Existence of Positive Solutions to Boundary Value Problem of Caputo Fractional Differential Equation

Abstract

We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problem , where is continuous, , and is the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.

1. Introduction

In this paper, we consider the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem (BVP): where is continuous, , and is the standard Caputo differentiation.

Owing to the rapid development of the theory of fractional calculus itself as well as its applications, fractional differential equations have attracted intensive study recently. There has been especially an increased interest in studying the existence of positive solutions for the continuous fractional calculus concerning the Riemann-Liouville and Caputo derivatives; see [1–7] and references therein. For example, if , using the cone expansion or the cone compression fixed point theorem, Bai and Lü [1] studied the existence of positive solutions. If , by the similar methods as [1], Xu et al. [2] obtained the existence of multiple positive solutions. As far as we know, there are many papers of fractional order which have allowed the boundary value conditions to depend on ; see [2–6]. However, to the authors’ knowledge, if , there are few papers of fractional order subjected to the boundary value conditions where the first order is not involved. Motivated by the above results and [8–10], to cover up this gap, if , we mainly discuss the existence of positive solutions to fractional differential equation which is under the boundary value conditions . In our paper, we firstly derive the corresponding Green’s function which is different from these Green’s functions that appeared in the references here and give some properties. Finally, based on Schauder’s fixed point theorem, the cone expansion or the cone compression fixed point theorem, and an extension of Krasnoselskii’s fixed point theorem, we obtain the existence of positive solutions and give some examples to illustrate our results. Here the solutions especially are the interior points of cone; thus the solutions have better properties.

2. Preliminary

In this section (refer to [11, 12]), we list some necessary notations, lemmas, and theorems.

Definition 1 (see [12]). The Caputo fractional derivative of order of a continuous function is given by where and denotes the integer part of the real number , provided that the right side is pointwise defined on .

Lemma 2 (see [12]). Let ; if or , then where and denotes the Riemann-Liouville fractional integral of order .

Lemma 3. Let . Given , the unique solution of ishere is Green’s function of (4)-(5).

Proof. We apply Lemma 2 to reduce (4) to an equivalent equation,for some . ThenBy (5) and the above equalities, we get andTherefore, the unique solution of problem (4)-(5) is The proof is completed.

Lemma 4. has following properties:(i), and ,  ;(ii),  .

Proof. Since , thenthis implies that properties (i) and (ii) hold. The proof is completed.

The following fixed point theorems are fundamental in the proofs of our main results.

Theorem 5 (see [11]). Let be a real Banach space, a cone, and , two bounded open subsets of centered at the origin with . Assume that is a completely continuous operator such that either of the following holds:(i), and , ,(ii), and , .Then has at least one fixed point in .

Let be a real Banach space and a cone. Suppose are two continuous convex functionals satisfyingand there exists a constant such thatand , , and .

Theorem 6 (see [10, Theorem ]). Let , be constants and two bounded open sets in . Let . Assume is a completely continuous operator satisfying, ; , ;, ;there is a such that and , , .Then has at least one fixed point in .

From now on, we assumeIt is well known that for all , where . Define functionals , , ; then is a real Banach space with the equivalent norm and .

Define operators , , and , respectively, by and . It is clear that the solution of BVP (1) is equivalent to the fixed point of in . We will find the nonzero fixed point of by using the fixed point theory in cone. For this, we choose cone of bywhere

Lemma 7. Assume is continuous. Then () is completely continuous.

Proof. For , from the second inequality of property (ii) of Lemma 4, we haveFrom (19) and the first inequality of property (ii) of Lemma 4, we have then It follows from Lemma 3 that By direct calculation, we have Since is continuous, it is easy to see from (21)–(23) and Lemma 4 that is continuous. Now, we only need to show that is compact. Let be bounded; that is, there exists a constant number such that for . By the definition of , we know , . Let . Then for , by (19), we haveBy (23) and Lemma 4, we havethus is bounded. Let with , for ; we have The Arzela-Ascoli theorem guarantees that is relatively compact, which means is compact. Hence is completely continuous. The proof is completed.

Lemma 8. Assume is continuous. If is the solution of BVP (1), then .

Proof. If is the solution of BVP (1), it follows from the condition and (22) that we have By (23), we have By (28), there exist and such thatBy (27), there exists such that , Letting , for , , we haveSo, we have , . Therefore, . The proof is completed.

3. Main Results

In this section, we impose some growth conditions on which allow us to apply Theorems 5 and 6 to establish the existence of positive solutions to BVP (1).

Theorem 9. Assume is continuous and there exist positive constants and such that and , . Then BVP (1) has at least one positive solution.

Proof. Let , where , and . We now show that . In fact, if , then , . By condition By (25) and (31), we have this means . By applying Schauder’s fixed point theorem, the condition implies that has at least one nontrivial fixed point in , which is a positive solution of BVP (1). The proof is completed.

Theorem 10. Assume is continuous and there exist two constants such that,  ;,  Then BVP (1) has at least one positive solution.

Proof. Take ; then, for , we have , . From condition , , By (23), we haveTake ; then, for , we have , . From condition and Lemma 4, we haveConsequently, It follows from (35) that for . Therefore, by Theorem 5, has at least one fixed point in , which is the positive solution of BVP (1). The proof is completed.

To be convenient, we denote

We will suppose that there are such that satisfies the following growth conditions:, ;, ;, ,where is defined by (18).

Let Then for . Define

Theorem 11. Assume is continuous and conditions hold. Then BVP (1) has at least one positive satisfying , .

Proof. Take Since is continuous, by Lemma 7 and (37)-(38), it is obvious that is also completely continuous, and there is a such that , , . For , then , . From condition , we havewhereas, for , according to the definition of that , . From condition , we haveFinally, for , from condition and Lemma 4, we haveAccording to Theorem 6, has at least one fixed point . Noticing that for , we know that is also one fixed point of . Obviously, is a positive solution of BVP (1). The proof is completed.

We now present some simple examples. Consider the following (BVP):

Example 12. Let , . It is clear that satisfies all the conditions of Theorem 9. Then, by Theorem 9, BVP (43) has at least one positive solution.

Example 13. Let , . Choosing , , then satisfies all the conditions of Theorem 10. By Theorem 10, BVP (43) has at least one positive solution.

Example 14. Let , . By simple calculation, we have , , . Choosing , , and , it is easy to verify that satisfies all the conditions of Theorem 11. By Theorem 11, BVP (43) has at least one positive solution.

Conflict of Interests

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

Authors’ Contribution

Shugui Kang conceived the study and participated in its design. Yaqiong Cui drafted the paper and participated in its design and coordination. Zhiping Liu participated in the sequence correction. All authors read and approved the final paper.

Acknowledgments

The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. The project is supported by National Nature Science Foundation of China (11271235) and The Foundation of Datong University (2013K5).