The Scientific World Journal

Volume 2014 (2014), Article ID 817542, 11 pages

http://dx.doi.org/10.1155/2014/817542

Research Article

Multiple Positive Solutions to Nonlinear Boundary Value Problems of a System for Fractional Differential Equations

Chengbo Zhai and Mengru Hao

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 5 August 2013; Accepted 3 November 2013; Published 23 January 2014

Academic Editors: L. Acedo and N. Hussain

Copyright © 2014 Chengbo Zhai and Mengru Hao. 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.

By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by where is the standard Riemann-Liouville fractional derivative, for and , subject to the boundary conditions , for , and , for , or , for , and , , for , Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.

1. Introduction

The purpose of this paper is to consider the existence of multiple positive solutions for the following system of nonlinear fractional differential equations:

where , for and , and , subject to a couple of boundary conditions. In particular, we first consider (1) subject to

where , and . We then consider the case in which the boundary conditions are changed to

where .

Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, and engineering and biological sciences; see [1–11] for example. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. For a small sample of such work, we refer the reader to [12–20] and the references therein. The situation of at least one positive solution has been studied in many excellent monograph; see [12–19, 21] and other references therein. In [22], by means of Schauder fixed point theorem, Su investigated the existence of one positive solution to the following boundary value problem for a coupled system of nonlinear fractional differential equations:

where .

In [21], Goodrich established the existence of one positive solution to problems (1)-(2) and (1), (3) by using Krasnoselskii's fixed point theorem. Different from the above works mentioned, in this paper we will present the existence of at least two positive solutions to problems (1)-(2) and (1), (3) by using the similar method presented in [21]. Moreover, under different conditions, we also present the existence of at least one positive solution to problems (1)-(2) and (1), (3) with .

2. Preliminaries

For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our theorems.

Definition 1 (see [23]). Let with . Suppose that . Then the th Riemann-Liouville fractional integral is defined to be

whenever the right-hand side is defined. Similarly, with and , we define the th Riemann-Liouville fractional derivative to be

where is the unique positive integer satisfying and .

Lemma 2 (see [24]). Let be given. Then the unique solution to problem together with the boundary conditions , where and , is

where

is the Green function for this problem.

Lemma 3 (see [24]). Let be as given in the statement of Lemma 2. Then one finds that

(i) is a continuous function on the unit square ;(ii) for each ;(iii), for each .

Lemma 4 (see [24]). Let be as given in the statement of Lemma 2. Then there exists a constant such that

To prove our results, we need the following Krasnoselskii's fixed point theorem which can be seen in Guo and Lakshmikantham [25].

Lemma 5 (see [25]). Let be a Banach space, and let be a cone. Assume that are open bounded subsets of with , , and let be a completely continuous operator such that

(i), and ; or(ii), and .

Then has a fixed point in .

3. Main Results

In this section, we apply Lemma 5 to study problems (1)-(2) and (1), (3), and we obtain some new results on the existence of multiple positive solutions.

3.1. Problem (1)-(2) in the General Case

In our considerations, let represent the Banach space of when equipped with the usual supremum norm, . Then put , where is equipped with the norm for . Observe that is also a Banach space (see [26]). In addition, we define two operators by

where is the Green function of Lemma 2 with replaced by and, likewise, is the Green function of Lemma 2 with replaced by . Now, we define an operator by

We claim that whenever is a fixed point of the operator defined in (11), it follows that and solve problems (1)-(2). That is, a pair of functions is a solution of problems (1)-(2) if and only if is a fixed point of the operator defined in (11) (see [26]).

In the following, we will look for fixed points of the operator , because these fixed points coincide with solutions of problems (1)-(2). For use in the sequel, let and be the constants given by Lemma 4 associated, respectively, with the Green functions and , and define by , and notice that .

For the sake of convenience, we set

Now we list some assumptions:

();();()there are numbers , where

such that , .

Next, we define the cone by

Lemma 6 (see [21]). Let be the operator defined by (11). Then .

Lemma 7. is a completely continuous operator.

Proof. The operator is continuous in view of nonnegativeness and continuity of , and .

Let be bounded; that is, there exists a positive constant such that , for all . Let ; then, for , we have

Hence, is bounded.

On the other hand, given , setting , then, for each , , , and , one has . That is to say, is equicontinuity. In fact,

In the following, we divide the proof into two cases.

Case 1. If , then we have

where .

Case 2. If , then we have

By the means of the Arzela-Ascoli theorem, we have that is completely continuous. Similarly, is completely continuous. Consequently, is a completely continuous operator. This completes the proof.

In [21], Goodrich established the following result.

Theorem 8 (see Theorem 3.3 in [21]). Suppose that are satisfied. Then problem (1)-(2) has at least one positive solution.

From Theorem 8, the following problem is natural: whether we can obtain some conclusions or not, if   or  In the rest of this paper, we give some answers to this problem.

For the sake of convenience, we make some assumptions:

there exist constants , such that

there exist constants , such that

there are numbers , where

 such that ;there are numbers , where

 such that .

Theorem 9. Suppose that and are satisfied. Then problem (1)-(2) has at least two positive solutions , , such that .

