x)= \Vert x \Vert ,\qquad \theta(x)=\psi(x)=\max_{t \in[0,1]} \bigl\vert x(t) \bigr\vert . $$

Define continuous nonnegative concave functionals as

$$ \varphi(x)=\min_{t \in[0,\eta]} \bigl\vert x(t) \bigr\vert . $$

Thus

$$ \rho\theta(x) \leq\varphi(x)\leq\theta(x)=\psi(x),\qquad \Vert x \Vert \leq M \gamma(x), $$

where \(M=1\).

Let

$$\begin{gathered} J_{1}= \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds , \\ J_{2}=\frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{1}(1-s)^{\alpha-\beta-1} \varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds ,\end{gathered} $$

and

$$ J_{3}= \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{\eta}H(s,\tau)\,d\tau \biggr)\,ds. $$

Theorem 3.1

Suppose (L0) hold, and there exist constants \(A, B, D \geq\omega (1) \)with \(A < B <\rho D \min\{\frac{J_{3}}{J_{1}},\frac {J_{3}}{J_{2}}\} \)and \(C=\frac{B}{\rho}\), such that

\((L_{1})\):

\(f(t,x,y) \leq\min \{\varphi_{p}(\frac{D-\omega (1)}{J_{1}}), \varphi_{p}(\frac{D}{J_{2}}) \}\), \((t,x,y)\in[0,1] \times [0,D] \times[0,D]\);

\((L_{2})\):

\(f(t,x,y) > \varphi_{p}(\frac{B-\rho\omega(1)}{\rho J_{3}})\), \((t,x,y)\in[0,\eta]\times[B,\frac{B}{\rho}]\times[0,D]\);

\((L_{3})\):

\(f(t,x,y) < \varphi_{p}(\frac{A-\omega(1)}{J_{1}})\), \((t,x,y)\in[0,1]\times[0,A]\times[0,D]\).

Then BVP (1) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\), satisfying

$$\begin{aligned}& \Vert x_{i} \Vert \leq D\quad (i=1,2,3), \end{aligned}$$
(18)
$$\begin{aligned}& \min_{t\in[0,\eta]} \bigl\vert x_{1}(t) \bigr\vert > B,\qquad A< \min_{t\in[0,\eta]} \bigl\vert x_{2}(t) \bigr\vert , \qquad\max_{t\in[0,1]} \bigl\vert x_{2}(t) \bigr\vert < B, \qquad\max_{t\in[0,1]} \bigl\vert x_{3}(t) \bigr\vert < A. \end{aligned}$$
(19)

Proof

Obviously, the function x is a positive solution of BVP (1) if and only if x is a fixed point of the operator T in P.

For \(x \in\overline{P(\gamma;D)}\), we get

$$ \max_{t \in[0,1]} \bigl\vert x(t) \bigr\vert \leq D,\qquad \max _{t \in[0,1]} \bigl\vert D_{0+}^{\beta}x(t) \bigr\vert \leq D, $$

this implies

$$ 0 \leq x(t),\qquad D_{0+}^{\beta}x(t) \leq D. $$

From \((L_{1})\), we get

$$\begin{aligned} \max_{t \in[0,1]} \bigl\vert Tx(t) \bigr\vert \leq& \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds+\omega(1) \\ \leq& \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl(\varphi_{p} \biggl( \frac{D-\omega(1)}{J_{1}} \biggr) \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds+\omega(1) \\ =& \frac{D-\omega(1)}{J_{1}} \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds+\omega(1) \\ =& D \end{aligned}$$

and

$$\begin{aligned}& \max_{t \in[0,1]} \bigl\vert D_{0+}^{\beta}Tx(t) \bigr\vert \\& \quad= \max_{t \in[0,1]} \biggl\vert \frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{t}(t-s)^{\alpha-\beta-1} \varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds \biggr\vert \\& \quad\leq \frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{1}(1-s)^{\alpha-\beta-1} \varphi_{q} \biggl(\varphi_{p} \biggl(\frac{D}{J_{2}} \biggr) \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds \\& \quad= \frac{D}{J_{2}} \frac{1}{\varGamma(\alpha-\beta)} \int_{0}^{1}(1-s)^{\alpha-\beta-1} \varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds \\& \quad= D, \end{aligned}$$

so that

$$ \gamma(Tx)= \Vert Tx \Vert = \max \Bigl\{ \max_{t \in[0,1]} \bigl\vert Tx(t) \bigr\vert , \max_{t \in[0,1]} \bigl\vert D_{0+}^{\beta}Tx(t) \bigr\vert \Bigr\} \leq D. $$

Therefore \(T:\overline{P(\gamma;D)} \rightarrow\overline{P(\gamma;D)}\).

