Journal of Applied Mathematics

Volume 2013 (2013), Article ID 349025, 11 pages

http://dx.doi.org/10.1155/2013/349025

Research Article

Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

Dandan Yang and Chuanzhi Bai

Department of Mathematics, Huaiyin Normal University, Huai’an, Jiangsu 223300, China

Received 14 August 2013; Revised 25 October 2013; Accepted 27 October 2013

Academic Editor: Saeid Abbasbandy

Copyright © 2013 Dandan Yang and Chuanzhi Bai. 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 establish the existence of unbounded solutions for nonlinear fractional boundary value problems on the half-line. By the upper and lower solution method technique, sufficient conditions for the existence of solutions for the fractional boundary value problems are established. An example is presented to illustrate our main result.

1. Introduction

Boundary value problems on the half-line arise in the study of radially symmetric solutions of nonlinear elliptic equations and various physical phenomena, such as the theory of drain flows and plasma physics; see [1–10] and the references therein. In 2006, Lian and Ge in [11] investigated the following boundary value problem on the half-line for the second-order differential equation:

where, , and are given. Based on Leray-Schauder continuation theorem, some suitable conditions for the existence of solutions to (1) are established.

On the other hand, fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary noninteger order. Fractional calculus is a wonderful technique to understand memory and hereditary properties of materials and processes. Some recent contributions to fractional differential equations are present in the monographs [12–19]. Very recently, Chen and Tang in [20] considered the following fractional differential boundary value problem on the half-line:

where and is the standard Riemann-Liouville fractional derivative. By the recent Leggett-Williams norm-type theorem, the existence of positive solutions is obtained. In 2011, [21] set up the global existence results of solutions of initial value problems on the half-axis as follows:

where is the standard Riemann-Liouville fractional derivative. By constructing a special Banach space and employing fixed point theorems, some sufficient conditions for the existence of solutions are obtained. In [22], the authors studied the following boundary value problem of fractional order on the half-line:

where, , and are the standard Riemann-Liouville fractional derivatives. By Schauder’s fixed point theorem on an unbounded domain, they obtain the existence result for (4). Some papers have recently been done for fractional boundary value problem on the half-line or unbounded domain, see [22–31].

Inspired by the above-mentioned works, in this paper, we study the existence of solutions to the following fractional differential equations with boundary value problems:

where, and is the standard Riemann-Liouville fractional derivative. By upper and lower solution method techniques, the sufficient conditions for solutions to (5) are obtained. Our main findings given in this paper have some new features. Firstly, the like Nagumo condition defined by us plays an important role in the nonlinear term involving the standard Riemann-Liouville derivatives. Secondly, to the best of our knowledge, no work has been done concerning fractional boundary value problems (5) and our method is different from that of [22, 26, 31]. Thirdly, the nonlinear term may take negative values, and depends on allowed to be quadratic, referring to our example. The rest of this paper is organized as follows: in Section 2, we present some preliminaries and some lemmas that will be used in Section 3. The main result and proof will be given in Section 3. In addition, an example is given to demonstrate the application of our main result.

2. Preliminaries

We first present some basic definitions and preliminary results about fractional calculus; we refer the reader to [17, 18] for more details.

Definition 1 (see [18]). The integral

where, is called the Riemann-Liouville fractional integral of order and is the Euler gamma function defined by , .

Definition 2 (see [17]). A function given in the interval , the expression

where,denotes the integer part of number, is called the Riemann-Liouville fractional derivative of order.

Lemma 3 (see [17]). Letand Then the fractional differential equation

has

as a unique solution.

Lemma 4 (see [17]). Assume thatwith a fractional derivative of orderthat belongs to; then

for some, .

Denote bythe space of all continuous real functions defined onand the space of all real functions which are Lebesgue integrable on every bounded subinterval of . Denote

with the norm , where

By standard arguments, it is easy to prove thatis a Banach space.

Proposition 5 (see [32]). Assume thatis inwith a fractional derivative of order that belongs to . Then

for some. When the function is in , then .

Definition 6. A functionis called a lower solution of (5) if

Similarly, we can define an upper solutionof (5) by reversing the above inequalities.

Remark 7. If

and, , then we havefor all.

We consider the following two cases.

Case 1. Consider . The inequality (15) reduces to. Then

that is,

From the boundary conditionand we obtain, and

Thus,for.

Case 2. Consider. If (15) holds, then

Noting thatand, from the above inequality and Proposition 5, we have thatfor.

For example, consider,. Obviously,, . Thusand

We also getfor each.

Definition 8. Let be lower and upper solutions to (5) and suppose that (15) holds. A continuous function is said to satisfy the like Nagumo condition with respect to the pair of functions, if there exist a nonnegative functionand a positive onesuch that

for all, , , and

We list some assumptions related to, and as follows.

