The Scientific World Journal

Volume 2013 (2013), Article ID 473828, 9 pages

http://dx.doi.org/10.1155/2013/473828

Research Article

Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems

Daliang Zhao and Yansheng Liu

School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China

Received 23 August 2013; Accepted 19 September 2013

Academic Editors: A. Atangana, A. Kilicman, S. C. O. Noutchie, S. S. Ray, and A. Secer

Copyright © 2013 Daliang Zhao and Yansheng Liu. 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.

This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem , , , where is a real number, is the Caputo fractional derivative, and is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii’s fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii’s fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.

1. Introduction

This paper investigates the existence of multiple positive solutions for the following nonlinear fractional boundary value problem (BVP, for short):

where is a real number, is the Caputo fractional derivative, and is continuous.

Recently, fractional differential equations have gained considerable importance due to their wide applications [1–8] in various sciences such as mechanics, chemistry, physics, control theory, and engineering. Much attention has been focused on the solutions of differential equations of fractional order. Some kinds of methods are presented, such as the upper and lower method [9–11], the Laplace transform method [12, 13], the iteration method [14], the Fourier transform method [15], the homotopy analysis method [16, 17], and the Green function method [18–20]. In recent years, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional BVP; for details, see [8, 9, 11, 21] and references therein.

In [21], Bai and Qiu investigated the following nonlinear fractional BVP:

where is a real number, is the Caputo fractional derivative, and is continuous. By using Guo-Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder, they established the existence and multiplicity of solutions to the above fractional BVP.

In [11], Zhao et al. established the existence of multiple positive solutions for the nonlinear fractional BVP:

where is a real number, is the Riemann-Liouville fractional derivative, and . The authors obtained the existence of positive solutions by the lower and upper solution method and fixed-point theorem.

In [8], Yang et al. investigated the existence of positive solutions of the BVP for differential equation of fractional order:

where is a real number, is the Caputo fractional derivative, and is continuous. By means of a new fixed point theorem and Schauder fixed theorem, some results on the existence of positive solutions are obtained.

Though the fractional boundary value problems have been studied by lots of authors, there are few pieces of work considering the case that the nonlinear term depends on the first order derivative . In addition, to the best of our knowledge, there is no paper discussing the existence of multiple positive solutions for BVP (1). By constructing a special cone, using Guo-Krasnoselskii and Leggett-Williams fixed point theorems, two sufficient conditions are established for the existence of multiple positive solutions to BVP (1). In addition, by virtue of a new extension of Krasnoselskii’s fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions of BVP (1) from its auxiliary problem. Finally, some illustrative examples are worked out to demonstrate the main results.

The organization of this paper is as follows. Section 2 contains some definitions and lemmas of fractional calculus theory which will be used in the next two sections. In Section 3, we establish the existence results on multiple positive solutions to BVP (1) by Guo-Krasnoselskii, Leggett-Williams fixed point theorem, and another new extension of Krasnoselskii’s fixed point theorem. Finally, some examples are presented to support the obtained results in Section 4.

2. Preliminary Results

In this section, we introduce some necessary definitions and preliminary facts which will be used throughout this paper.

Definition 1 (see [15]). The Caputo fractional derivative of order of a continuous function is given by

where , denotes the integer part of the real number and provided that the right side integral is pointwise defined on .

Definition 2 (see [15]). The Riemann-Liouville standard fractional derivative of order of a continuous function is given by

where , denotes the integer part of the real number , and provided that the right side integral is pointwise defined on .

Definition 3 (see [15]). The Riemann-Liouville standard fractional integral of order of a continuous function is given by

provided that the right side integral is pointwise defined on .

Lemma 4 (see [15]). Let . Then,

for some , , .

Lemma 5 (see [15]). Let and . Then,

holds on .

Lemma 6 (see [15]). Let . If one assumes that , then .

Lemma 7 (see [15]). Let . The fractional differential equation has solution

for some , , .

Lemma 8. For any and , the unique solution of problem

is

where

Here, is said to be the Green function of BVP (11).

Proof. In view of Lemma 4, (11) is equivalent to the integral equation

for some , . So, we have

From the boundary condition , one has

Therefore, by Definition 3, we conclude that the unique solution of BVP (11) is

The proof is completed.

The following properties of the Green function play an important role in this paper.

Lemma 9. Green function defined as (13) satisfies the following conditions.

(i) and for any .(ii)There exist a positive number and a positive function such that

Proof. (i) It is obvious that is continuous on . For , we have

Similarly, we can obtain that

Hence, for all . In addition, it is clear that for . Therefore, we get that for any .

(ii) In the following, we consider the existence of and .

Firstly, if , then by the definition of , we have

If , then by an argument similar to the case , we also have .

Secondly, for given , it is obvious that

As , we also have that

Thus, setting

we immediately obtain that

The proof is completed.

Now, we list the following fixed point theorems which will be used in the next section.

Lemma 10 (see [22], (Guo-Krasnoselskii’s fixed point theorem)). Let be a Banach space, a cone, and , two bounded open balls of centered at the origin with . Suppose that is a completely continuous operator such that either

(i), and , , or (ii), and , ,

holds. Then, has a fixed point in .

