A Note on Fractional Equations of Volterra Type with Nonlocal Boundary Condition

We deal with nonlocal boundary value problems of fractional equations of Volterra type involving Riemann-Liouville derivative. Firstly, by defining a weighted norm and using the Banach fixed point theorem, we show the existence and uniqueness of solutions. Then, we obtain the existence of extremal solutions by use of the monotone iterative technique. Finally, an example illustrates the results.

Motivated by [3], in this paper we investigate the following nonlocal boundary value problem:  Firstly, the nonlocal condition can be more useful than the standard initial condition to describe many physical and chemical phenomena. In contrast to the case for initial value problems, not much attention has been paid to the nonlocal fractional boundary value problems. Some recent results on the existence and uniqueness of nonlocal fractional boundary value problems can be found in [1,2,12,14,18]. However, discussion on nonlocal boundary value problems of fractional equations of Volterra type involving Riemann-Liouville derivative is rare. Secondly, in [3], in order to discuss the existence and uniqueness of problem (1), Jankowski divided ∈ (0, 1) into two situations to discuss; one is 0 < ≤ 1/2 with an additional condition and the other is 1/2 < < 1. In this paper, we unify the two situations without using the additional condition. Thirdly, for the study of differential equation, monotone iterative technique is a useful tool (see [9,10,16,17]). We know that it is important to build a comparison result when we use the monotone iterative technique. We transform the differential equation into integral equation and use the integral equation to build the comparison result which is different from [3]. It makes the calculation easier and is suitable for the more complicated forms of equations.
The paper is organized as follows. In Section 2, we present some useful definitions and fundamental facts of fractional calculus theory. In Section 3, by applying Banach fixed point theorem, we prove the existence and uniqueness of solution for problem (2). In Section 4, by the utility of the monotone iterative technique, we prove that (2) has extremal solutions. At last, we give an example to illustrate our main results.

Preliminaries
where is a fixed positive constant which will be fixed in Section 3. Obviously, the space 1− ( , ) is a Banach space. Now, let us recall the following definitions from fractional calculus. For more details, one can see [5,11].
is called the Riemann-Liouville fractional integral of order .

Existence and Uniqueness of Solutions
In what follows, to discuss the existence and uniqueness of solutions of nonlocal boundary value problems for fractional equations of Volterra type involving Riemann-Liouville derivative, we suppose the following.
(H1) There exist nonnegative constants 1 , 2 , and such that | ( , )| ≤ , for all ( , ) ∈ Δ, and (H2) There exists a nonnegative constant 3 ∈ (0, 1) such that Lemma 4. Let (H1) hold. ∈ 1− ( ) and is a solution of the following problem: if and only if ( ) is a solution of the following integral equation: Proof. Assume that ( ) satisfies (8). From the first equation of (8) and Lemma 3, we have Conversely, assume that ( ) satisfies (9). Applying the operator to both sides of (9), we have In addition, by calculation, we can concludẽ(0) The proof is completed. Proof. Define the operator It is easy to check that the operator is well defined on 1− ( ). Next we show that is a contradiction operator on 1− ( ). For convenience, let and choose where is a positive constant defined in the norm of the space Abstract and Applied Analysis 3 Then, for any , ∈ 1− ( ), we have from (H1), (H2), and the Hölder inequality According to > and the Banach fixed point theorem, the problem (2) has a unique solution. The proof is completed.
In the following discussion, we need the following assumptions.
(H3) Assume that : 1− ( ) → is a nondecreasing continuous function, Let Proof. This proof consists of the following three steps.
For any ∈ [ 0 , 0 ], we consider the following linear problem: Define an operator : It is easy to check that the operator is well defined on [ 0 , 0 ].