Boundary Value Problems of Nonlinear Mixed-Type Fractional Differential Equations

Abstract

In this paper, by means of a fixed point theorem for monotone decreasing operators on a cone, we discuss the existence of positive solutions for boundary value problems of nonlinear fractional singular differential equation. The proof of the main result is based on Gatica–Oliker–Waltman fixed-point theorem. At last, an example is given to illustrate our main conclusion.

1. Introduction

Fractional differential equations with boundary conditions have been studied by many researchers because of their wide applications in many fields of mechanics, engineering, robotics, electrical networks, and so on [1–4]. Many authors consider the applications of fractional differential equations in different fields and obtain some basic results of fractional differential equations (see [5–9] and its references for details). In particular, the singularity of the nonlinear fractional boundary value problem of spatial variables and its references are discussed in literature [10–15].

In the paper [10], by Guo-krasnosl’skil fixed point theorem, the authors proved the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with singular nonnegative nonlinearities and p-Laplacian operators:where , the functions and are nonnegative, and they may be singular at and/or .

In [14], the authors studied the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a sign-changing nonlinearity:where is a positive parameter, for all denotes the Riemann–Liouville derivative of order , and the nonlinearity f may change sign and may be singular at or . The authors also proved relevant conclusion by Guo-krasnosl’skil fixed point theorem.

On the basis of the above literatures, the singular fractional boundary value problem (BVP) in this text,is studied, where ; ; ; ; are continuous with ; and and denote the Riemann–Stieltjes integrals, in which is the function of bounded variation. And, is the fractional derivative of Riemann–Liouville. At , is singular. Let us make the following hypothesis: (B1) is a continuous function (B2) decreases monotonically with respect to x for each fixed t (B3) and , uniformly on compact subsets of (0,1) (B4) for all and

The layout of this paper is as follows: Section 1 is a brief introduction of the research. Section 2 introduces some definitions and basic properties of fractional order derivative and integral. And, the Green function is proposed for the BVP (3) (4). Section 3 is the main results. Section 4 presents an example to illustrate the correctness of our main conclusion.

2. Preliminaries

For convenience, in this section, we present some definitions and lemmas which will be used in the following conclusions.

Definition 1 (see [6]). The Riemann–Liouville type integral of order of a function is defined bywhere and is the Gamma function.

Definition 2 (see [6]). On the basis of Riemann–Liouville, the Riemann–Liouville fractional derivative of order of a function that is considered as antifractional integral of a function is given as

Lemma 1 (see [6]). Let . If we assume , then the fractional differential equation has , as the unique solution, where is the smallest integer greater than or equal to .

Lemma 2 (see [6]). We assume that with a fractional derivative of order of belonging to . Thenfor some , where is the smallest integer greater than or equal to .

Lemma 3 (see [16]). Given , then for ,is the unique solution of the following equation:satisfying the boundary condition (4), where and

Apparently, ,

Obviously, is continuous on .

Lemma 4. Suppose that , then the positive function defined in Lemma 3 is nondecreasing on .

3. Main Conclusions

To simplify the proof of our main results, in this section, we firstly present and prove some lemmas.

Lemma 5. Let and is increasing on . So, the Green function of equation (10) satisfies the following:(i), when (ii), for , where and

Proof. (i)For , For ,(ii)If , we have If , we have

Assume that is a Banach space that the supremum norm is . Let . It is easy to see that is a cone in . For , let , where is given in Lemma 5. In what follows, a set and an integral operator are given, respectively, as

Since is singular at , it suffices to define as above. Moreover, note that, for , there exists such that for all . We get for , because decreases with respect to . Therefore,

Likewise, is decreasing with respect to .

Lemma 6 (see [17]). Assume that is a Banach space and is a normal cone on . And, is a subset in when with ; therefore, . Assume that is a continuous and decreasing mapping which is compact on every closed order interval contained in , and suppose there is an such that is defined and are orders comparable to ; Therefore, has a fixed point in in case that either(1), (2), or(3)Define a complete sequence of iterates and there is so as to with for each .

Lemma 7. is a solution of BVP (3) (4) if and only if .

Proof. First, we find that one direction of the lemma is clearly correct. Now let us prove the other side. Let . Then, , and satisfies BVP (3) (4). Furthermore, by Lemma 5, for each , we haveTherefore, there exists some such that , which implies that . That is, .

Now, we present another lemma which establishes a priori upper bound and a priori lower bound on solutions to prepare for Lemma 6.

Lemma 8. Suppose that satisfies , then there are constants such that for any solution of BVP (3) (4).

Proof. Prove the lemma by the inverse method.
Firstly, let us prove . Suppose a solution sequence of BVP ((3) (4)) such that with . For any solution of BVP ((3) (4)), from equation (20), for , we haveHence, assumptions and yield, for and all :for some . Particularly, , for each , which contradicts .
Secondly, let us prove . And, let us suppose the conclusion is not true. Therefore, there is a solution sequence of BVP (3) (4) such that uniformly on as . Let . From , there exists some such that for and , we have . On the other hand, there is an such that implies , for . Then, we have, for and ,It is easy to see that this conclusion contradicts the hypothesis that uniformly on as . Then, there is an such that .

Next let us prove the main conclusion of the article.

Theorem 1. Assume that satisfies , then BVP ((3) (4)) has at least one positive solution.

Proof. Define byBy conditions , for all ,Next define a sequence of functions , with .
So, is continuous and satisfies , for . Moreover,It is easy to see that effectively eliminates the singularity in at . Next, we defined a sequence of operators , by , . By the Arzela–Ascoli Theorem, is a compact mapping on . In addition, and . From Lemma 6, for all , there is such that . Then, conforms to the conditions of BVP (3) (4). Moreover, for all ,which implies , . By Lemma 8, there exists constants such that for all . Then, we can follow the argument of equation (20) to show that , on , for . Because is a compact mapping, there exists a subsequence of which converges to some . We relabel the subsequence as the original sequence such that as .
And then, Let us prove thatLet and . From , there exists such thatFurthermore, combining equation (26), there exists an such that, for , on , so that on . Thus, for and and for ,Then, for ,Then, , we conclude that for all . Therefore, and for

4. Example

Example 1. We consider the following fractional equations:where andIt is obvious that the hypotheses are satisfied, that is to say, the conditions of Theorem 1 are satisfied. Then, the BVP (35) has at least one positive solution.

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

The authors contributed equally to the manuscript. They read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of Anhui Province of China (2008085QA19), Key Natural Science Research Project of Education Department of Anhui Province (KJ2020A0779, KJ2018A0584, and KJ2019A0875), Science Foundation of Anhui Xinhua University of China (2019zr005 and 2019zr018), and National Natural Science Foundation of China (11601003, 11771001, and 11371027).