For the sake of stating Leggett-Williams fixed point theorem, we first give the definition of concave functions.

Definition 11 (see [11]). The map is said to be a nonnegative concave functional on a cone of a real Banach space provided that is continuous and

for all and .

Lemma 12 (see [23], (Leggett-Williams fixed point theorem)). Let be a cone in a real Banach space , a nonnegative continuous concave functional on such that for all , and . Suppose that is completely continuous and there exist constants such that

and for ; for ; for with .

Then, has at least three fixed points , , and with , , with .

Remark 13. If holds, then condition of Lemma 12 implies condition .

Finally, in this section, we give a new extension of Krasnoselskii’s fixed point theorem, which is developed in [24].

Let be a Banach space and a cone. Suppose that are two continuous convex functions satisfying

for , ; for ; and for , , where is a constant.

Lemma 14 (see [24]). Let , be constants and , two bounded open sets in . Set . Assume that is a completely continuous operator satisfying

, ; , ;, ; there is a such that and for all and .

Then, has at least one fixed point in .

3. Main Results

In this section, we assume that is continuous and satisfies some specific growth conditions, which allows us to apply Lemmas 10–14 to establish the existence of multiple positive solutions for BVP (1).

First, let be endowed with the norm

Define the set by

It is easy to verify that is a cone in the space .

Let the nonnegative continuous concave function on the cone be defined by

Define an operator on by the formula

Lemma 15. The operator is completely continuous.

Proof. For any , we have that in view of nonnegativeness of and . It is obvious that

By Lemma 9, for any and , we obtain that

Hence, .

The operator is continuous in view of continuity of and .

Let be bounded; that is, there exists a positive constant such that for all . By means of the definition of , we have and for . Let . Then, for , by Lemma 9, we have

Hence, is bounded.

For each , with , then

By means of the Arzela-Ascoli theorem, one can obtain that the operator is completely continuous. The proof is completed.

Now, we are in a position to state the main results.

For convenience, denote

Theorem 16. Assume that there exist two positive constants such that

for ; for .

Then, BVP (1) has at least one solution such that .

Proof. By Lemma 15, we know that the operator defined by (31) is completely continuous.

(i) Let . For any , we have , for all . It follows from condition and Lemma 9 that, for ,

which implies that , .

(ii) Let . For any , we have , for all . It follows from condition and Lemma 9 that, for ,

which implies that , .

In view of Lemma 10, has a fixed point in which is a solution of BVP (1). The proof is completed.

Theorem 17. Suppose that there exist constants such that the following assumptions hold:

for ; for ; for .

Then, BVP (1) has at least three positive solutions , , and with

Proof. We will show that all the conditions of Lemma 12 are satisfied.

First, if , then . By condition and Lemma 9, we have

which implies that for . Hence, .

Next, by using the analogous argument, it follows from condition that for .

Choose for . It is easy to see that

consequently, . Hence, if , then for . By condition , we have for . So,

which implies that for .

By Lemma 12, BVP (1) has at least three positive solutions , , and with

The proof is completed.

Theorem 18. Assume that there exist constants such that

for ; for ; for ,

where

Then, BVP (1) has at least one positive solution satisfying , for .

Proof. In order to apply the new extension of Krasnoselskii’s fixed point theorem, we consider the following auxiliary BVP:

where

It is obvious that is continuous according to the continuity of . By using the similar proof of Lemma 15, one can obtain that the operator given by

is also completely continuous on and maps into . Let

It is obvious that there exists a nonnegative function such that for all , . We divide the proof into the following three steps.

Step 1. By virtue of condition and , , we have

Step 2. For , , it follows from , , and condition that

Step 3. In view of condition , for , we have

Hence, . By Lemma 14, there exists such that . Consequently, is a positive solution for the auxiliary BVP (45) satisfying , . In addition, by virtue of the definition of , we know that , . Therefore, is a positive solution of BVP (1). The proof is completed.

4. Examples

Example 1. Consider the following fractional BVP:

By a simple calculation, one can obtain that , , and . Choosing , , we have

for and

for .

With the use of Theorem 16, BVP (52) has at least one solution such that .

Example 2. Consider the following fractional BVP:

where

Choosing , , and , then, there hold

for

for ; and

for . Hence, all the conditions of Theorem 17 are satisfied. By Theorem 17, BVP (55) has at least three positive solutions , , and such that

Example 3. Consider the following fractional BVP:

where

After a simple computation, one can find that

Choosing , , and , we know that

for ;

for ; and

for . So, all the conditions of Theorem 18 are satisfied. By Theorem 18, BVP (61) has at least one positive solution satisfying , .

5. Conclusion

In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem (1) in the Caputo sense. Using Guo-Krasnoselskii and Leggett-Williams fixed point theorems, we establish the existence of multiple positive solutions to BVP (1). By virtue of a new extension of Krasnoselskii’s fixed point theorem, we obtain a sufficient condition for the existence of multiple positive solutions of BVP (1) from its auxiliary problem. As applications, examples are presented to demonstrate the main results.

Acknowledgments

The authors wish to thank the referees for their helpful comments. This research is supported by NNSF of China (11171192) and the Natural Science Foundation of Shandong Province (ZR2013AM005).

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