Existence of Positive Solutions to Semipositone Fractional Differential Equations

In this paper, by means of constructing a special cone, we obtain a sufficient condition for the existence of positive solution to semipositone fractional differential equation.


Introduction
The aim of this paper is to investigate the existence of positive solutions to the semipositone fractional differential equation where 2 3 is the standard Riemann-Liouville fractional derivative of order α which is defined as follows: where Γ denotes the Euler gamma function and [ ] α denotes the integer part of number α , provided that the right side is pointwise defined on ( ) 0, ∞ , see [1].Here, by a positive solution to the problem (1), we mean a function [ ] 0,1 u C ∈ , which is positive in ( ) 0,1 , and satisfies (1).

X. S. Du
Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modelling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology.In recent years, there exist a great deal of researches on the existence and/or uniqueness of solutions (or positive solutions) to boundary value problems for fractional-order differential equations.Sun [2] studied the existence of positive solutions for the following boundary value problems: on [ ] 0,1 .But paper [2] did not give the results of the existence of positive solution when the nonlinearity can take negative value, i.e. semipositone problems.
The purpose of the present paper is to apply the method of varying translation together with the fixed point theorems in cone to discuss (1) without nonnegativity imposed on the nonlinearity.Meanwhile, we also allow the nonlinearity to have many finite singularities on [ ] 0,1 t ∈ .

Preliminaries and Lemmas
In this section, we present several lemmas that are useful to the proof of our main results.For the forthcoming analysis, we need the following assumptions: ( ) , L G s will be defined in the following text.
In [3], the authors obtained the Green function associated with the problem (1).More precisely, the authors proved the following lemma.
Lemma 2.1 [3].For any , the unique solution of the boundary value problem where here , , lim min .
Let X be a real Banach space, Ω be a bounded open subset of X with θ ∈ Ω and : is a completely continuous operator, where P is a cone in X.
and define the cone , then ( ) w t is the unique solution to (2) for ( ) ( ) . Now we first consider the singular nonlinear boundary value problem where Proof.In fact, if u is a positive solution to (6) such that ( ) ( ) . Since ( ) w t is the unique solution to (2) for ( ) ( ), is positive solution to (1).This complete the proof of Lemma 2.4.
For any u X ∈ , define an operator

Tu t
G t s f s u s w s q s s t Since for any fixed u X ∈ , we can choose 0  so by (H 1 ), we have Therefore, the operator T is well defined and : .T X X → Lemma 2.5.Assume that (H 1 ), (H 2 ) hold.Then : T P P → is a completely continuous operator.Proof.For any u P ∈ , in view of (2) we conclude that

Tu t G t s f s u s w s q s s p t G s f s u s w s q s s t
Thus, we have r f s u s w s q s M f s q s , , 0,1 , by ( 9), (11) and the Lebesgue control convergence theorem, and noticing the continuity of ( ) .
Thus, ( ) T D is equicontinuous on [0,1].The Arezlà-Ascoli Theorem guarantees that ( ) T D is relatively compact set.Therefore : T P P → is completely continuous operator.
Proof.Assume that there exists

Thus we have
This contradiction shows that ( ) Now we show that

Main Results
Theorem 3.1.Suppose that (H 1 ), (H 2 ) hold.Then, the boundary value problems (1) has at least one positive solution ( ) Applying Lemma 2.6 and Lemma 2.7 and the definition of the fixed point index, we