Open Access

Solvability of fractional boundary value problems with p-Laplacian operator

Advances in Difference Equations20152015:352

https://doi.org/10.1186/s13662-015-0648-7

Received: 20 July 2015

Accepted: 27 September 2015

Published: 17 November 2015

Abstract

This paper studies the existence of solutions for two boundary value problems for the fractional p-Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, two new existence results are obtained by using Schaefer’s fixed point theorem.

Keywords

fractional differential equation p-Laplacian operator boundary value problem Schaefer’s fixed point theorem

MSC

34A08 34B15

1 Introduction

The fractional calculus is a generalization of the ordinary differentiation and integration on an arbitrary order that can be noninteger. In the last two decades, the theory of fractional calculus has gained importance and popularity due to its wide range of applications in varied fields of sciences and engineering. In [18], the applications are mentioned to fluid flow, rheology, dynamical processes in self-similar and porous structures, electrical networks, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, chemical physics, optics and signal processing, and so on. Recently, many important results about the fractional differential equations have been given. For example, for fractional initial value problems, the existence and multiplicity of solutions were discussed in [4, 911]. In addition, for fractional boundary value problems (BVPs for short), Agarwal et al. (see [12]) considered a two-point BVP at nonresonance, and Bai (see [13]) considered an m-point BVP at resonance. For more papers on the fractional BVPs, see [1422] and the references therein.

The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [23]) introduced the p-Laplacian equation as follows:
$$\begin{aligned} \bigl(\phi_{p} \bigl(x'(t)\bigr) \bigr)'=f\bigl(t,x(t),x'(t)\bigr), \end{aligned}$$
(1.1)
where \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\). Obviously, \(\phi_{p}\) is invertible and its inverse operator is \(\phi_{q}\), where \(q>1\) is a constant such that \(1/p+1/q=1\). In the past few decades, many important results about equation (1.1) with certain boundary value conditions have been obtained. We refer the readers to [2431] and the references cited therein.
Motivated by the works mentioned previously, in this paper, we investigate the existence of solutions for the fractional p-Laplacian equation of the form
$$\begin{aligned} D_{0^{+}}^{\beta}\phi_{p} \bigl(D_{0^{+}}^{\alpha}x(t)\bigr)=f\bigl(t,x(t),D_{0^{+}}^{\alpha}x(t)\bigr),\quad t\in[0,1] \end{aligned}$$
(1.2)
subject to the boundary value conditions either
$$\begin{aligned} D_{0^{+}}^{\alpha}x(0)=x(1)=0 \end{aligned}$$
(1.3)
or
$$\begin{aligned} x(0)=x(1)=0, \end{aligned}$$
(1.4)
where \(0<\alpha,\beta\leq1\), \(D_{0^{+}}^{\alpha}\) is a Caputo fractional derivative, and \(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is continuous. Note that the nonlinear operator \(D_{0^{+}}^{\beta}\phi _{p}(D_{0^{+}}^{\alpha})\) reduces to the linear operator \(D_{0^{+}}^{\beta}D_{0^{+}}^{\alpha}\) when \(p=2\) and the additive index law
$$\begin{aligned} D_{0^{+}}^{\beta}D_{0^{+}}^{\alpha}u(t)=D_{0^{+}}^{\alpha+\beta}u(t) \end{aligned}$$
holds under some reasonable constraints on the function u (see [32]).

The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, based on Schaefer’s fixed point theorem, we establish two theorems on the existence of solutions for BVP (1.2) (1.3) (Theorem 3.1) and BVP (1.2) (1.4) (Theorem 3.2). Finally, in Section 4, an explicit example is given to illustrate the main results.

2 Preliminaries

For convenience of the readers, we present here some necessary basic knowledge and definitions as regards the fractional calculus theory, which can be found, for instance, in [33, 34].

