Abstract

In this paper, we investigate the existence of multiple positive solutions or at least one positive solution for fractional three-point boundary value problem with p-Laplacian operator. Our approach relies on the fixed point theorem on cones. The results obtained in this paper essentially improve and generalize some well-known results.

1. Introduction

Nowadays, fractional calculus has been adapted to numerous fields, such as engineering, mechanics, physics, chemistry, and biology. Many essays and mongraphs studying various issues in fractional calculus have been researched (see [1–6]). In particular, fractional differential equations have been found to be a powerful tool in modeling various phenomena in many areas of science and engineering such as physics, fluid mechanics, and heat conduction. More details about research achievement on fractional differential equations and their applications are shown in [7–10].

Recently, fractional differential equations have gained considerable attention (see [11–15] and the references therein). Fractional differential equations and differential equations with p-Laplacian operators have attracted much attention from many mathematicians. As a result, meaningful research results have been drawn [16–20]. Fractional-order boundary value problems involving classical, multipoint, high-order, and integral boundary conditions have extensively been studied by many researchers and a variety of results can be found in recent literature on the topic [21–25].

In [15], Chai studied the boundary value problems of fractional differential equations with p-Laplacian operator as follows:where , , , , and , , are the standard Riemann–Liouville derivatives. Some existence results of positive solutions are obtained by using the monotone iterative method.

In [16], by using Krasnosel'skii’s fixed point theorem, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:where , , , , and , are the Riemann–Liouville fractional derivatives.

In [17], by using the monotone iterative method, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:where , , and , , , and is the Riemann–Liouville fractional derivative.

In [18], by means of the p-Laplacian operator, Han et al. obtained the existence of positive solutions for the boundary value problem of fractional differential equation as follows:where is continuous, and , are the Riemann–Liouville fractional derivatives.

Based on the above research, this paper analyzed the following fractional three-point boundary value problem with the p-Laplacian operator:where , and .

The aim is to establish some existence and multiplicity results of positive solutions for BVP (5). This paper is organized as follows. In Section 2, some properties of Green’s function will be given, which are needed later. In Section 3, the existence of multiplicity results of positive solutions of BVP will be discussed (5).

2. Preliminary Knowledge and Lemmas

Lemma 1. (see [14]). Assume that with a fractional derivative of order . Then,for some , where n is the smallest integer greater than or equal to _.

Lemma 2. If then the fractional boundary value problem is

The unique solution is where ,

Proof. The general solution to the problem (7) isFrom the boundary value condition of (7), ,Therefore,

Lemma 3. If then the fractional boundary value problem is

The unique solution is where

Proof. Problem (12) is equivalent toFrom the boundary value condition of (12), and ; then,Therefore,Based on Lemma 2,

Lemma 4. The properties of and are(i), for (ii), for (iii)(see [13]). , for (iv)(see [13]). , for

Proof. (i)For , it is easy to show that ; for , So, for , Then, for , .(ii)For , it is easy to show that ; for ,and for ,

Lemma 5. (see [20]). Let be a Banach space and let be a cone in E. Assume and are open subsets of E with and and let be a continuous and completely continuous. In addition, suppose either(1), , and , , or(2), , and ,

Lemma 6. (see [16]). Let K be a cone in a real Banach space E, be nonnegative continuous concave functional on K such that , and

Suppose is completely continuous and there exist constants such that(i) and for (ii) for (iii) for with

Then, has at least three fixed points , and with , , and with .

3. Main Results

When , any , , then E is a real Banach space. is a cone, which can be defined as . Defining the operator , for any ,and for convenience, the following notation is introduced:

Theorem 1. If there are two positive numbers such that the following conditions hold:  , for   , for then the fractional three-point boundary value problem (5) has at least one positive solution and .

Proof. From the continuity of G, H, f, it can be concluded that is continuous. For , , by Lemma 4, we haveIt means that . Therefore, the Arzela–Ascoli theorem can prove that the operator is completely continuous.
Let , for . From Lemma 4 and , we can conclude thatThen, when , .
Let , for . Then, we also can conclude from Lemma 4 and thatTherefore, for , . In summary, by Lemma 5, the fractional three-point boundary value problem (5) has at least one positive solution and .