Existence and Uniqueness Results for Fractional Differential Equations with Riemann-Liouville Fractional Integral Boundary Conditions

We prove the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions. The first existence and uniqueness result is based on Banach’s contraction principle. Moreover, other existence results are also obtained by using the Krasnoselskii fixed point theorem. An example is given to illustrate the main results.


Introduction
Very recently, fractional differential equations have gained much attention due to extensive applications of these equations in the mathematical modelling of physical, engineering, biological phenomena and viscoelasticity (see, e.g., [1,2]).There has been a significant development in fractional differential equations.One can see the monographs of Kilbas et al. [3], Miller and Ross [4], Lakshmikantham et al. [5], and Podlubny [6].In particular, Agarwal et al. [7] establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative in finite dimensional spaces.
Moreover, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics.For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, population dynamics [8], and cellular systems [9] can be reduced to the nonlocal problems with integral boundary conditions.For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [10][11][12][13] and so forth.
In this paper, we study the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions of the following form: () =  (,  ()) , where    is the Caputo fractional derivative of order ,   is the Riemann-Liouville fractional integral of order , and   ,   ,   ( = 1, 2) are real constants.Moreover,  :  ×  →  is a continuous function satisfying some assumptions that will be specified later and  is a Banach space with norm ‖ ⋅ ‖.

Preliminaries
We now gather some definitions and preliminary facts which will be used throughout this paper.Denote by (, ) the Banach space of continuous functions  :  → , with the usual supremum norm ( We need some basic definitions and properties [6,14] of fractional calculus which are used in this paper. Definition 1.The Riemann-Liouville fractional integral of order  > 0 of the function ℎ ∈  1 ([, ]) is defined as Definition 2. For a function ℎ defined on the interval [, ], the Caputo fractional order derivative of ℎ is defined by where  = [] + 1 and [] denotes the integer part of .
From the definition of the Caputo derivative, the following auxiliary results have been established in [3].
Lemma 4. Let  > 0; then To study the boundary value problem (1), we first consider the associated linear problem and obtain its solution.
Lemma 5.For a given  ∈ (, ), the unique mild solution of the fractional boundary value problem, satisfies the following integral equation: where Proof.By Lemma 4, we reduce (7) to an equivalent integral equation Then we have where the integral is evaluated with the help of the substitution  =  + ( − ) and the definition of the beta function Applying the boundary conditions in (7), we have Abstract and Applied Analysis 3 Using ( 13) in ( 14) together with (9), we obtain ) . ( Substituting the values of  0 and  1 in (10), we get (8).This completes the proof.
Lemma 6 (Arzelà-Ascoli, [15]).If a sequence {  } in a compact subset of  is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence.
Lemma 7 (Krasnoselskii, [15]).Let Ω be a closed convex and nonempty subset of a Banach space .Let  and  be two operators such that (ii)  is compact and continuous; (iii)  is a contraction mapping.

Main Results
In view of Lemma 5, we define the operator  : (, ) → (, ) as follows: For the forthcoming analysis we impose suitable conditions on the functions involved in the boundary value problem (1).Namely, we assume the following: On   = { ∈  : ‖‖ ≤ }, we define the two operators  and  as follows: For ,  ∈   , we have which is independent of  and tends to zero as  2 −  1 → 0. Thus,  is equicontinuous.Hence, by the Arzelà-Ascoli Theorem (Lemma 6),  is compact on   .Thus, all the assumptions of Lemma 7 are satisfied.So the conclusion of the Krasnoselskii fixed point theorem implies that the boundary value problem (1) has at least one solution on .This completes the proof of Theorem 9.

An Example
Consider the following fractional boundary value problem: where

Conclusion
In this paper, we study the existence and uniqueness for fractional order differential equations with Riemann-Liouville integral boundary conditions (1) in Banach spaces.Existence and uniqueness results of solutions are established by virtue of fractional calculus, Banach's contraction principle, and the Krasnoselskii fixed point theorem.As applications, an example is presented to illustrate the main results.