Definition 2.1

The Riemann-Liouville fractional integral operator of order \(\alpha>0\) of a function \(u:(0,+\infty )\rightarrow\mathbb{R}\) is given by
$$\begin{aligned} I_{0^{+}}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t} (t-s)^{\alpha-1}u(s)\,ds, \end{aligned}$$
provided that the right-hand side integral is pointwise defined on \((0,+\infty)\).

Definition 2.2

The Caputo fractional derivative of order \(\alpha>0\) of a continuous function \(u:(0,+\infty)\rightarrow \mathbb{R}\) is given by
$$\begin{aligned} D_{0^{+}}^{\alpha}u(t) =&I_{0^{+}}^{n-\alpha} \frac{d^{n}u(t)}{d t^{n}}\\ =&\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-s)^{n-\alpha-1}u^{(n)}(s)\,ds, \end{aligned}$$
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on \((0,+\infty)\).

Lemma 2.1

(see [8])

Let \(\alpha>0\). Assume that \(u,D_{0^{+}}^{\alpha}u\in L([0,1],\mathbb{R})\). Then the following equality holds:
$$\begin{aligned} I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}u(t)=u(t)+c_{0}+c_{1}t+ \cdots+c_{n-1}t^{n-1}, \end{aligned}$$
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), and n is the smallest integer greater than or equal to α.

Lemma 2.2

For fixed \(l\in C([0,1],\mathbb{R})\), let us define
$$\begin{aligned} G_{l}(a)=\int_{0}^{1}(1-s)^{\alpha-1} \phi_{q}\bigl(a+l(s)\bigr)\,ds, \end{aligned}$$
then the equation \(G_{l}(a)=0\) has a unique solution \(\widetilde{a}(l)\), and the function \(\tilde{a}:C[0,1]\rightarrow\mathbb{R}\) is continuous and sends bounded sets into bounded sets.

Proof

The proof is similar to the proof of Proposition 2.2 in [35], so we omit the details. □

In this paper, we take \(Y=C([0,1],\mathbb{R})\) with the norm \(\|y\| _{0}=\max_{t\in[0,1]}|y(t)|\), and \(X=\{x|x,D_{0^{+}}^{\alpha}x\in Y\}\) with the norm \(\|x\|_{X}=\max\{\|x\|_{0},\|D_{0^{+}}^{\alpha}x\|_{0}\}\). By means of the linear functional analysis theory, we can prove that X is a Banach space.

3 Main results

In this section, two theorems on the existence of solutions for BVP (1.2) (1.3) and BVP (1.2) (1.4) will be given under nonlinear growth restrictions of f.

As a consequence of Lemma 2.1, we have the following results that are useful in what follows.

Lemma 3.1

Given \(h\in Y\), the unique solution of
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x(t))=h(t),\quad t\in[0,1],\\ D_{0^{+}}^{\alpha}x(0)=x(1)=0 \end{array}\displaystyle \right . \end{aligned}$$
(3.1)
is
$$\begin{aligned} x(t) =&I_{0^{+}}^{\alpha}\phi_{q}\bigl(I_{0^{+}}^{\beta}h\bigr) (t)+Ah(t)\\ =&\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)}\int_{0}^{s}(s- \tau)^{\beta-1}h(\tau)\,d\tau \biggr)\,ds+Ah(t), \end{aligned}$$
where
$$\begin{aligned} Ah(t) =&-I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}h\bigr) (1)\\ =&-\frac{1}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)}\int_{0}^{s}(s- \tau)^{\beta-1}h(\tau)\,d\tau \biggr)\,ds, \end{aligned}$$
and \(\phi_{q}\) is understood as the operator \(\phi_{q}:Y\rightarrow Y\) defined by \(\phi_{q}(x)(t)=\phi_{q}(x(t))\).

