Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

Abstract

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) , is established, where , is the Caputo fractional derivative, and is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

1. Introduction

In this paper, we concentrate on the existence and multiplicity of positive solutions for the following problem: where is the Caputo fractional derivative, and is a continuous function.

In the past twenty years, the fractional differential equation has aroused great consideration [1–21] not only in its application in mathematics but also in other applications in science and engineering, for example, fluid mechanics, viscoelastic mechanics, electroanalytical chemistry, and biological engineering. Bai and Qiu [22, 23] have investigated the existence and multiplicity of positive solutions of (1) and (2) by using the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem in a cone, but they did not consider its eigenvalue criteria.

The rest of the paper is organized as follows. In Section 2, we recall some concepts relative to fractional calculus and give some lemmas with respect to the corresponding Green function. In Section 3, with the use of the fixed-point theory, some existence and multiplicity results of positive solutions are obtained. At last, two examples are given.

2. Background Materials

For the convenience of the reader, we give some definitions and lemmas.

Definition 1 (see [23]). The Caputo’s fractional derivative of order of a continuous function is given by where and , provided that the right-hand side is pointwise defined on .

Lemma 2 (see [15]). Given , the unique solution of is given by where

Lemma 3 (see [23]). The Green function defined by (6) satisfies the following properties: (i) for all (ii)

Lemma 4 (see [24]). Let be a cone in a Banach space , and be a bounded open set in . Suppose that is a completely continuous operator. If there exists such that then the fixed-point index .

Lemma 5 (see [24]). Let be a cone in a Banach space . Suppose that is a completely continuous operator. If there exists a bounded open set such that each solution of satisfies , then the fixed-point index .

Lemma 6 (see [25]). Suppose that is a completely continuous linear operator and . If there exist and a constant such that , then the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .

3. Existence and Multiplicity

Let be endowed with the maximum norm and the ordering if for all . Define

Given . Let be the operators defined by and

It is well known that are all completely continuous [23].

Denote where are positive constants.

Lemma 7. Suppose is defined by (11), then the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .

Proof. The operator is a completely continuous linear operator and (see [23]). Choose and such that for ; for . By the use of Lemma 2, for , there holds So, we can choose so large that By Lemma 6, we complete the proof.

Theorem 8. Suppose the following conditions hold: where is the first eigenvalue of the operator defined by (11). Then, BVP (1) and (2) have at least one positive solution.

Proof. By condition , there exists small enough such that Let be the positive eigenfunction of corresponding to , thus .
For every , for Suppose without loss of generality that has no fixed point on (otherwise, the proof is completed). We claim that In fact, if there exist and such that , then Let It is easy to see that and . Taking into account that is a linear positive operator, we have Therefore, by (17), which contradicts the definition of . Hence (18) holds and we have from Lemma 3 that On the other hand, by , there exist and such that Define as . Then is a bounded linear operator and . Denote It is clear that . Let In the following, we firstly prove that the set is bounded.
For any , set for and denote , then for , Thus . Since is the first eigenvalue of and , the first eigenvalue of . Therefore, the inverse operator exists and It follows from that . So we have and the set is bounded.
Choose . Then by Lemma 4, we have By (23) and (29), one has Then, has at least one fixed point on . This means that problem (1) and (2) have at least one positive solution. The proof is complete.

Theorem 9. Suppose the following conditions are met: where is the first eigenvalue of the operator defined by (11). Then BVP (1) and (2) have at least one positive solution.The proof is similar to Theorem 8.

Theorem 10. Suppose there exist two numbers such that the following conditions are met:

Then, BVP (1) and (2) have at least one positive solution.

Proof. If C₁ and C₂ hold, similar to Lemma 3 [6], we have Consequently, the additivity of the fixed-point index implies Consequently, has a fixed point in .

Theorem 11. The problem in (1) and (2) has at least two positive solutions if conditions , , and C₁ hold, where is the first eigenvalue of the operator defined by (11).

Proof. Because and hold, there exist such that On the other hand, C₁ implies . So we have therefore, has two fixed points, .

Theorem 12. The problem in (1) and (2) has at least two positive solutions if conditions , , and C₂ hold, where is the first eigenvalue of the operator defined by (10).

Proof. Because and hold, there exist such that On the other hand C₂ implies . So we have Therefore, has two fixed points, .

4. Example

To illustrate the main points, we give two examples.

Example 13. Let

Consider the BVP

It is not difficult to see that

Then where is the first eigenvalue of the operator defined by (11). By Theorem 11, BVP (40) and (41) have at least one positive solution.

Example 14. Let

Consider the BVP

It is not difficult to see that

Then where is the first eigenvalue of the operator defined by (11). By Theorem 12, BVP (45) and (46) have at least one positive solution.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

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

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by NSFC (11571207) and the Taishan Scholar Project.