  Consider

  Let be a continuous function satisfying the like Nagumo condition with respect to the pair of functions, , such that

Lemma 9. Let, then the fractional boundary value problem

has a unique solution, where

Proof. By Lemma 4, the solution of (25) has the following form:

By boundary value condition, one has. Thus, we have by (27) that

Substituting the conditioninto (28), we obtain . Together with , we get

Hence, we have

Lemma 10. The functiondefined by (26) satisfies the following properties:

(1)is a continuous function and for ;(2), for .

The proof is easy, so we omit it here.

SetFor, defineThenis a Banach space.

Lemma 11 (see [33]). LetThenis relatively compact if the following conditions hold:

(a)is bounded in;(b)the functions belonging toare locally equicontinuous on;(c)the functions fromare equiconvergent; that is given, there correspondssuch that , and.

By Lemma 11, similar to the proof of Theoremin [34], we can easily have the following lemma.

Lemma 12. LetThen is relatively compact if the following conditions hold:

(1) is bounded in ;(2)the functions belonging to,, and , , are locally equicontinuous on;(3)the functions from , , and , are equiconvergent at.

Define the auxiliary functions

Consider the following boundary value problem:

where

For each, it follows from (21), (23), and (33) that

where. From (34) and Lemma 9, we know that u is a solution of (32) if and only ifsolves the operator equation , and is defined by

Lemma 13. Suppose thathold; then (the operator defined in (35)) is completely continuous.

Proof. Consider the following.

Step 1. is well defined. For, it follows from (34) that

which implies

We also have

Combining (36) with (38), one has

If we apply Lebesgue dominated convergence theorem with (37) and (39), then

which yields

that is,

By virtue of (34), we have

It follows from (39) that

Therefore, we have

Thus, we conclude that.

Step 2. is continuous. For any convergent sequence in , we find

From continuity of, we obtain

Since, we have Set

which follows that

Thus, applying the Lebesgue dominated convergence theorem and then using (49), we have

as. On the other hand, we have

From (49), it is clear that

Moreover, by (52), we have

Therefore, combining (50), (51), and (53), we get, as; then we claim thatis continuous.

Step 3. is compact. Let be any bounded subset of ; then, for, set

in a similar manner as (50), (51), and (53); we have by (21) and (23) that

which implies that. Hence, is uniformly bounded. Meanwhile, for any , for , we have

We also have

which approaches, asFurthermore, we have

which approaches, asHence, we get that is equicontinuous. From (41), we have

Since

Thus, we have

which approaches, asWe also have by (45) that

which approachesasThus,is equiconvergent atThenis relatively compact. Therefore,is completely continuous. The proof is complete.

3. Main Result

We are in the position to state the main existence result.

Theorem 14. Letbe lower and upper solutions to (5), and suppose that (15) holds. Moreover,hold. Then fractional boundary value problem (5) has at least one solutionsatisfying

where is a constant dependent only on, and.

Proof. By Lemma 13, we know that is completely continuous. By the Schauder fixed point theorem, we can easily obtain thathas at least one fixed point . Thus, is a solution of (32). Next, we will show that satisfies the inequalities

which implies that is a solution of (5). First of all, we will show that , for all .

If not, then

Note that ; then there are two cases.

Case 1. There exists asuch that

Thenand

On the other hand, in view of (32), (33), and, we have

which contradicts (67).

Case 2. Consider −. By the boundary condition, we have the contradiction −. Consequently, holds for all . By using the similar method, we also can prove that for all . By Remark 7, we have

If, we choose

and such that

where . From (23), it is clear that . Hence,.

Finally, we will check thatforIf, for every , then . If, for all , then for any ; using (70), we obtain

which is a contradiction. If , for every , we can also have a similar contradiction. There exists such that . Thus, there exists such that ,  ,  . Otherwise, , , . Without loss of generality, we assume that , , . Therefore, by a convenient change of variable and using (21) and (71), we get

which implies that , since can be arbitrarily as long as ; we have , for any , which follows that . By a similar analysis, we can also obtain that if ,  ,  , then ,  . Therefore,

that is, is a solution of (5), which completes the proof.

Example 15. Consider the boundary value problem of the fractional differential equation on the half-line

In this case,, . Obviously, and are a pair of lower and upper solutions of (75). Furthermore, we have , , for . It is clear that is continuous on . If , , , we have

where and ; we have

andsatisfies the like Nagumo condition with respect to . Furthermore, we have

Thus, we conclude by Theorem 14 that there exists at least one solution to boundary value problem (75) such that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their valuable and helpful suggestions and comments that improved the paper. This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).

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