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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 758390, 11 pages

http://dx.doi.org/10.1155/2014/758390
Research Article

Existence and Uniqueness of the Solutions for Fractional Differential Equations with Nonlinear Boundary Conditions

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 3 March 2014; Accepted 29 May 2014; Published 19 June 2014

Academic Editor: Abdelghani Bellouquid

Copyright © 2014 Xiping Liu et al. 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 study the existence and uniqueness of the solutions for the boundary value problem of fractional differential equations with nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we obtain the sufficient conditions of both the existence of the maximal and minimal solutions between an upper solution and a lower solution and the uniqueness of the solutions for the boundary value problem and present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results.

1. Introduction

In this paper, we study the following fractional differential equation with nonlinear boundary conditions: where , . is a constant. , are given functions. is the standard Caputo fractional derivative of order with .

With the development of mathematics, fractional derivative occurs more and more frequently in different research areas, such as physics, mechanics, electricity, and economics (see [1, 2]). At the same time, significant progress has also been made on the studies of fractional differential equations (see [312]).

The upper and lower solutions method and monotone iterative techniques have been widely used in the studies for the boundary value problems of integer-order differential equations (see [1315] and the references therein). In [13], the authors studied first order differential equation By using the upper and lower solutions method and the monotone iterative techniques, the authors concluded that (2) exists the maximal and minimal solutions in , where , are the lower and upper solutions of (2).

As for the fractional differential problems, upper and lower solutions and monotone iterative techniques have also been widely used in the studies of boundary value problem (see [1618]). For example, in [16], by using the method, the authors discussed the existence and uniqueness of solutions for the fractional differential equation with linear boundary condition where , .

In this paper, we study the existence and uniqueness of the solutions for the boundary value problem of the fractional differential equation (1), which has nonlinear boundary conditions. It is very difficult to get the iterative sequence which is used to obtain the solutions since the boundary value problem has nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we not only obtain the sufficient conditions that the boundary value problem (1) has the maximal and minimal solutions between an upper solution and a lower solution as well as having the unique solution, but also present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results.

The organization of this paper is as follows. In Section 2, we provide the necessary background and comparison principles which are used to prove our main results. In Section 3, we consider a linear problem associated with (1). In Section 4, by using the method of upper and lower solutions and the monotone iterative techniques, we obtain the existence and uniqueness solutions of (1). In Section 5, the calculation method of the approximate analytical solutions for the boundary value problems (1) is obtained. In Section 6, an example is presented to illustrate our main results.

2. Preliminaries and Comparison Principles

In this section, we present some basic definitions, lemmas, and comparison principles which play important roles in our investigation.

Definition 1 (see [3, 4]). Let and function . The Riemann-Liouville fractional integral operator of order of is defined by provided the integral exists.

The Caputo derivative of order for function is given by provided the right side is pointwise defined on , where with .

If , then .

Definition 2. Let be the space of the functions which are absolutely continuous on . We denote by the set of the functions which have continuous derivatives up to order on such that . In particular, .

Lemma 3 (see [3]). If , then the Caputo fractional derivative exists almost everywhere on , where is the smallest integer greater than or equal to .

Lemma 4 (see [3]). Suppose and . Then where is the smallest integer greater than or equal to .

Definition 5. One says is a solution of the boundary value problem (1) if and and satisfies (1).

Definition 6 (see [4]). Let the real . The function is defined by whenever the series converges called the Mittag-Leffler function of order .

Definition 7 (see [4]). Let . The function is defined by whenever the series converges called the two-parameter Mittag-Leffler function with parameters and .

Remark 8 (see [4]). It is evident that the one-parameter Mittag-Leffler functions may be defined in terms of their two-parameter counterparts via the relation .

Lemma 9 (see [4], Theorem 4.1). The two-parameter Mittag-Leffler function for some . The power series defining is convergent for all .

Lemma 10 (see [3], Theorem 4.3). Let . The Cauchy problem with and has a unique solution

Lemma 11 (see [19]). Let be a partially ordered Banach space, , if , . Then .

In this paper, we assume the following conditions are satisfied.(H0)A given function and . is monotone decreasing on , and , .(H1) , , and .

Definition 12. Let and ; we say that is a lower solution of the boundary value problem (1), if where Let and ; we say that is an upper solution of the boundary value problem (1), if where and is defined in (H0).

Lemma 13. Let be any function with and . If and and satisfies then for .

Proof. If , by the boundary conditions, we have where is a constant between and .

Because and , then and ; that is,

Let , , and , and then for and .

By Lemma 10, we can get that the Cauchy problem has a unique solution So we can obtain that which is contradictory to (17).

Hence, . By (19), we can obtain that for .

Let the function in Lemma 13; we can obtain Corollary 14.

Corollary 14. If there exists a constant with such that and and satisfies then on .

Corollary 15. Suppose (H0) holds, and there exists a constant such that and and satisfies then for .

Proof. Let and , .

Since is monotone decreasing and , then on . So .

By (22), we have and .

Because and , then .

So .

By Lemma 13, we can get that , which implies that for .

3. Boundary Value Problems for the Linear Equation

In this section, we consider the existence and uniqueness of solutions for the linear fractional differential equation with nonlinear boundary conditions where .

Theorem 16. Assume that (H0) and (H1) hold; there exist a lower solution and an upper solution of the boundary value problem (24) with on . Then the boundary value problem (24) has a unique solution . Moreover, ; that is, for .

Proof. (1) We show that the solution of (24) is unique if it exists.

Suppose that , are two solutions of (24) and let . Then, where is a constant between and and is a constant.