Proof

Assume that \(x(t)\) satisfies the equation of BVP (3.1), then Lemma 2.1 implies
$$\begin{aligned} \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t) \bigr)=c_{1}+I_{0^{+}}^{\beta}h(t),\quad c_{1}\in \mathbb{R}. \end{aligned}$$
From the boundary value condition \(D_{0^{+}}^{\alpha}x(0)=0\), one has \(c_{1}=0\). Thus, we have
$$\begin{aligned} x(t)=c_{2}+I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}h\bigr) (t),\quad c_{2}\in\mathbb{R}. \end{aligned}$$
By the condition \(x(1)=0\), we get
$$\begin{aligned} c_{2}=-I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}h\bigr) (1)=Ah(t). \end{aligned}$$
The proof is completed. □
Define the operator \(\mathcal{K}_{1}:X\rightarrow X\) by
$$\begin{aligned} &\mathcal{K}_{1}x(t)\\ &\quad=I_{0^{+}}^{\alpha}\phi_{q}\bigl(I_{0^{+}}^{\beta}N_{f}x\bigr) (t)+AN_{f}x(t)\\ &\quad=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)} \int_{0}^{s}(s- \tau)^{\beta-1}f\bigl(\tau,x(\tau),D_{0^{+}}^{\alpha}x(\tau) \bigr)\,d\tau \biggr)\,ds\\ &\qquad{}-\frac{1}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)} \int_{0}^{s}(s- \tau)^{\beta-1}f\bigl(\tau,x(\tau),D_{0^{+}}^{\alpha}x(\tau) \bigr)\,d\tau \biggr)\,ds, \end{aligned}$$
where \(N_{f}:X\rightarrow Y\) is the Nemytskii operator defined by
$$\begin{aligned} N_{f}x(t)=f\bigl(t,x(t),D_{0^{+}}^{\alpha}x(t)\bigr),\quad \forall t\in[0,1]. \end{aligned}$$
(3.2)
Clearly, the fixed points of \(\mathcal{K}_{1}\) are the solutions of BVP (1.2) (1.3).

Lemma 3.2

Given \(h\in Y\), the unique solution of
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x(t))=h(t), \quad t\in[0,1],\\ x(0)=x(1)=0 \end{array}\displaystyle \right . \end{aligned}$$
(3.3)
is
$$\begin{aligned} x(t) =&I_{0^{+}}^{\alpha}\phi_{q}\bigl(I_{0^{+}}^{\beta}h+Bh\bigr) (t)\\ =&\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)}\int_{0}^{s}(s- \tau)^{\beta-1}h(\tau)\,d\tau +Bh(s) \biggr)\,ds, \end{aligned}$$
where
$$\begin{aligned} Bh(t) =&-I_{0^{+}}^{\beta}h\bigl(\eta(h)\bigr)\\ =&-\frac{1}{\Gamma(\beta)} \int_{0}^{\eta(h)}\bigl( \eta(h)-s\bigr)^{\beta-1}h(s)\,ds, \end{aligned}$$
and \(\eta(h)\in(0,1)\) is a unique constant dependent on \(h(t)\).

