Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions

and Applied Analysis 3 where 1 < α ≤ 2, 0 < β < 1, 0 < ξi < 1, ζi ∈ 0, ∞ with ∑m−2 i 1 ζiξ α−β−1 i < 1. By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for the BVP 1.5 provided that the nonlinear function f : 0, 1 ×R×R is continuous and satisfies certain growth conditions. Since ∫1 0 Dtx s dA s covers the multipoint BVP and integral BVP as special case, the fractional differential equations with the Riemann-Stieltjes integral condition also were extensively studied by many authors, see 22, 23 . In 23 , Zhang and Han considered the existence of positive solution of the following singular fractional differential equation: Dtx t f t, x t 0, 0 < t < 1, n − 1 < α ≤ n, x k 0 0, 0 ≤ k ≤ n − 2, x 1 ∫1 0 x s dA s , 1.6 where α ≥ 2 and dA s can be a signed measure. Some growth conditions were adopted to guarantee that 1.6 has an unique positive solution, moreover, the authors also gave the iterative sequence of the solution, an error estimation, and the convergence rate of the positive solution. Fractional differential equations like 1.1 , with nonlinearities which are allowed to change sign and boundary conditions which contain nonlocal condition given by a RiemannStieltjes integral with a signed measure, are rarely studied. This type problems are referred to as semipositone problems in the literature, which arise naturally in chemical reactor theory 24 . In the recent work 25 , by constructing a modified function, Zhang and Liu studied the existence of positive solution of a class of semipositone singular second-order Dirichlet boundary value problem, and when f is superlinear, a sufficient condition for the existence of positive solution is obtained under the simple assumptions. Motivated by the above work, in this paper, we establish the existence of positive solutions for the semipositone fractional differential equations 1.1 when f is superlinear and involves fractional derivatives of unknown functions. 2. Preliminaries and Lemmas Definition 2.1 see 4, 5 . The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by Ix t 1 Γ α ∫ t 0 t − s α−1x s ds, 2.1 provided that the right-hand side is pointwise defined on 0, ∞ . Definition 2.2 see 4, 5 . The Riemann-Liouville fractional derivative of order α > 0 of a function x : 0, ∞ → R is given by Dtx t 1 Γ n − α ( d dt )n ∫ t 0 t − s n−α−1x s ds, 2.2 4 Abstract and Applied Analysis where n α 1, α denotes the integer part of number α, provided that the right-hand side is pointwise defined on 0, ∞ . Remark 2.3. If x, y : 0, ∞ → R with order α > 0, then Dt ( x t y t ) Dtx t Dty t . 2.3 Proposition 2.4 see 4, 5 . (1) If x ∈ L1 0, 1 , ν > σ > 0, then IIx t I x t , DtIx t Iν−σx t , DtIx t x t . 2.4 (2) If ν > 0, σ > 0, then Dttσ−1 Γ σ Γ σ − ν t σ−ν−1. 2.5 Proposition 2.5 see 4, 5 . Let α > 0, and f x is integrable, then IDtx t f x c1xα−1 c2xα−2 · · · cnxα−n, 2.6 where ci ∈ R i 1, 2, . . . , n , n is the smallest integer greater than or equal to α. Let x t Iy t , y t ∈ C 0, 1 , by standard discuss, one easily reduces the BVP 1.1 to the following modified problems, − Dty t f ( t, Iy t ,−y t ) q t , y 0 y′ 0 0, y 1 ∫1 0 y s dA s , 2.7 and the BVP 2.7 is equivalent to the BVP 1.1 . Lemma 2.6 see 26 . Given y ∈ L1 0, 1 , then the problem, Dtx t y t 0, 0 < t < 1, x 0 x′ 0 0, x 1 0, 2.8 has the unique solution x t ∫1 0 G t, s y s ds, 2.9 where G t, s is the Green function of the BVP 2.8 and is given by G t, s 1 Γ ( α − β) { t 1 − s α−β−1, 0 ≤ t ≤ s ≤ 1, t 1 − s α−β−1 − t − s α−β−1, 0 ≤ s ≤ t ≤ 1. 2.10 Abstract and Applied Analysis 5 Lemma 2.7 see 26 . For any t, s ∈ 0, 1 , G t, s satisfies: tα−β−1 1 − t s 1 − s α−β−1 Γ ( α − β) ≤ G t, s ≤ s 1 − s α−β−1 Γ ( α − β − 1) , or ( tα−β−1 1 − t Γ ( α − β − 1) ) . 2.11and Applied Analysis 5 Lemma 2.7 see 26 . For any t, s ∈ 0, 1 , G t, s satisfies: tα−β−1 1 − t s 1 − s α−β−1 Γ ( α − β) ≤ G t, s ≤ s 1 − s α−β−1 Γ ( α − β − 1) , or ( tα−β−1 1 − t Γ ( α − β − 1) ) . 2.11 By Lemma 2.6, the unique solution of the problem, Dtx t 0, 0 < t < 1, x 0 x′ 0 0, x 1 1, 2.12 is tα−β−1. Let C ∫1 0 tα−β−1dA t , B ∫1 0 tα−β−1 1 − t dA t 2.13


