Positive Solutions of Fractional Differential Equation with p-Laplacian Operator

and Applied Analysis 3 Proof. By applying Lemma 4, wemay reduce (12) to an equivalent integral equation x (t) = −I α y (t) + c1t α−1 + c2t α−2 , c1, c2 ∈ R. (16) From x(0) = 0 and (16), we have c2 = 0. Consequently the general solution of (12) is x (t) = −I α y (t) + c1t α−1 = −∫ t


Introduction
It is well known that differential equation models can describe many nonlinear phenomena such as applied mathematics, economic mathematics, and physical and biological processes.Undoubtedly, the application of differential equation in the economics, management science, and engineering that is most successful especially plays an important role in the construction of the model for the corresponding phenomenon.In fact, many economic processes such as ecological economics model, risk model, the CIR model, and the Gaussian model in [1] can be described by differential equations.Recently, fractional-order models have proved to be more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models.So complicated dynamic phenomenon of fractional-order calculus system has received more and more attention; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15].
The upper and lower solutions method is a powerful tool to achieve the existence results for boundary value problem; see [2][3][4][5][6].Recently, Zhang and Liu [2] considered the existence of positive solutions for the singular fourth-order -Laplacian equation with the four-point boundary conditions where   () = || −2 ,  > 1, 0 < ,  < 1, and  ∈ ((0, 1) × (0, +∞), [0, +∞)) may be singular at  = 0 and/or 1 and  = 0.By using the upper and lower solutions method and fixed-point theorems, the existence of positive solutions to the boundary value problem is obtained.In [2], a upper and lower solution condition (H3) is used.There exist a continuous function () and some fixed positive number , such that () ≥ (1 − ),  ∈ [0, 1], and where (, ), (, ) are the associated Green's functions for the relevant problems.And then, the condition (H3) was also adopted by Wang et al. [3] to deal with the -Laplacian fractional boundary value problem (1).By using similar method as [2], the existence results of at least one positive solution for the above fractional boundary value problem are established.
Recently, replaced (H3) with a simple integral condition, Jia et al. [8] studied the existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal fractional differential equation by using upper and lower solutions method.
In this paper, we restart to establish the existence of positive solutions for the BVP (1) when the nonlinearity  may be singular at both  = 0, 1 and  = 0.By finding more suitable upper and lower solutions of (1), we completely omit the condition (H3) in [2,3] and integral condition in [8], thus our work improves essentially the results of [2,3,8].

Basic Definitions and Preliminaries
In this section, we present some necessary definitions and lemmas from fractional calculus theory, which can be found in the recent literatures [7,16,17].
Definition 1.The Riemann-Liouville fractional integral of order  > 0 of a function  : (0, +∞) → R is given by provided that the right-hand side is pointwise defined on (0, +∞).
Definition 2. The Riemann-Liouville fractional derivative of order  > 0 of a function  : (0, +∞) → R is given by where  = [] + 1, [] denotes the integer part of number , provided that the right-hand side is pointwise defined on (0, +∞).
So, the unique solution of problem (12) is The proof is completed.
(S2) For any  > 0, From Lemmas 7 and 9, it is easy to obtain the following conclusion.
In fact, for any  ∈ , by the definition of , there exist two positive numbers 0 <   < 1,   > 1, such that   () ≤ () ≤   () for any  ∈ [0, 1].It follows from Lemma 9 and (S1)-( S3) that On the other hand, by Lemma 9, we also have then by ( 40) and (41), which implies that  is well defined, and () ⊂ .It follows from (S1) that the operator  is decreasing in .And by direct computations, we have Next we focus on lower and upper solutions of the fractional boundary value problem (1).Let We will prove that the functions () = (), () = () are a couple of lower and upper solutions of the fractional boundary value problem (1), respectively.
From the uniform continuity of (, ) and Lebesgue dominated convergence theorem, we easily obtain that B is equicontinuous.Thus by the means of the Arzela-Ascoli theorem, we have B :  →  is completely continuous.The Schauder fixed point theorem implies that B has at least a fixed point , such that  = B.
At the end, we claim that In fact, since  is fixed point of B and (44), we get =  () , =  () , Otherwise, suppose that () > ().According to the definition of , we have On the other hand, it follows from  is an upper solution to where Remark 12.In Theorem 11, we find more suitable lower and upper solutions, then we refine the proved process, and the key condition (H3) in [2,3] is removed, but the existence of positive solution is still obtained, thus our result is essential improvement of [2,3].(65) Proof.The proof is similar to Theorem 11, so we omit it here.At the end of this work we also remark that the extension of the pervious results to the nonlinearities depending on the time delayed differential system for energy price adjustment or impulsive differential equation in financial field requires some further nontrivial modifications, and the reader can try to obtain results in our direction.We also anticipate that the methods and concepts here can be extended to the systems with economic processes such as risk model, the CIR model, and the Gaussian model as considered by Almeida and Vicente [1].

) Definition 6 .
A continuous function () is called an upper solution of the BVP (1), if it satisfies