Proof

Assume that \(x(t)\) satisfies the equation of BVP (3.3), by Lemma 2.1, we get
$$\begin{aligned} x(t)=c+I_{0^{+}}^{\alpha}\phi_{q}\bigl(a+I_{0^{+}}^{\beta}h\bigr) (t),\quad a,c\in\mathbb{R}. \end{aligned}$$
From the boundary value conditions \(x(0)=x(1)=0\), one has \(c=0\) and
$$\begin{aligned} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q}\bigl(a+I_{0^{+}}^{\beta}h(s)\bigr)\,ds=0. \end{aligned}$$
(3.4)
Then, basing on Lemma 2.2, we know equation (3.4) has a unique solution \(\widetilde{a}(I_{0^{+}}^{\beta}h)\). Moreover, by the integral mean value theorem, there exists a unique constant \(\eta(h)\in(0,1)\) dependent on \(h(t)\) such that \(\phi_{q}(\widetilde{a}(I_{0^{+}}^{\beta}h)+I_{0^{+}}^{\beta}h)(\eta(h))=0\), which implies \((\widetilde {a}(I_{0^{+}}^{\beta}h)+I_{0^{+}}^{\beta}h)(\eta(h))=0\). Thus, we have
$$\begin{aligned} \widetilde{a}\bigl(I_{0^{+}}^{\beta}h\bigr)=-I_{0^{+}}^{\beta}h\bigl(\eta(h)\bigr)=Bh(t). \end{aligned}$$
The proof is completed. □
Define the operator \(\mathcal{K}_{2}:X\rightarrow X\) by
$$\begin{aligned} &\mathcal{K}_{2}x(t)\\ &\quad=I_{0^{+}}^{\alpha}\phi_{q}\bigl(I_{0^{+}}^{\beta}N_{f}x+BN_{f}x\bigr) (t)\\ &\quad=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \phi_{q} \biggl(\frac{1}{\Gamma(\beta)}\int_{0}^{s}(s- \tau)^{\beta-1}f\bigl(\tau,x(\tau ),D_{0^{+}}^{\alpha}x(\tau) \bigr)\,d\tau\\ &\qquad{} -\frac{1}{\Gamma(\beta)} \int_{0}^{\eta(x)}\bigl( \eta(x)-\tau\bigr)^{\beta-1}f\bigl(\tau,x(\tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)\,d\tau \biggr)\,ds, \end{aligned}$$
where \(\eta(x)\in(0,1)\) is a unique constant dependent on \(x(t)\) and \(N_{f}\) is the Nemytskii operator defined by (3.2). Clearly, the fixed points of \(\mathcal{K}_{2}\) are the solutions of BVP (1.2) (1.4).

Our first result, based on Schaefer’s fixed point theorem and Lemma 3.1, is stated as follows.

Theorem 3.1

Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous. Assume that
  1. (H)
    there exist nonnegative functions \(a,b,c\in Y\) such that
    $$\begin{aligned} \bigl|f(t,u,v)\bigr|\leq a(t)+b(t)|u|^{p-1}+c(t)|v|^{p-1},\quad \forall t\in [0,1],(u,v)\in\mathbb{R}^{2}. \end{aligned}$$
     
Then BVP (1.2) (1.3) has at least one solution provided that
$$\begin{aligned} \gamma_{1}:=\frac{1}{\Gamma(\beta+1)} \biggl[\frac{2^{p-1}\|b\|_{0}}{(\Gamma(\alpha+1))^{p-1}}+ \|c\|_{0} \biggr]< 1. \end{aligned}$$
(3.5)

Proof

The proof will be given in the following two steps.

Step 1: \(\mathcal{K}_{1}:X\rightarrow X\) is completely continuous.

By the definition of \(\mathcal{K}_{1}\), we obtain
$$\begin{aligned} D_{0^{+}}^{\alpha}\mathcal{K}_{1}x(t)=\phi_{q} \bigl(I_{0^{+}}^{\beta}N_{f}x\bigr) (t). \end{aligned}$$
Obviously, the operators \(\mathcal{K}_{1}\) and \(D_{0^{+}}^{\alpha}\mathcal {K}_{1}\) are compositions of the continuous operators. So \(\mathcal{K}_{1}\) and \(D_{0^{+}}^{\alpha}\mathcal{K}_{1}\) are continuous in Y. Hence, \(\mathcal{K}_{1}\) is a continuous operator.

Let \(\Omega\subset X\) be an open bounded set, then \(\mathcal {K}_{1}(\overline{\Omega})\) and \(D_{0^{+}}^{\alpha}\mathcal{K}_{1}(\overline {\Omega})\) are bounded. Moreover, for \(\forall x\in\overline{\Omega }\), \(t\in[0,1]\), there exists a constant \(T>0\) such that \(|I_{0^{+}}^{\beta}N_{f}x(t)|\leq T\). Thus, in view of the Arzelà-Ascoli theorem, we need only to prove that \(\mathcal{K}_{1}(\overline{\Omega})\subset X\) is equicontinuous.