Introduction
In this paper, we discuss the existence of positive solutions for the following singular semipositone fractional differential equation with nonlocal condition: where 2 < α ≤ 3, 0 < β < 1, and α − β > 2, D t is the standard Riemann-Liouville derivative.
Differential equations of fractional order have been recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.Indeed, we can find numerous applications in physics, engineering like traffic, transportation, logistics, etc. , mechanics, chemistry, and so forth, see 1-5 .There has been a significant development in the study of fractional differential equations in recent years, see the monographs of Kilbas et 9 and some recent results 10-14 .
On the other hand, the nonlocal condition given by a Riemann-Stieltjes integral is due to Webb and Infante in 15-17 and gives a unified approach to many BVPs in 15, 16 .Motivated by [15][16][17]Hao et al. 18 studied the existence of positive solutions for nth-order singular nonlocal boundary value problem: x n t a t f t, x t 0, 0 < t < 1, where a can be singular at t 0, 1, f also can be singular at x 0, but there is no singularity at t 0, 1.The existence of positive solutions of the BVP 1.2 is obtained by means of the fixed point index theory in cones.
More recently, Zhang 19 considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of unknown functions where D α is the standard Riemann-Liouville fractional derivative of order α, q may be singular at t 0 and f may be singular at x 0, x 0, . . ., x n−2 0. By using fixed point theorem of the mixed monotone operator, the unique existence result of positive solution to problem 1.3 was established.And then, Goodrich 20 was concerned with a partial extension of the problem 1.3 by extending boundary conditions and the author derived the Green's function for the problem 1.4 and showed that it satisfies certain properties, then by using cone theoretic techniques, a general existence theorem for 1.4 was obtained when f t, x satisfies some growth conditions.Recently, Rehman and Khan 21 investigated the multipoint boundary value problems for fractional differential equations of the form: Abstract and Applied Analysis . By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for the BVP 1.5 provided that the nonlinear function f : 0, 1 ×R×R is continuous and satisfies certain growth conditions.Since 1 0 D t β x s dA s covers the multipoint BVP and integral BVP as special case, the fractional differential equations with the Riemann-Stieltjes integral condition also were extensively studied by many authors, see 22, 23 .In 23 , Zhang and Han considered the existence of positive solution of the following singular fractional differential equation: where α ≥ 2 and dA s can be a signed measure.Some growth conditions were adopted to guarantee that 1.6 has an unique positive solution, moreover, the authors also gave the iterative sequence of the solution, an error estimation, and the convergence rate of the positive solution.
Fractional differential equations like 1.1 , with nonlinearities which are allowed to change sign and boundary conditions which contain nonlocal condition given by a Riemann-Stieltjes integral with a signed measure, are rarely studied.This type problems are referred to as semipositone problems in the literature, which arise naturally in chemical reactor theory 24 .In the recent work 25 , by constructing a modified function, Zhang and Liu studied the existence of positive solution of a class of semipositone singular second-order Dirichlet boundary value problem, and when f is superlinear, a sufficient condition for the existence of positive solution is obtained under the simple assumptions.
Motivated by the above work, in this paper, we establish the existence of positive solutions for the semipositone fractional differential equations 1.1 when f is superlinear and involves fractional derivatives of unknown functions.

Preliminaries and Lemmas
Definition 2.1 see 4, 5 .The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by provided that the right-hand side is pointwise defined on 0, ∞ .
Definition 2.2 see 4, 5 .The Riemann-Liouville fractional derivative of order α > 0 of a function x : 0, ∞ → R is given by where n α 1, α denotes the integer part of number α, provided that the right-hand side is pointwise defined on 0, ∞ .