By Corollary 14, we have that ; that is, for .

Similarly, we can also obtain that , for .

Hence, .

(2) We prove that if is a solution of the boundary value problem (24).

Let .

If , then on . We have where is a constant between and .

We denote . Hence, .

By Corollary 14, we can obtain that on .

If , then . We have that where is a constant between and .

We denote . Hence, .

We can show , .

By Corollary 15, we have ; that is, on . So on .

Similarly, we can obtain that for . Therefore, .

(3) We prove that the problem (24) has a unique solution.

Let

It is obvious that , , , and , . If , we have . If , we have ; that is, Similarly, we can get

If , we have If , since is a lower solution of boundary value problem (24), then . And by (12), we can obtain that Hence, we obtain Using the same way as mentioned above, we can get that Since (H0) holds, it is easy to see that for . We can easily get that and for .

Let . It follows from (34) and (35) that where is a constant between and and is a constant.

By Corollary 14, we have for ; that is,

According to Lemma 10, for each , the Cauchy problem has a unique solution

In the following, we show that for each , where is defined by (39).

If , we denote , and then So , for any .

It follows that from (H0), and by (29), we have that . Then And from (35), we have By Corollary 14, we can obtain that , for all , which is a contradiction to (40). So Similarly, we can get

Let . Since and then   is strictly monotone decreasing and continuous for .

Hence, we can get that has a unique solution with .

It is easy to see that is the unique solution of the boundary value problem (24).

4. Existence and Uniqueness of the Solutions for Boundary Value Problem

In this section, we study the existence and uniqueness of the solutions for fractional differential equation with nonlinear boundary conditions (1).

Let endowed with norm for . Then is a Banach space. Denote and if and only if on .

Theorem 17. Suppose that (H0) and (H1) hold; there exist , which are lower and upper solutions of the boundary value problem (1) with for . And satisfies(H2) for any , .Then there exist monotone sequences , such that converging uniformly on , and , are the minimal and the maximal solutions of (1) on .

Proof. We denote . For any , we consider the following boundary value problem: Since , are lower and upper solutions of the boundary value problem (1), by (H2), we have for .

Therefore, , are also the lower and upper solutions of the boundary value problem (49), respectively.

In view of Theorem 16, the boundary value problem (49) has a unique solution and .

Define by . Hence, , .

We will show that if .

Let . By (H2) and (49), we have where is a constant between and , and denote . By Corollary 14, , which implies . Hence, is monotone increasing in .

Let and for . We can get monotone iterative sequences Therefore, there exist , such that

It is easy to see that satisfies By Lemma 10, we have Because is continuous and , there exists a constant such that By Lebesgue dominated convergence theorem, we can get that That is,

Therefore, It is similar to show that

It is clear that , are solutions of the boundary value problem (1).

Assume is a solution of the boundary value problem (1). We can easily obtain that by the fact that is increasing in . That is, . Doing this repeatedly, we have , for . From Lemma 11, we obtain that , as .

Hence, , are the maximal and the minimal solutions of the boundary value problem (1), respectively.

Theorem 18. Suppose that the conditions (H0) and (H1) hold; there exist lower and upper solutions of the boundary value problem (1) with for , respectively. If there exists a constant with , where , and satisfies(H3) for any , ,then the boundary value problem (1) has a unique solution on and for any , the iterative sequence , , converging uniformly to on and its error estimate is

Proof. It is easy to check that the conditions of Theorem 17 are satisfied. Then the boundary value problem (1) has the maximal solution and the minimal solution which are denoted by , , respectively.

For , , we have and from (H3), where and is defined in Theorem 17.

It follows that from for . Hence,

We can easily get Because , we have It implies that is contraction mapping. By using the contraction mapping principle, the has a unique fixed point.

Therefore, the boundary value problem (1) has a unique solution; that is, , and we denote it by .

For any , let the iterative sequence , . Similar to (65), we can get It follows that converging uniformly to on from (66).

Since , we can obtain that the error estimate of is

5. The Calculation Method of the Approximate Analytical Solutions

In this section, we give the calculation method of the approximate analytical solutions for the boundary value problem (1).

Theorem 19. Suppose (H0) and (H1) hold; there exist a lower solution and an upper solution of the boundary value problem (1) with for . And there exists a constant with , where , such that satisfies(H4) for any , .Then for any initial value , the sequence is defined by converging uniformly to the unique solution of the boundary value problem (1) on the . Furthermore, the error estimate is where

Proof. Because , where , then and , where is defined by Theorem 18.

Hence, the conditions of Theorem 18 hold, and we can get that the boundary value problem (1) has a unique solution on and for any , the iterative sequence , , converging uniformly to on . That is, By Lemma 10 and the definition of operator , we can easily obtain that is equivalent to and , .

Let , . By Lemma 10, the Cauchy problem has a unique solution which is (69).

In the following, we prove converging uniformly to on .

For , we have where is between and .

Therefore,

Because , we can show

On the other hand, by (73), we have where is between and .

In view of , we can get It follows that from .

By (80), we can easily get

Substituting (81) into (77), we have It is easy to show that

We claim from (71).

If , we have So which is a contradiction to the hypothesis of the theorem.

Therefore, it follows from (78) that By (72) and (86), we can show that Then for any initial value , the sequence is defined by (69) converging uniformly to the unique solution of the boundary value problem (1) on the .

In view of (61), we have According to (86), we can show

It is easy to see that By (88) and (89), we can obtain Hence, the error estimate of is

6. Example

We consider the boundary value problem

Let and