For \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline{\Omega}\), we have
$$\begin{aligned} &\bigl|\mathcal{K}_{1}x(t_{2})-\mathcal{K}_{1}x(t_{1})\bigr|\\ &\quad=\frac{1}{\Gamma(\alpha)} \biggl\vert \int_{0}^{t_{2}}(t_{2}-s)^{\alpha-1} \phi_{q}\bigl(I_{0^{+}}^{\beta}N_{f}x(s) \bigr)\,ds -\int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1} \phi_{q}\bigl(I_{0^{+}}^{\beta}N_{f}x(s) \bigr)\,ds\biggr\vert \\ &\quad\leq\frac{T^{q-1}}{\Gamma(\alpha)} \biggl\{ \int_{0}^{t_{1}} \bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1}\bigr]\,ds +\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}\,ds \biggr\} \\ &\quad=\frac{T^{q-1}}{\Gamma(\alpha+1)}\bigl[t_{1}^{\alpha}-t_{2}^{\alpha} +2(t_{2}-t_{1})^{\alpha}\bigr]. \end{aligned}$$
Since \(t^{\alpha}\) is uniformly continuous in \([0,1]\), we can obtain that \(\mathcal{K}_{1}(\overline{\Omega})\subset Y\) is equicontinuous. A similar proof can show that \(I_{0^{+}}^{\beta}N_{f}(\overline{\Omega })\subset Y\) is equicontinuous. This, together with the uniform continuity of \(\phi_{q}(s)\) on \([-T,T]\), yields that \(D_{0^{+}}^{\alpha }\mathcal{K}_{1}(\overline{\Omega})\subset Y\) is also equicontinuous. Thus, we find that \(\mathcal{K}_{1}:X\rightarrow X\) is compact.

Step 2: A priori bounds.