From \(\frac{B}{\rho} \in P(\gamma,\theta,\varphi;B,C,D)\) and \(\varphi (\frac{B}{\rho}) > B \), we have

$$ \bigl\{ x \in P(\gamma,\theta,\varphi;B,C,D):\varphi(x) > B \bigr\} \neq {\O}. $$

For \(x \in P(\gamma,\theta,\varphi;B,C,D)\), we know that \(B \leq x(t) \leq C=\frac{B}{\rho} \) for \(t\in[0,\eta]\) and \(0 \leq{ D_{0+}^{\beta}x(t) \leq D}\).

By \((L_{2})\),

$$\begin{aligned} \varphi(Tx) =& \min_{t \in[0,\eta]} \bigl\vert Tx(t) \bigr\vert \\ \geq& \int_{0}^{1} \rho \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{\eta}H(s,\tau) \varphi_{p} \biggl( \frac{B-\rho\omega(1)}{\rho J_{3}} \biggr)\,d\tau \biggr)\,ds+\rho\omega (1) \\ =& \rho\frac{B-\rho\omega(1)}{\rho J_{3}} \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{\eta}H(s,\tau)\,d\tau \biggr)\,ds+\rho \omega(1) \\ =& B. \end{aligned}$$

So \(\varphi(Tx)>B\) for all \(x \in P(\gamma,\theta,\varphi;B,C,D)\). Hence, the condition \((H1)\) of Lemma 2.8 is satisfied.

For all \(x \in P(\gamma,\varphi;B,D) \) with \(\theta(Tx)>C=\frac{B}{\rho }\), we have

$$ \varphi(Tx) \geq\rho\theta(Tx) > \rho C= \rho\frac{B}{\rho} =B. $$

Thus, the condition \((H2)\) of Lemma 2.8 holds.

Because of \(\psi(0) = 0 < A \), then \(0 \notin P(\gamma,\psi;A,D)\). For \(x \in P(\gamma,\psi;A,D)\) with \(\psi(x)=A\), we know \(\gamma(x) \leq D \). It means that \(\max_{t \in[0,1]}x(t)=A\) and \(0 \leq D_{0+}^{\beta }x(t) \leq D\).

From \((L_{3})\), we can obtain

$$\begin{aligned} \psi(Tx) =& \max_{t \in[0,1]} \bigl\vert Tx(t) \bigr\vert \\ \leq& \max_{t \in[0,1]} \int_{0}^{1}G(t,s)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)f \bigl(\tau,x( \tau),D_{0+}^{\beta}x(\tau) \bigr)\,d\tau \biggr)\,ds+\omega(1) \\ < & \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\varphi_{p} \biggl( \frac{A-\omega(1)}{J_{1}} \biggr)\,d\tau \biggr)\,ds+\omega(1) \\ =& \frac{A-\omega(1)}{J_{1}} \int_{0}^{1} \bigl(G_{1}(0,s)+G_{2}(1,s) \bigr)\varphi_{q} \biggl( \int_{0}^{1}H(s,\tau)\,d\tau \biggr)\,ds+\omega(1) \\ =& A. \end{aligned}$$

Therefore, the condition \((H3)\) of Lemma 2.8 holds.

