Positive Solutions to Periodic Boundary Value Problems of Nonlinear Fractional Differential Equations at Resonance

Abstract

By Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima, we discuss the existence of positive solutions to fractional order with periodic boundary conditions at resonance. At last, an example is presented to demonstrate the main results.

1. Introduction

Fractional differential equations are generalizations of ordinary differential equations to an arbitrary order. It has played a significant role in many fields, such as viscoelasticity, engineering, physics, and economics; see [1–5]. During the last ten years, there are a large number of papers dealing with the existence of solutions boundary value problem for fractional differential equations; see [6–10].

Recently, there is an increasing tendency on discussion for the existence of positive solutions to boundary value problems of fractional differential equations which enriched many previous results.

In [9], Yang and Wang considered the existence of solutions for the following two-point boundary value problems for fractional differential equations:where , denoting the Caputo fractional derivative. By using the Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima, the authors obtained the existence of positive solutions to the above problem.

In [10], Hu discussed the existence of positive solutions for a boundary value problem of fractional differential inclusions with resonant boundary conditions:where , , denotes the Caputo fractional derivative, , and donates the family of nonempty compact and convex subsets of . By using the Leggett-Williams theorem for coincidences of multivalued operators due to O’Regan and Zima, results on the existence of positive solutions were established.

Periodic boundary value problems have profound practical background and wide range of applications, such as mechanics, biology, and engineering; see [11–14]. Recently, periodic boundary value conditions of fractional order have been studied by some authors, such as [15–17].

In [16], Chen et al. studied the following periodic boundary value problem for fractional -Laplacian equation:where , is a Caputo fractional derivative, and is continuous.

In [17], Hu et al. considered the existence of solutions for the following periodic boundary value problem for fractional differential equation:where , denotes the Caputo fractional derivative, and is continuous. By using the coincidence degree theory, the authors obtained the existence of solutions.

From the above works, we notice that the study of positive solutions to periodic boundary value problems of fractional order at resonance is poor. Now, the question is as follows: though the existence of solutions to (4) is obtained, how can we get the positive solutions of it? The aim of this paper is to fill the gap in the relevant literature. Our main tool is the recent Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima [18].

The rest of this paper is organized as follows. Section 2, we give some necessary notations, definitions, and lemmas. In Section 3, we obtain the existence of positive solutions of (4) by Theorem 9. Finally, an example is given to illustrate our results in Section 4.

2. Preliminaries

First of all, we present the necessary definitions and lemmas from fractional calculus theory. For more details, see [1].

Definition 1 (see [1]). The Riemann-Liouville fractional integral of order of a function is given byprovided that the right-hand side is pointwise defined on .

Definition 2 (see [1]). The Caputo fractional derivative of order of a continuous function is given bywhere , provided that the right-hand side is pointwise defined on .

Lemma 3 (see [1]). The fractional differential equationhas solution , , , and .
Furthermore, for ,

Lemma 4 (see [1]). The relation is valid in the following case: , , and .

In the following, let us recall some definitions on Fredholm operators and cones in Banach space (see [19]).

Let , be real Banach spaces. Consider a linear mapping and a nonlinear operator . Assume that(A1) is a Fredholm operator of index zero; that is, is closed and .This assumption implies that there exist continuous projections and such that and . Moreover, since , there exists an isomorphism . Denote by the restriction of to . Clearly, is an isomorphism from to ; we denote its inverse by . It is known that the coincidence equation is equivalent toLet be a cone in such that(i) for all and ,(ii) implies .It is well known that induces a partial order in by

The following property is valid for every cone in a Banach space .

Lemma 5 (see [18]). Let be a cone in . Then for every there exists a positive number such that

Let be a retraction, that is, a continuous mapping such that for all . Set

We use the following result due to O’Regan and Zima.

