Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.


Introduction
We consider the nonlinear fractional differential equation was investigated in the paper [1]. The last condition in (BC 1 ) can be written as is the Caputo fractional derivative, and the operators A and B are defined as the operators from our problem, given above. In the paper [3], the authors studied the existence and multiplicity of positive solutions for the Riemann-Liouville fractional differential equation f t x t t α + + = ∈ , subject to the boundary conditions (BC), where f is a sign-changing function that can be singular in the points 0,1 t = and/or in the variable x. In addition, the methods used in the proofs of the main results in [3] are different than those used in the present paper, namely, in [3] the authors used various conditions which contain height functions of the nonlinearity defined on special bounded sets, and two theorems from the fixed point index theory. For some recent results on the existence, nonexistence and multiplicity of solutions for fractional differential equations and systems of fractional differential equations subject to various boundary conditions we refer the reader to the monographs [4] [5] and the papers [6]- [14].
We also mention the books [15]- [21], and the papers [22]- [28] for applications of the fractional differential equations in various disciplines.

Preliminary Results
We present in this section some auxiliary results from [3] that we will use in the proof of the main results. We consider the fractional differential equation Lemma 2 If 0 ∆ ≠ , then the solution x of problem (1)-(BC) given by (2) can be written as and By using some properties of the functions 1 2 , , 1, , given by (5) from [29], we obtain the following lemma.
Lemma 3 We suppose that 0 ∆ > . Then the function G given by ( Then the solution x of problem (1)-(BC) given by (3) satisfies the , and so In the proof of our main theorems, we use the Banach contraction mapping principle and the Krasnosel'skii fixed point theorem for the sum of two opera-Open Journal of Applied Sciences tors presented below.
Theorem 1 (see [30]) If ( ) , Y d is a nonempty complete metric space with the metric d, and : Theorem 2 ([31]) Let M be a closed, convex, bounded and nonempty subset of a Banach space X. Let 1 A and 2 A be two operators such that A is a completely continuous operator (continuous, and compact, that is, it maps bounded sets into relatively compact sets);

Main Results
In this section we study the existence of nonnegative solutions for our problem (E)-(BC). We present now the assumptions that we will use in the sequel. : (I3) There exist the functions and for all , , , , , x y z x y z + ∈  .
(I4) There exists the function We denote by By Lemma 2 we easily deduce that if x is a solution of Equation (6) (or equivalently (7)), then x is a solution of problem (E)-(BC).
We define the operator  on

t s f s x s Ax s Bx s s = ∫
 If x is a fixed point of operator  , then x is a solution of Equation (6) (or (7)), and hence x is a solution of problem (E)-(BC). Therefore we will study the existence (and uniqueness) of the fixed points of operator  by using the Banach contraction mapping principle.

G t s f s x s Ax s Bx s f s x s Ax s Bx s s G t s a s x s x s b s Ax s Ax s c s Bx s Bx s s
Therefore we obtain the inequality σω < , we deduce that  is a contraction mapping. By Theorem 1, we conclude that  has a unique fixed point, which is a nonnegative solu-

Conclusion
In this paper, we investigated the existence of nonnegative solutions for the Riemann-Liouville fractional differential equation with integral terms (E) supplemented with the boundary conditions (BC) which contain Riemann-Liouville fractional derivatives of different orders and Riemann-Stieltjes integrals, by using the Banach contraction mapping principle and the Krasnosel'skii fixed point theorem for the sum of two operators. For some future research directions, we have in mind the study of the existence, nonexistence and multiplicity of solutions or positive solutions for fractional differential equations subject to other boundary conditions.