On Multiple Positive Solutions for Singular Fractional Boundary Value Problems with Riemann-Stieltjes Integrals

Abstract

In this paper, the existence result of at least two positive solutions is obtained for a nonlinear Riemann-Liouville fractional differential equation subject to nonlocal boundary conditions, where fractional derivatives and Riemann-Stieltjes integrals are involved. The nonlinearity possesses singularities on both its time and space variables. The discussion is based on the fixed point index theory on cones.

1. Introduction

We consider the existence of at least two positive solutions for the following nonlinear fractional differential equation with integral boundary value conditions

where for all represents the Riemann-Liouville derivative of order (for ). The integrals involved in the boundary conditions are Riemann-Stieltjes integrals; here, are functions of bounded variation. For more general integral conditions, see [1] and the references therein. The nonlinearity permits singularities at both and .

In recent years, there has been a gradual increase in the investigation of fractional differential equations and systems of fractional differential equations with nonlocal boundary value conditions owing to their better descriptions in very important phenomena in science and technology than in integers. For fractional calculus and its applications in nonlocal problems, see monographs [2–6] and papers [7–25] and the references therein. Very recently, by means of the fixed point theory, principal characteristic value, and fixed point theorems together with height functions, Tudorache et al. [7, 9] investigated the existence of one, two, or three positive solutions for BVP (1). Existence results can be found in [15–19] for the system of fractional differential equations with boundary conditions related to BVP (1).

As is well known, Riemann-Stieltjes integral boundary conditions are more general, and they include many special cases such as two-point, three-point, and other classical integral conditions or a combination of them. As a consequence, boundary conditions in BVP (1) are generalizations of those adopted in the literature [10–14], which can be listed below:

We make an effort in this paper to investigate the existence of multiple positive solutions for BVP (1). By a positive solution of BVP (1), we mean a function satisfying BVP (1) with for all This paper admits the following features. Firstly, compared with [16–18], the nonlinearity in this paper possesses singularities not only on the time but also on the space variables. Secondly, compared with [7–9], super linear conditions on the nonlinearity at 0 and are imposed to obtain the existence of at least two positive solutions. Thirdly, conditions given in this paper are shown to be easy to verify by an example. The main tools employed in this paper are the cone theory and fixed point index theorems on cones.

2. Preliminaries and Several Lemmas

First, we introduce some useful lemmas from [7, 9] which will be used in the latter. For notational convenience, denote

Consider the fractional differential equation where

Lemma 1 (see [7, 9]). If , then the unique solution of problems (4) is given by where for all

Lemma 2 (see [7, 9]). We suppose that . Then, the Green function given by (6) is a continuous function on and satisfies the following inequalities: (i) for all , where(ii) for all (iii) for all whereWe make the following assumptions:
(H₁) for all are nondecreasing functions and
(H₂) .
(H₃) There exist such that for any where (H₄) There exists such that uniformly for , and (H₅) There exists such that uniformly for , and In addition, considering the boundedness of , it is easy to know that where Let be the traditional Banach space of all continuous functions defined on with the maximum norm and the cone Denote for
Define the operator as follows: Clearly,

Lemma 3. Suppose that (H₁)–(H₃) hold; then, for any , is completely continuous.

Proof. For any , we have . It follows from the definition of cone that By (H₂), (H₃), and Lemma 2, we get that which means that is well defined. For any , we have from (23) that Therefore, On the other hand, it follows from Lemma 2 and (25) that Thus, we have proven that maps into .
In the following, we are in the position to show that is completely continuous. First, we prove that is continuous. For with , we have . By (H₁), we know Similar to (22), for , we have Thus, It follows from (27), (29), (H₃), and the Lebesgue-dominated convergence theorem that which means that is continuous.
Next, we will show that is a compact operator. Let be a bounded set in . For any , we have . Similar to (23), we know which means that is bounded uniformly. In the following, we shall prove that is equicontinuous. To this end, we estimate for . Thus, for any and , one has Thus, is equicontinuous. It follows from the Arzelà-Ascoli theorem that is relatively compact, and then, is a compact operator. Hence, is completely continuous.

Lemma 4 (see [26]). Let be a Banach space and a cone in . Assume that is a compact map such that for , (i)If then(ii)If then

3. Main Result

Theorem 5. Assume that (H₁)–(H₅) hold. In addition, there exists such that Then, the BVP (1) has at least two positive solutions and with .

Proof. It follows from Lemma 3 that for any , the operator is completely continuous. In the following, we shall prove that has two different fixed points and in satisfying .
Choose . We know from (H₄) that there exists such that Let For , we have, by the construction of cone , that It follows from (37) to (39) that Thus, Hence, by Lemma 4, By condition (H₄), there exists such that Choose For , we have Consequently, we have from (43) to (45) and (H₅) that Thus, As a consequence, we get On the other hand, for , by (H₃), Lemma 2, and (36), we get i.e., Then, Lemma 4 guarantees that It follows from (42), (48), (51) and the additivity of the fixed point index that Hence, has two distinct fixed points and belonging to and , respectively, with .

4. An Example

Example 1. Consider the following fractional differential equations with nonlocal boundary value problems Conclusion: BVP (53) has at least two positive solutions and with .

Proof. In this problem, for all , . By simple computation, we have . It is clear that (H₁) and (H₂) are satisfied. Furthermore, For any , (H₃) holds for and Thus, (H₃) is verified. Obviously, (H₄) and (H₅) are valid for
Next, we focus on checking (36). Take . By (57), we know that Hence, which implies that (36) holds. Consequently, our conclusion follows from Theorem 5.

5. Conclusions

In this paper, we focus on the existence and multiplicity of positive solutions for a class of a higher-order Riemann-Liouville fractional differential equation with Riemann-Stieltjes integrals. The nonlinearity possesses singularities on both its time and space variables. By means of the fixed point index theory on cones, the existence result of at least two positive solutions is obtained. Conditions imposed on the nonlinearity are shown to be easy to verify by an example.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The project is supported financially by the Project of Shandong Province Higher Educational Science and Technology Program (J18KA217), Supporting Fund for Teachers’ Research of Jining Medical University (JYFC2018KJ015), National Natural Science Foundation of China (11571296, 11571197, and 11871302), Foundation for NSFC Cultivation Project of Jining Medical University (2016-05), and Natural Science Foundation of Jining Medical University (JY2015BS07 and 2017JYQD22).