Solvability of fractional boundary value problems with p-Laplacian operator
- Bo Zhang^{1}Email author
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 theoremMSC
34A08 34B151 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 [1–8], 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, 9–11]. 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 [14–22] and the references therein.
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
Definition 2.2
Lemma 2.1
(see [8])
Lemma 2.2
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
Proof
Lemma 3.2
Proof
Our first result, based on Schaefer’s fixed point theorem and Lemma 3.1, is stated as follows.
Theorem 3.1
- (H)there exist nonnegative functions \(a,b,c\in Y\) such that$$\begin{aligned} \bigl|f(t,u,v)\bigr|\leq a(t)+b(t)|u|^{p-1}+c(t)|v|^{p-1},\quad \forall t\in [0,1],(u,v)\in\mathbb{R}^{2}. \end{aligned}$$
Proof
The proof will be given in the following two steps.
Step 1: \(\mathcal{K}_{1}:X\rightarrow X\) is completely continuous.
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.
Step 2: A priori bounds.
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
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.
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
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
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