To sum up, the conditions of Lemma 2.8 are all verified and we notice that \(x_{i}(t)\geq{\omega(0)>0}\). Hence, BVP (1) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\) satisfying (18) and (19). □

Example

Consider the following boundary value problem:

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} D_{0+}^{\frac{7}{3}}\varphi_{\frac{3}{2}} ( D_{0+}^{\frac{11}{3}}x(t) ) =f (t,x(t),D_{0+} ^{\frac{7}{3}}x(t) ),\quad t \in(0,1),\\ ( \varphi_{\frac{3}{2}} ( D_{0+}^{\frac{11}{3}}x(0) ) )^{\prime} =\varphi_{\frac{3}{2}} ( D_{0+}^{\frac{11}{3}}x(1) )=0, \\ x(0)+x^{\prime}(0)= \int_{0}^{1}sx(s)\,ds+\frac{5}{3}, \\ x(1)+x^{\prime}(1) = \int_{0}^{1}(s^{2}+s)x(s)\,ds+\frac{7}{3}, \\ x^{\prime\prime}(0)=x^{(3)} = 0 , \end{array}\displaystyle \right . $$
(20)

where \(\alpha=\frac{11}{3}\), \(\beta=\frac{7}{3}\), \(p=\frac{3}{2}\), \(g_{0}(t)=t\), \(g_{1}(t)=t^{2}+t\), and

$$\begin{aligned} f(t,x,y) =& \left \{ \textstyle\begin{array}{l@{\quad}l} \tan(\frac{t}{100})+2^{x/10} x^{2}+\cos(y),& 0\leq x\leq65,\\ \tan(\frac{t}{100})+\cos(y)+160 \sqrt{5},&65< x\leq10{,}000. \end{array}\displaystyle \right . \end{aligned}$$

Choose \(A=3\), \(B=65\), \(D=25{,}000\), \(\eta=\frac{1}{4}\). By simple computation, we have

$$ \begin{gathered} \rho= 0.0384615,\qquad M=1,\qquad\omega(1)=2.83721, \\ M_{1} = \frac{1}{2},\qquad M_{2} = \frac{1}{3},\qquad N_{1} = \frac{5}{6},\qquad N_{2} =\frac{7}{12},\qquad\delta= \frac{72}{43}, \\ J_{1} =0.147109,\qquad J_{2} = 0.0722385,\qquad J_{3}=0.0310138. \end{gathered} $$

We can check that the nonlinear term \(f(t,x,y)\) satisfies

\((L_{1})\):

\(\max f(t,x,y)\approx383.413 \leq\min \{\varphi _{p}(\frac{D-\omega(1)}{J_{1}}), \varphi_{p}(\frac{D}{J_{2}}) \}\approx 412.216\), \((t,x,y)\in[0,1] \times [0,25{,}000] \times[0,25{,}000]\);

\((L_{2})\):

\(\min f(t,x,y) \approx381.413 > \varphi_{p}(\frac {B-\rho\omega(1)}{\rho J_{3}})\approx228.283\), \((t,x,y)\in[0,\frac {1}{3}]\times[65,\frac{B}{\rho}]\times [0,25{,}000]\);

\((L_{3})\):

\(\max f(t,x,y) \approx1.02108 < \varphi_{p}(\frac {A-\omega(1)}{J_{1}})\approx1.05195\), \((t,x,y)\in[0,1]\times[0,3]\times [0,25{,}000]\).

Thus, from Theorem 3.1, we know that BVP (20) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\), satisfying

$$\begin{gathered} \Vert x_{i} \Vert \leq25{,}000 \quad(i=1,2,3), \\ \min_{t\in[0,\eta]} \vert x_{1} \vert > 65,\qquad 3< \min _{t\in[0,\eta]} \vert x_{2} \vert ,\qquad \max _{t\in[0,1]} \vert x_{2} \vert < 65,\qquad \max _{t\in[0,1]} \vert x_{3} \vert < 3.\end{gathered} $$

References

  1. 1.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

  2. 2.

    Diethelm, K.: The Analysis of Fractional Differential Equations. Lectures Notes in Mathematics. Springer, Berlin (2010)

  3. 3.

    Bai, Z., Zhang, Y.: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 218(5), 1719–1725 (2011)

  4. 4.

    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

  5. 5.

    Jia, M., Liu, X.: Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions. Appl. Math. Comput. 232, 313–323 (2014)

  6. 6.

    Jia, M., Liu, X.: The existence of positive solutions for fractional differential equations with integral and disturbance parameter in boundary conditions. Abstr. Appl. Anal. 2014, Article ID 131548 (2014)

  7. 7.