Theorem 6 (see [18]). Let be a cone in and let , be open bounded subsets of with and . Assume (A1) and the following assumptions hold:(A2) is continuous and bounded and is compact on every bounded subset of .(A3) for all and .(A4) maps subsets of into bounded subsets of .(A5) .(A6) There exists such that for , where and such that for every .(A7) .(A8) .Then the equation has a solution in the set .

3. Main Results

In this section, we give the existence theorems for problem (4). We write Banach space with the norm .

Define the operator bywhereDefine the operatorbyThen problem (4) can be written by ,

LetLet be a constant, which is in and satisfies

Lemma 7. The mapping is a Fredholm operator of index zero. Furthermore, the operator can be written aswhere

Proof. By Lemma 3, has solutionAccording to the boundary value conditions of (4), we haveLet , so there exists a function which satisfies . By Lemma 3, we haveBy , we can obtain . On the other hand, suppose satisfies . Let . We can easily prove . Thus, we conclude thatConsider the linear operator defined byDefine the operator byFor , we getSo we have . In the same way, We notice that and . It follows from that is a Fredholm mapping of index zero.
Next, we will prove that the operator is the inverse of .
In fact, for , we have . By Lemma 3, we have . According to , we getBy , that is, , we haveThen, we haveIt is easy to see that . Hence, . This completes the proof of Lemma 7.

Lemma 8. Assume is an open bounded set such that ; then is -compact on

Proof. By the continuity of , we can obtain that and are bounded. Hence, for , , there exists a positive constant such thatThus, in the view of Arzela-Ascoli theorem, we need only to prove that is equicontinuous.
For , , we haveNotice that , are uniformly continuous on . Thus, we have that is equicontinuous on . The proof is completed.

Theorem 9. Assume that(H1)there exist positive constants , , , , , and with For all and , one has(H2)there exist , , , , and , on , such that and for . Moreover, is nonincreasing on andThen problem (4) has at least one positive solution on .

Proof. According to Lemmas 7 and 8, we have that conditions (A1) and (A2) of Theorem 6 are satisfied.
Consider the coneLetObviously, and are bounded andMoreover, . Let and for ; then is a retraction and maps subsets of into bounded subsets of , which means that (A4) holds.
Next, we will show that (A3) holds. Suppose that there exist and such that ; that is, , . In view of (H1), we getIn view of , from the definition of and (41), we obtainwhich givesFrom (40), we haveBy (43), we haveAccording to the function expression of , it is easy to see that , From (43) and the equation , we can getThen, we havewhich contradicts (H1). Hence (A3) holds.
To prove (A5), consider ; then . For and , we haveBy use of proof by contradiction, it is easy to show that implies . Suppose for some ; then we haveAccording to (H1), we havewhich is a contradiction. In addition, if , then , which is impossible. As a result, for and , we have . Thus,So (A5) holds.
Next, we prove (A6). Let , ; then , . We take . Let ; then and on .
By (H2), for every , we haveThus, for all , we have ; that is, (A6) holds.
For , by (H2) and (19), we haveThus, for . Then (A7) holds.
Next, we prove (A8). For , by (H2), we haveHence, ; that is, (A8) holds.
Hence, applying Theorem 6, BVP (4) has a positive solution on with . This completes the proof.

4. Example

To illustrate how our main result can be used in practice, we present here an example.

Let us consider the following fractional differential equation at resonance:whereCorresponding to BVP (4), we have that andBy simple calculation, we can get that if and , one hasSo, we can choose , , , , , , and . Furthermore, we can verifySo, (H1) is satisfied.

Taking , , and , we haveLet ; then

Moreover, take , and we have thatis nonincreasing on . So, conditions (H1)-(H2) of Theorem 9 are satisfied; then BVP (55) has a positive solution on .

Competing Interests

The author declares that he has no competing interests.

Authors’ Contributions

The author contributed to the writing of this paper. The author read and approved the final manuscript.

Acknowledgments

Research was supported by the Science Foundation of Shandong Jiaotong University (Z201429).