Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations

We present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problem , where is a real number and is the standard Riemann-Liouville differentiation. One example is also given to illustrate the main result.


Introduction
This paper is mainly concerned with the multiplicity of positive solutions of nonlinear fractional differential equation boundary value problem BVP for short D α 0 u t p t f t, u t − q t , 0 < t < 1, where 2 < α ≤ 3 is a real number and D α 0 is the standard Riemann-Liouville differentiation, and f, p, q is a given function satisfying some assumptions that will be specified later.
In the last few years, fractional differential equations in short FDEs have been studied extensively the motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering.For an extensive collection of such results, we refer the readers to the monographs by Kilbas et   Discrete Dynamics in Nature and Society Some basic theory for the initial value problems of FDE involving the Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala 6-8 , Babakhani and Daftardar-Gejji 9-11 , and Bai 12 , and others.Also, there are some papers that deal with the existence and multiplicity of solutions or positive solution for nonlinear FDE of BVPs by using techniques of nonlinear analysis fixed point theorems, Leray-Schauders theory, topological degree theory, etc. , see 13-22 and the references therein.
Bai and L ü 15 studied the following two-point boundary value problem of FDEs where D q 0 is the standard Riemann-Liouville fractional derivative.They obtained the existence of positive solutions by means of the Guo-Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem.
Zhang 22 considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem where 1 < q ≤ 2 is a real number, f : 0, 1 × 0, ∞ → 0, ∞ , and c D q 0 is the standard Caputo's fractional derivative.The author obtained the existence and multiplicity results of positive solutions by means of the Guo-Krasnosel'skii fixed point theorem.
From the above works, we can see the fact that although the fractional boundary value problems have been investigated by some authors to the best of our knowledge, there have been few papers that deal with the boundary value problem 1.1 for nonlinear fractional differential equation.Motivated by all the works above, in this paper we discuss the boundary value problem 1.1 , using the Guo-Krasnosel'skii fixed point theorem, and we give some new existence of multiple positive solutions criteria for boundary value problem 1.1 .
The paper is organized as follows.In Section 2, we give some preliminary results that will be used in the proof of the main results.In Section 3, we establish the existence of multiple positive solutions for boundary value problem 1.1 by the Guo-Krasnosel'skii fixed point theorem.In the end, we illustrate a simple use of the main result.

Preliminaries and Lemmas
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory.These definitions can be found in the recent literature such as 1, 4, 15 .
Definition 2.1 see 1, 4 .The Riemann-Liouville fractional integral of order α α > 0 of a function f : 0, ∞ → R is given by provided that the right side is pointwise defined on 0, ∞ , where Γ is the gamma function.
Definition 2.2 see 1, 4 .The Riemann-Liouville fractional derivative of order α α > 0 of a continuous function f : 0, ∞ → R is given by provided that the right side is pointwise defined on 0, ∞ , where n α 1 and α denotes the integer part of α.
as unique solutions, where N is the smallest integer greater than or equal to α.
In the following, we present Green's function of the fractional differential equation boundary value problem.
Lemma 2.5.Let h ∈ C 0, 1 and 2 < α ≤ 3, then the unique solution of is given by where G t, s is Green's function given by The following properties of Green's function form the basis of our main work in this paper.
Lemma 2.6.The function G t, s defined by 2.8 possesses the following properties: The following Krasnosel'skii's fixed point theorem will play a major role in our next analysis.
Lemma 2.7 see 23 .Let X be a Banach space, and let P ⊂ X be a cone in X. Assume Ω 1 , Ω 2 are open subsets of X with 0 ∈ Ω 1 ⊂ Ω 1 ⊂ Ω 2 , and let A : P → P be a completely continuous operator such that either Then A has a fixed point in P ∩ Ω 2 \ Ω 1 .

Main Results
In this section, we establish some new existence results for the fractional differential equation 1.1 .Given a ∈ L 1 0, 1 , we write a a 0, if a ≥ 0 for t ∈ 0, 1 , and it is positive in a set of positive measure.

3.8
For our constructions, we will consider the Banach space E C 0, 1 equipped with standard norm u Let the operator A : K → E be defined by the formula
Proof.Notice from 3.10 and Lemma 2.6 that, for x ∈ K, Ax t ≥ 0 on 0, 1 and

3.11
On the other hand, we have

3.12
Thus we have A K ⊂ K.The proof is finished.
It is standard that A : K → K is continuous and completely continuous.
Then the problem 1.1 has at least two positive solutions.
Proof.To show that 1.1 has at least two positive solutions, we will assume the problem 3.4 has at least two positive solutions x 1 and To see this, let , by Lemma 3.1 and A1 , we have

3.14
Thus, we see, from Lemma 2.6 and A1 , that

3.15
from which we see that Ax ≤ x , for x ∈ K ∩ ∂Ω 1 .
Next we now show

3.19
from which we see that Ax > x , for x ∈ K ∩ ∂Ω 2 .
On the other hand, let ε > 0, where

and so
Au max 0≤t≤1 1 0 G t, s p s g s, x s − γ s ds

3.23
from which we see that Ax ≤ x , for x ∈ K ∩ ∂Ω 3 .
In view of Lemma 2.7, the problem 3.4 has at least two positive solutions x 1 and

3.24
Therefore x 1 , x 2 are solutions of the problem 1.1 .This completes the proof.
Theorem 3.5.Suppose that (H1), (H2) are satisfied.Furthermore assume that Then the problem 1.1 has at least two positive solutions.

An Example
As an application of the main results, we consider It is clear that f : 0, 1 × 0, ∞ → 0, ∞ is continuous.Since all the conditions of Theorem 3.5 are satisfied, the problem 4.1 has at least two positive solutions.