Positive Solutions for a Fractional Boundary Value Problem with a Perturbation Term

Abstract

We obtain some new upper and lower estimates for the Green’s function associated with a fractional boundary value problem with a perturbation term. Criteria for the existence of positive solutions of the problem are then obtained based on it.

1. Introduction

In this paper, we are investigating the existence of positive solution for fractional differential equation with a perturbation termwith the boundary condition (BC)where , , and . Here, is the standard Riemann-Liouville derivative of order of a continuous function is given by provided that the right-hand side is pointwise defined on .

Spurred by the extensively applicability of fractional derivatives in a variety of mathematical models in science and engineering [1–3], the subject of fractional differential equations with boundary value problems, which emerged as a new branch of differential equations, have attracted a great deal of attention for decades. As a small sampling of recent development, we refer the reader to [4–14]. When one seeks the existence of solution of boundary value problems for fractional differential equations, the usual method is converted to a Fredholm integral equation and find the fixed points by using various techniques of nonlinear analysis such as Banach contraction map principle [13, 15], linear operator theory [16, 17], Leggett-Williams fixed point theorem [12, 18], Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory [19], and Krasnosel’skii fixed point theorem [20]. It should be noted that the Green’s functions play a vital role in the construction of an appropriate Fredholm integral equation. However, as a result of the unusual feature of the fractional calculus, the investigation on the Green’s functions for fractional boundary value problems is still in the initial stage. Recently, based on the spectral theory, the authors in [21] give an associated Green’s function for BVP (1) (2) as series of functions. This idea was also used in [22–24].

In the next section, we will study some new sharper upper and lower estimates for the Green’s function of BVP (1) (2) than the ones given in [21]. In Section 3, we employ the new estimate to obtain the existence of a positive solution of BVP (1) (2). The idea of this paper may trace to [21–27].

2. Some New Upper and Lower Estimates for the Green’s Function

Firstly, we present the Green’s function for BVP (1) (2) which is given in [21]. Let be defined by It is well known that the function is Green’s function for BVP (1) (2) with . In the following lemma we present some properties of Green’s function , see [28] for details. Listed properties will be used later for estimating the upper bound and the lower bound on Green’s function of BVP (1) (2).

Lemma 1 (see [28]). The function defined by (4) satisfies the following conditions:(i), ,  ,(ii),  for ,  .

Let ,  , . For  , define andIt follows from Theorem   in [21] that the function defined by (6) as a series of functions that converge uniformly is the Green’s function for BVP (1) (2) if holds. Furthermore, the function satisfies the following property: provided , where .

The uniform convergence of (6) follows from the fact that , where the operator is defined by the following form: Indeed, the uniform convergence of (6) can be obtained by (see [21, 29]), where is the spectral radius of .

Lemma 2. If holds, then .

Proof. By Lemma 1, for any , we have Hence, we conclude that By induction, one has which implies that Note that . Then by the Gelfand formula, we get

Lemma 3. If holds, then for any ,

Proof. By Lemma 1, for and , we haveThen introducing the above inequality into (6) can lead to It is easy to verify that if , then . Therefore, (14) follows from (6) and (16).

Similar to the proof of Lemmas 2 and 3, we can obtain the following results.

Lemma 4. If holds, then .

Lemma 5. If holds, then for any , where .

By the properties of definite integral and , we assert that that is Thus, we obtain that This means that Lemmas 2–5 is more general and complements many known results.

Combining Lemmas 1 and 3, we obtain the following result.

Theorem 6. If holds. Then for any ,

Theorem 7. If satisfies the boundary conditions (2), If holds, then

3. Existence Theorems

In this section, we shall employ Theorem 6 to investigate the existence results for BVP (1) (2). Let be the Banach space endowed with the maximum norm .

Theorem 8. Assume that there exist such that and where Then BVP (1) (2) has at least one positive solution in .

Proof. Define an operator by where is given by (6). Obviously, is a solution of BVP (1)(2) if and only if is a fixed point of . Moreover, we can show that is completely continuous.
For any given , by (24) and (25), we conclude that and Therefore, . By Schauder’s fixed point theorem, has a fixed point in which implies that BVP (1) (2) has at least one positive solution in .

The following corollaries are direct results of Theorem 8.

Corollary 9. Assume that there exist such that for any , is nondecreasing on , and Then BVP (1) (2) has at least one positive solution in .

Corollary 10. Assume that there exist such that for any , is nonincreasing on , and Then BVP (1) (2) has at least one positive solution in .

Example 11. Consider the BVP After simple computation, we have , .

Let . It is easy to see that (30) and (31) hold when is small enough and is large enough. Then, by Corollary 9, BVP (34) has at least one solution.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (11371221 and 51774197) and Shandong Natural Science Foundation (ZR2018MA011).