Set
$$\begin{aligned} \Omega_{1}=\bigl\{ x\in X|x=\lambda^{q-1}\mathcal{K}_{1}x, \lambda\in(0,1)\bigr\} . \end{aligned}$$
Now it remains to show that the set \(\Omega_{1}\) is bounded.
By Lemma 2.1 and the boundary value condition \(x(1)=0\), one has
$$\begin{aligned} x(t) =I_{0+}^{\alpha}D_{0^{+}}^{\alpha}x(t)-I_{0+}^{\alpha}D_{0^{+}}^{\alpha}x(1), \end{aligned}$$
which together with
$$\begin{aligned} \bigl|I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}x(t)\bigr| =& \frac{1}{\Gamma(\alpha)}\biggl\vert \int_{0}^{t}(t-s)^{\alpha-1}D_{0^{+}}^{\alpha}x(s)\,ds\biggr\vert \\ \leq&\frac{1}{\Gamma(\alpha)}\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}\cdot \frac{1}{\alpha }t^{\alpha} \\ \leq&\frac{1}{\Gamma(\alpha+1)}\bigl\| D_{0^{+}}^{\alpha}x \bigr\| _{0},\quad \forall t\in[0,1] \end{aligned}$$
(3.6)
yields
$$\begin{aligned} \|x\|_{0}\leq\frac{2}{\Gamma(\alpha+1)}\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}. \end{aligned}$$
(3.7)
For \(x\in\Omega_{1}\), we have
$$\begin{aligned}[b] \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr) &=\lambda I_{0^{+}}^{\beta}N_{f}x(t)\\ &=\frac{\lambda}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta -1}f \bigl(s,x(s),D_{0^{+}}^{\alpha}x(s)\bigr)\,ds. \end{aligned} $$
So, from (H), we obtain
$$\begin{aligned} &\bigl|\phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr)\bigr|\\ &\quad\leq\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1} \bigl(a(s)+b(s)\bigl|x(s)\bigr|^{p-1} +c(s)\bigl|D_{0^{+}}^{\alpha}x(s)\bigr|^{p-1}\bigr)\,ds\\ &\quad\leq\frac{1}{\Gamma(\beta)}\bigl(\|a\|_{0}+\|b\|_{0}\|x \|_{0}^{p-1} +\|c\|_{0}\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1}\bigr)\cdot\frac{1}{\beta}t^{\beta}\\ &\quad\leq\frac{1}{\Gamma(\beta+1)}\bigl(\|a\|_{0}+\|b\|_{0}\|x \|_{0}^{p-1} +\|c\|_{0}\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1}\bigr), \quad\forall t\in[0,1], \end{aligned}$$
which, together with \(|\phi_{p}(D_{0^{+}}^{\alpha}x(t))|=|D_{0^{+}}^{\alpha}x(t)|^{p-1}\) and (3.7), yields
$$\begin{aligned} \bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1} \leq \frac{1}{\Gamma(\beta+1)} \biggl[\|a\|_{0} + \biggl(\frac{2^{p-1}\|b\|_{0}}{(\Gamma(\alpha+1))^{p-1}}+\|c \|_{0} \biggr)\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1} \biggr]. \end{aligned}$$
Hence, in view of (3.5), we can get
$$\begin{aligned} \bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0} \leq \biggl[ \frac{\|a\|_{0}}{ (1-\gamma_{1})\Gamma(\beta+1)} \biggr]^{q-1}:=M_{11}. \end{aligned}$$
(3.8)
Thus, from (3.7), one has
$$\begin{aligned} \|x\|_{0}\leq\frac{2M_{11}}{\Gamma(\alpha+1)}:=M_{12}. \end{aligned}$$
(3.9)
Therefore, combining (3.8) with (3.9), we have
$$\begin{aligned} \|x\|_{X} =&\max\bigl\{ \|x\|_{0},\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}\bigr\} \\ \leq&\max\{M_{11},M_{12}\}. \end{aligned}$$

As a consequence of Schaefer’s fixed point theorem, we deduce that \(\mathcal{K}_{1}\) has a fixed point which is the solution of BVP (1.2) (1.3). The proof is completed. □

Our second result, based on Schaefer’s fixed point theorem and Lemma 3.2, is stated as follows.

Theorem 3.2

Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous. Suppose (H) holds, then BVP (1.2) (1.4) has at least one solution provided that
$$\begin{aligned} \gamma_{2}:=\frac{2}{\Gamma(\beta+1)} \biggl[ \frac{\|b\|_{0}}{(\Gamma(\alpha+1))^{p-1}}+\|c\|_{0} \biggr]< 1. \end{aligned}$$
(3.10)

Proof

The proof of complete continuity of \(\mathcal {K}_{2}\) is similar to the proof of complete continuity of \(\mathcal {K}_{1}\), so we omit the details.