Proof. From Lemma 7, is a completely continuous operator. At first, in view of , we have , for ; , for , where satisfies . Set . So we define by . Then for each , we find that

So for .

Similarly, we find that for . Consequently,

whenever . Thus, is cone expansion on .

Next, since , we have for ; for , where satisfies . Set . Let and . Then implies

So we obtain

So for .

Similarly, we find that for .

Consequently, , whenever . Thus, is cone expansion on .

Finally, let . For , from , , we have

Similarly, we find that for .

Consequently, , whenever . Thus, is cone compression on .

So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with

The proof is complete.

Theorem 10. Suppose that and , are satisfied. Then problem (1)-(2) has at least two positive solutions , , such that .

Proof. At first, in view of , we have , , for , where satisfies . Let .

Then for each , we find that

Like Theorem 9, we get for .

Next, in view of , we have , , for , where satisfies . We consider two cases.

Case 1. Suppose that is unbounded; there exists such that

Since , one has for . Then, for and , we obtain

Case 2. Suppose that is bounded; there exists such that for all . Taking , for and , we obtain

Hence, in either case, we always may set such that for . Like Theorem 9, we get , for .

Finally, set . Then implies

Hence we have

Consequently, for . Like Theorem 9, we get for .

So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with

which complete the proof.

3.2. Problem (1)(3) in Case

In the following, for the sake of convenience, set

Assume that there exist two positive constants such that

, for ;, for .

Theorem 11. Suppose that and are satisfied. Then problem (1)-(2), in the case where , has at least one positive solution such that between and .

Proof. With loss of generality, we may assume that .

Let . For , one has

Like Theorem 9, we get for .

Now, set . Then for , one has

Thus, we get

Like Theorem 9, we get for . Hence, from Lemma 5, we complete the proof.

Remark 12. In [21], problem (1)-(2) with is not considered.

3.3. Problem (1), (3) in the General Case

Consider the following.

Lemma 13 (see [21]). A pair of functions is a solution of (1), (3) if and only if is a fixed point of the operator defined by

where are defined by

Lemma 14 (see [21]). Each of and is strictly increasing in t and satisfies and . Moreover, there exist constants and satisfying such that and .

Let one define a new cone by

where . It is obvious that .

Lemma 15 (see [21]). is a completely continuous operator.

Now, one assumes

, for each ;There are numbers , where

 such that .There are numbers , where

 such that .

Theorem 16. Suppose that and , , are satisfied. Then problem (1), (3) has at least two positive solutions ,, such that

Proof. At first, in view of , we have , , for , where satisfies .

Let . Then for each , we find that

So for .

Similarly, we find that for . Consequently,

whenever . Thus, is cone expansion on .

Next, since , we get , , for , where satisfies . Let and ; then, implies

So for , we obtain

That is, for .

Similarly, we find that for . Consequently, , whenever . Thus, is cone expansion on .

Finally, let . For , from , and , we have

Similarly, we find that for . Consequently, , whenever . Thus, is cone compression on .

So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1), (3) with

The proof is complete.

Theorem 17. Suppose that and , , are satisfied. Then problem (1), (3) has at least two positive solutions , , such that

Proof. At first, in view of , we have , for , where satisfies . Let . Then for each , we find that

Like Theorem 16, we get for .

Next, in view of , we have , , for , where satisfies . We consider two cases.

Case 1. Suppose that is unbounded and there exists such that

Since , one has for .

Then, for and , we obtain

Case 2. Suppose that is bounded; there, exists such that for all . Taking , for and , we obtain

Hence, in either case, we always may set such that for .

Like Theorem 16, we get , for .

Finally, set . Then implies

Hence we have

So, for .

Like Theorem 16, we get for . So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1), (3) with , which complete the proof.

3.4. Problem (1), (3) in Case

In [21], the author obtained that problem (1), (3) with having at least one positive solution. In the following, we also establish the existence of one positive solution to problem (1), (3) with under different conditions.

For the sake of convenience, set

Assume that there exist two positive constants such that

for ; for .

Theorem 18. Suppose that ,, and are satisfied. Then problem (1), (3), in the case where , has at least one positive solution such that between and .

Proof. With loss of generality, we may assume that . Let . For , from , , one has

Like Theorem 16, we get for .

Now, set . For , one has

Thus, from , we get

Like Theorem 16, we get for . Hence, from Lemma 5, we complete the proof.

4. An Example

To illustrate how our main results can be used in practice, we present one example.

Example 1. Consider the following BVP, for :

subject to the boundary conditions

Obviously, problem (63)–(65) fits the framework of problem (1)-(2) with

In addition, we have set

We can see that and are continuous. The functions and are obviously nonnegative.

Now, observe that holds. Again set , because , is monotone increasing function for , taking ; then, when , we get

which implies that holds.

On the other hand, to calculate the admissible range of the eigenvalues , , as given by condition , observe by numerical approximation, we find that

Thus, for any , satisfying , , condition will be satisfied.

Consequently, by Theorem 9, problem (63)–(65) has at least two positive solutions.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgments

The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).

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