Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with p-Laplacian Operator

This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with p-Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective.


Introduction
Fractional differential equation has recently attracted many scholars' interest due to its wide applications [1][2][3] in engineering, technology, biology, chemical process, and so on.The first issue for the theory of fractional differential equations is the existence of solutions to kinds of boundary value problems (BVPs), which has been studied recently by many scholars, and lots of excellent results have been obtained [4][5][6][7][8][9][10][11][12][13][14][15][16][17] by means of fixed point theorems, upper and lower solutions technique, and so forth.
As an important branch of BVPs, -Laplacian equation was firstly introduced in [18] to model the following turbulent flow in a porous medium: (  (  ())) =  (,  () ,   ()) , where Φ  () = || −2 ,  > 1, Φ −1  = Φ  , and 1/ + 1/ = 1.Then, it was investigated in both integer-order BVPs [19,20] and fractional BVPs [21][22][23][24][25].In [21], Chen and Liu considered the antiperiodic boundary value problem of fractional differential equation with -Laplacian operator and obtained the existence of one solution by using Schaefer's fixed point theorem under certain nonlinear growth conditions.Han et al. [22] investigated a class of fractional boundary value problem with -Laplacian operator and boundary parameter and presented several existence results for a positive solution in terms of the boundary parameter.It is noted that although there exist several results on the existence of one solution to fractional -Laplacian BVPs, there are, to our best knowledge, relatively few results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to fractional -Laplacian BVPs.
The main contributions of this paper are as follows.(i) We systematically study the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and propose some techniques to deal with fractional -Laplacian BVPs, which enriches this academic area.(ii) We present a sufficient condition for the existence of three positive solutions to BVP (2) by introducing a free constant to construct a proper concave functional.The main feature of this free constant is that one can weaken the conditions by regulating the free constant (see Remark 12).
The rest of this paper is structured as follows.Section 2 contains some preliminaries on the Caputo derivative.Section 3 investigates the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and presents the main results of this paper.In Section 4, three illustrative examples are worked out to support our obtained results.

Preliminaries
In this section, we give some necessary preliminaries on the Caputo derivative, which will be used in the sequel.For details, please refer to [1][2][3] and the references therein.
Definition 2 (see [3]).The Caputo fractional derivative of order  > 0 of a continuous function  : (0, +∞) → R is given by where  is the smallest integer greater than or equal to , provided that the right side is pointwise defined on (0, +∞).
One can easily obtain the following property from the definition of the Caputo derivative.

Main Results
In this section, we first convert BVP (2) into an equivalent operator equation and then present some new results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2).Firstly, we convert BVP (2) into an equivalent operator equation.
Proof.The necessity is obvious.Next, we prove the sufficiency.
Next, we study the existence of a unique positive solution to BVP (2).To this end, we need the following fixed point theorem [26].Definition 7. Let  be a solid cone in a real Banach space ,  :  ∘ →  ∘ an operator, and 0 ≤  < 1.Then,  is called a -concave operator if Lemma 8. Assume that  is a normal solid cone in a real Banach space , 0 ≤  < 1, and  :  ∘ →  ∘ is a -concave increasing operator.Then  has only one fixed point in  ∘ .
Based on Lemma 8, we have the following result.
In fact, from (19), for any 0 <  < 1,  ∈  ∘ , it is easy to see that which implies that  is a -concave operator.
Finally, we investigate the multiplicity of positive solutions to BVP (2).
We first recall the famous Leggett-Williams fixed point theorem [26].
Proof.Let us divide the proof into 4 steps.
To sum up, all conditions of Lemma 10 hold.By Lemma 10, BVP (2) has at least three positive solutions.(31)

Illustrative Examples
In this section, we give three illustrative examples to support our new results.Let  =  = 1 and  =  = 1/2; we can draw the graph of the unique positive solution to BVP (34) by using MATLAB, shown in Figure 1.