Set
$$\begin{aligned} \Omega_{2}=\bigl\{ x\in X|x=\lambda^{q-1}\mathcal{K}_{2}x, \lambda\in(0,1)\bigr\} . \end{aligned}$$
From Lemma 2.1 and \(x(0)=0\), one has
$$\begin{aligned} x(t) =I_{0+}^{\alpha}D_{0^{+}}^{\alpha}x(t), \end{aligned}$$
which together with (3.6) yields
$$\begin{aligned} \|x\|_{0}\leq\frac{1}{\Gamma(\alpha+1)}\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}. \end{aligned}$$
(3.11)
Then, by the hypothesis (H) and (3.11), we obtain
$$\begin{aligned} &\bigl|I_{0^{+}}^{\beta}N_{f}x(t)\bigr| \leq\frac{1}{\Gamma(\beta+1)} \biggl[ \biggl(\frac{\|b\|_{0}}{(\Gamma(\alpha+1))^{p-1}}+\|c\|_{0} \biggr)\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1} +\|a\|_{0}\biggr], \\ &\quad\forall t\in[0,1]. \end{aligned}$$
(3.12)
For \(x\in\Omega_{2}\), we have
$$\begin{aligned} \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr)=\lambda I_{0^{+}}^{\beta}N_{f}x(t)-\lambda I_{0^{+}}^{\beta}N_{f}x\bigl(\eta(x)\bigr),\quad \eta(x)\in(0,1), \end{aligned}$$
which, together with \(|\phi_{p}(D_{0^{+}}^{\alpha}x(t))|=|D_{0^{+}}^{\alpha}x(t)|^{p-1}\) and (3.12), yields
$$\begin{aligned} \bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1} \leq \frac{2}{\Gamma(\beta+1)} \biggl[\|a\|_{0} + \biggl(\frac{\|b\|_{0}}{(\Gamma(\alpha+1))^{p-1}}+\|c \|_{0} \biggr)\bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0}^{p-1} \biggr]. \end{aligned}$$
Thus, in view of (3.10), we can get
$$\begin{aligned} \bigl\| D_{0^{+}}^{\alpha}x\bigr\| _{0} \leq \biggl[ \frac{2\|a\|_{0}}{ (1-\gamma_{2})\Gamma(\beta+1)} \biggr]^{q-1}:=M_{21}. \end{aligned}$$
(3.13)
So, from (3.11), one has
$$\begin{aligned} \|x\|_{0}\leq\frac{M_{21}}{\Gamma(\alpha+1)}:=M_{22}. \end{aligned}$$
(3.14)
Hence, combining (3.13) with (3.14), we obtain
$$\begin{aligned} \|x\|_{X}\leq\max\{M_{21},M_{22}\}. \end{aligned}$$
Therefore, \(\Omega_{2}\) is bounded.

As a consequence of Schaefer’s fixed point theorem, we deduce that \(\mathcal{K}_{2}\) has a fixed point which is the solution of BVP (1.2) (1.4). The proof is completed. □

4 An example

In this section, we give an example to illustrate our main results.

Example 4.1

Consider the following fractional p-Laplacian equation:
$$\begin{aligned} D_{0^{+}}^{\frac{3}{4}}\phi_{3} \bigl(D_{0^{+}}^{\frac{1}{2}}x(t)\bigr) =\sin t+\frac{1}{6}x^{2}(t)+t \cos\bigl(D_{0^{+}}^{\frac{1}{2}}x(t)\bigr),\quad t\in[0,1]. \end{aligned}$$
(4.1)
Corresponding to equation (1.2), we get \(p=3\), \(\alpha=1/2\), \(\beta=3/4\) and
$$\begin{aligned} f(t,u,v)=\sin t+\frac{1}{6}u^{2}+t\cos v. \end{aligned}$$
Choose \(a(t)=2\), \(b(t)=1/6\), \(c(t)=0\). By a simple calculation, we obtain \(\| b\|_{0}=1/6\), \(\|c\|_{0}=0\) and
$$\begin{aligned}& \gamma_{1}=\frac{1}{\Gamma(\frac{3}{4}+1)} \biggl[\frac{2^{2}}{ (\Gamma(\frac{1}{2}+1) )^{2}}\times \frac {1}{6}+0 \biggr]< 1,\\& \gamma_{2}=\frac{2}{\Gamma(\frac{3}{4}+1)} \biggl[\frac{1}{ (\Gamma(\frac{1}{2}+1) )^{2}}\times \frac {1}{6}+0 \biggr]< 1. \end{aligned}$$
Obviously, equation (4.1) subject to the boundary value conditions (1.3) (or (1.4)) satisfies all assumptions of Theorem 3.1 (or Theorem 3.2). Hence, BVP (4.1) (1.3) (or BVP (4.1) (1.4)) has at least one solution.

Declarations

Acknowledgements

The author is grateful for the valuable comments and suggestions of the referees.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Huaibei Normal University

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