Abstract and Applied Analysis

Volume 2013 (2013), Article ID 432941, 8 pages

http://dx.doi.org/10.1155/2013/432941

Research Article

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

Zhenhai Liu1,2 and Rui Wang²

¹Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, China

²College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 6 March 2013; Accepted 24 June 2013

Academic Editor: Zhanbing Bai

Copyright © 2013 Zhenhai Liu and Rui Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, and so forth. There has been a significant theoretical development in fractional differential equations in recent years (see [1–18]). Monotone iterative technique is a useful tool for analyzing fractional differential equations.

In [3], Jankowski considered the existence of the solutions of the following problem:

where , by using the Banach fixed point theorem and monotone iterative technique.

Motivated by [3], in this paper we investigate the following nonlocal boundary value problem:

where , is a continuous functional, ,, and ; here .

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 into two situations to discuss; one is with an additional condition and the other is . 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.

2. Preliminaries

Let with the norm , where is a fixed positive constant which will be fixed in Section 3. Obviously, the space is a Banach space. Now, let us recall the following definitions from fractional calculus. For more details, one can see [5, 11].

Definition 1. For , the integral

is called the Riemann-Liouville fractional integral of order .

Definition 2. The Riemann-Liouville derivative of order can be written as

Lemma 3 (see [5]). Let . If and , then one has the following equality:

3. 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 , , and such that , for all , and

(H2) There exists a nonnegative constant such that

Lemma 4. Let (H1) hold. 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 conclude . The proof is completed.

Theorem 5. Let (H1), (H2) hold, . Then problem (2) has a unique solution.

Proof. Define the operator by

It is easy to check that the operator is well defined on . Next we show that is a contradiction operator on . For convenience, let

and choose

where is a positive constant defined in the norm of the space .

Then, for any , 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.

Remark 6. Theorem 5 is an essential improvement of [3, Theorem 1].

4. The Monotone Iterative Technique for Problem (2)

In this section, the monotone iterative technique is presented and constructed for problem (2). This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a solution of the problem.

Let . We may assume , , for all , . Then, according to Lemma 4 and Theorem 5, the following linear problem

has a unique solution which satisfies

Lemma 7. Let , , , . Suppose that

and satisfies the problem

Then for all .

Proof. Suppose that the inequality , for all , is not true. Therefore, there exists at least a such that . Without loss of generality, we assume .

We obtain that

Let ; we have

So

This is a contradiction. Hence for all . The proof is completed.

Definition 8. We say that is called a lower solution of problem (2) if

We say that is called an upper solution of problem (2) if

In the following discussion, we need the following assumptions.

(H3) Assume that is a nondecreasing continuous function, for all , , . and are lower and upper solutions of problem (2), respectively, and .(H4) Consider

 where , . .

Let .

Theorem 9. Let inequality (17), (H2)–(H4) hold. Then there exist monotone sequences , which converge uniformly to the extremal solutions of (2) in , respectively.

Proof. This proof consists of the following three steps.

Step 1. Construct the sequences .

For any , we consider the following linear problem:

By Theorem 5, (25) has a unique solution which satisfies

Define an operator by . It is easy to check that the operator is well defined on . Let with .

Setting , , and , by (26), we obtain

Besides,

By Lemma 7, we know , . It means that is nondecreasing. Obviously, we can easily get that is a continuous map. Let , , .

Step 2. The sequences , converge uniformly to , , respectively.

In fact, , satisfy the following relation:

Setting and is a lower solution of problem (2):

Besides,

By Lemma 7, we can obtain that for all . Similarly, we can show that for all . Applying the operator to both sides of , , and , we can easily get (29). Obviously, the sequences , are uniformly bounded and equicontinuous. Then by using the Ascoli-Arzela criterion, we can conclude that the sequences , converge uniformly on with , uniformly on .

Step 3. , are extremal solutions of (1).

, are solutions of (1) on , because of the continuity of operator . Let be any solution of (1). That is,

Suppose that there exists a positive integer such that on . Let ; we have

By Lemma 7, we know that on , which implies on . Similarly, we obtain that on . Since on , by induction we get that on for every . Therefore, on by taking . Thus, we completed this proof.

5. An Example

Example 1. Consider the following problem:

Obviously, , , , and .

Let , , , and .

It is easy to check that

So, (H1) and (H2) are satisfied. By the choice of , , we can get that and . According to Theorem 5, the problem (34) has a unique solution.

Consider the same equation as (34), taking , , and then we have .

Moreover,

On the other hand, it is easy to check that (H3) holds. And let , , and then we have

where ,  . So (H4) is satisfied. Obviously, , , and then we can get

Inequality (17) holds. All conditions of Theorem 9 are satisfied, so problem (34) has extremal solutions.

Acknowledgments

The project is supported by NNSF of China Grant nos. 11271087, 61263006 and Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116.

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