    Shen, C., Zhou, H., Yang, L.: Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter. Math. Methods Appl. Sci. 39, 4504–4511 (2016)

  8. 8.

    Li, X., Liu, X., Jia, M., Zhang, L.: The positive solutions of infinite-point boundary value problem of fractional differential equations on the infinite interval. Adv. Differ. Equ. 2017, Article ID 126 (2017)

  9. 9.

    Liu, L., Li, H., Liu, C., Wu, Y.: Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions. J. Nonlinear Sci. Appl. 10(1), 243–262 (2017)

  10. 10.

    Song, Q., Bai, Z.: Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, Article ID 183 (2018)

  11. 11.

    Wei, Y., Song, Q., Bai, Z.: Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 87, 101–107 (2019)

  12. 12.

    Liu, X., Jia, M.: Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives. Appl. Math. Comput. 353, 230–242 (2019)

  13. 13.

    Leibenson, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSR 9, 7–10 (1983) (in Russian)

  14. 14.

    Liu, X., Jia, M., Ge, W.: Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013, Article ID 126 (2013)

  15. 15.

    Liang, S., Zhang, J.: Existence and uniqueness of positive solutions for integral boundary problems of nonlinear fractional differential equations with p-Laplacian operator. Rocky Mt. J. Math. 44(3), 953–974 (2014)

  16. 16.

    Mahmudov, N.I., Unul, S.: Existence of solutions of fractional boundary value problems with p-Laplacian. Bound. Value Probl. 2015, Article ID 99 (2015)

  17. 17.

    Han, Z., Lu, H., Zhang, C.: Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian. Appl. Math. Comput. 257, 526–536 (2015)

  18. 18.

    Liu, X., Jia, M., Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem whit p-Laplacian operator. Appl. Math. Lett. 65, 56–62 (2017)

  19. 19.

    Zhang, L., Zhang, W., Liu, X., Jia, M.: Existence of positive solutions for integral boundary value problems of fractional differential equations with p-Laplacian. Adv. Differ. Equ. 2017, Article ID 36 (2017)

  20. 20.

    Liu, X., Jia, M.: The positive solutions for integral boundary value problem of fractional p-Laplacian equation with mixed derivatives. Mediterr. J. Math. 14(2), 1–13 (2017)

  21. 21.

    Dong, X., Bai, Z., Zhang, S.: Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Probl. 2017, Article ID 5 (2017)

  22. 22.

    Sheng, K., Zhang, W., Bai, Z.: Positive solutions to fractional boundary-value problems with p-Laplacian on time scales. Bound. Value Probl. 2018, Article ID 70 (2018)

  23. 23.

    Liu, X., Jia, M.: Method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian. Adv. Differ. Equ. 2018, Article ID 28 (2018)

  24. 24.

    Ahmad, M., Zada, A., Alzabut, J.: Stability of a coupled implicit switched nonlinear fractional differential system with p-Laplacian. Adv. Differ. Equ. 2019, Article ID 436 (2019)

  25. 25.

    Wang, X., Liu, X., Deng, X.: Existence and nonexistence of positive solutions for fractional integral boundary value problem with two disturbance parameters. Bound. Value Probl. 2015, Article ID 186 (2015)

  26. 26.

    Hao, X., Zhang, L., Liu, L.: Positive solutions of higher order fractional integral boundary value problem with a parameter. Nonlinear Anal., Model. Control 24(2), 210–223 (2019)

  27. 27.

    Avery, R.I., Peteson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001)

Download references

Acknowledgements

The author would like to thank the anonymous referees and the editor for their constructive suggestions on improving the presentation of the paper.

Availability of data and materials

Not applicable.

Funding

This research was supported by National Natural Science Foundation of China (Grant No. 11471215).

Author information

All authors jointly worked on the results and they read and approved the final manuscript.

Correspondence to Weiguo Zhang.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Zhang, W., Liu, X. et al. Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters. J Inequal Appl 2020, 2 (2020). https://doi.org/10.1186/s13660-019-2273-6

Download citation

Keywords

  • Caputo fractional derivative
  • Positive solutions
  • p-Laplacian operator
  • Integral boundary conditions
  • Parameter