2.5
Proposition 2.5 see 4, 5 .Let α > 0, and f x is integrable, then where has the unique solution where G t, s is the Green function of the BVP 2.8 and is given by

2.11
By Lemma 2.6, the unique solution of the problem, and define as in 25 , one can get that the Green function for the nonlocal BVP 2.7 is given by Throughout paper one always assumes the following holds.H0 A is a increasing function of bounded variation such that G A s ≥ 0 for s ∈ 0, 1 and 0 ≤ C < 1, where C is defined by 2.13 .Define q t max q t , 0 , q − t max −q t , 0 .

2.16
One has the following Lemma.

2.19
Proof.By 2.11 and that A t is a increasing function of bounded variation, we have Consequently,
In fact, since 1 < α − β − 1 < 2, the left side of 1 clearly holds.For right side of 1 , from B ≤ C, one gets Abstract and Applied Analysis 7 thus we have 2 is obvious from 2.11 .Now define a function • * for any z ∈ C 0, 1 by and consider the following approximate problem of the BVP 2.7 :

2.27
Lemma 2.10.Suppose v is a positive solution of the problem 2.27 and satisfies v t ≥ w t , t ∈ 0, 1 , then v − w is a positive solution of the problem 2.7 , consequently, I β v t − w t also is a positive solution of the BVP 1.1 .
Proof.In fact, if v is a positive solution of the BVP 2.27 such that v t ≥ w t for any t ∈ 0, 1 , then, from 2.27 and the definition of z t * , we have

2.28
Let y v − w, then we have

2.30
Since q t q t − q − t , then 2.27 is transformed to 2.7 , that is, v − w is a positive solution of the BVP 2.27 .By 2.7 , I β v t − w t is a positive solution of the BVP 1.1 .
It is well known that the BVP 2.27 is equivalent to the fixed points of the mapping T given by Tv t 1 0 H t, s f s, I β v s − w s * , − v s − w s * q s ds.

2.31
The basic space used in this paper is E C 0, 1 ; R , where R is a real number set.Obviously, the space E is a Banach space if it is endowed with the norm as follows: For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.

2.39
Thus f t, u, v is nondecreasing in u on 0, ∞ .On the other hand, for any

2.40
Thus f t, u, v is nonincreasing in v on −∞, 0 .Now choose u > 1 and v < −1, then by Remark 2.11, we have

2.41
Thus for any a, b ⊂ 0, 1 , and any t ∈ a, b , we have

2.43
Lemma 2.13.Assume that H0 -H2 holds.Then T : P → P is well defined.Furthermore, T : P → P is a completely continuous operator.
Proof.For any fixed v ∈ P , there exists a constant L > 0 such that v ≤ L. And then,

10
Abstract and Applied Analysis By 2.44 and H1 -H2 , we have

2.45
which implies that the operator T : P → E is well defined.
Next let f * s f s, I β v s − w s * , − v s − w s * q s , for any v ∈ P , by 2.11 , we have

2.46
On the other hand, by 2.11 , 2.22 , and 2.46 , we also have

2.47
So we have Then T has a fixed point in P Ω 2 \ Ω 1 .

3.4
Therefore, On the other hand, choose a real number M > 0 such that From 2.38 , there exists N > r such that, for any t ∈ a, b , Abstract and Applied Analysis 13 then R > r.Let Ω 2 {v ∈ P : v < R}, for any v ∈ P ∩ ∂Ω 2 and for any t ∈ a, b , we have 3.9 So for any v ∈ P ∩ ∂Ω 2 , t ∈ a, b , by 3.7 -3.9 , we have

3.11
By Lemma 2.14, T has at least one fixed points v such that r ≤ ||v|| ≤ R.
In the end,

14
Abstract and Applied Analysis Equation 3.12 implies that v t > w t , t ∈ 0, 1 , and

3.17
In the end, we notice
So by Lemma 2.10, the BVP 1.1 has at least one positive solution x, and x satisfies 3.13 .
≈ 4.5688 × 10 −11 , 3.25 which implies that H2 holds.According to Theorem 3.1, the BVP 3.14 has at least one positive solution x t , and x t satisfies