Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

Abstract

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation and where and are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

1. Introduction

In this paper, we devote to the investigation of the following nonlinear fractional differential equationwhere and are the standard Riemann-Liouville derivatives, , . The nonlinear term is continuous and may be singular on both and ; permits sign-changing.

Differential equation models can describe many nonlinear phenomena in applied mathematics, economics, finance, engineering, and physical and biological processes [1, 2]. In recent years, there has been a great deal of research on the existence and/or uniqueness of solution in studying FDEs nonlocal problems for their wide applications in modeling some important physical laws (see [3–16], for instance).

In [3], the authors were concerned with the existence of monotone positive solutions to the following fractional-order multipoint boundary value problems where , , , and . The authors obtained the existence of monotone positive solutions and establish iterative schemes for approximating the solutions.

In [4], the authors investigated the existence of positive solutions of the following fractional differential equation multipoint boundary value problems with changing sign nonlinearitywhere is a positive parameter, , , and may change sign and may be singular at . By employing the cone expansion and compression fixed point theorem, the existence of positive solutions was obtained.

In [5], the authors established the uniqueness of a positive solution to the following higher-order fractional differential equation: where is continuous and may be singular at , and is continuous and may be singular at and/or . By using the fixed point theorem for the mixed monotone operator, the existence of unique positive solutions for above singular nonlocal boundary value problems of fractional differential equations is established. The nonlinear term in [11] is nonnegative.

In [11], the authors studied the existence of positive solutions for the following nonlocal fractional-order differential equations with sign-changing singular perturbation. where , , is continuous and may be singular near the zero for the third argument, and may be sign-changing. By means of Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established, respectively, for the cases where the nonlinearity is positive, negative, and semipositone.

Motivated by the work mentioned above, we consider the fractional-order singular nonlocal BVP (1) and establish the existence results of positive solutions for (1). The main tools used in this paper are Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem. For the concepts and properties about the cone theory and the fixed point theorem, one can refer to [17–21].

The rest of this paper is organized as follows: in Section 2, we present some useful preliminaries and lemmas. The main results are given in Section 3 and Section 4, in which the singular cases with respect to the time variables and space variables are discussed, respectively.

2. Preliminaries and Some Lemmas

Definition 1 (see [1, 2]). The Riemann-Liouville fractional integral of order of a function is given byprovided the right-hand side is pointwisely defined on .

Definition 2 (see [1, 2]). The Riemann-Liouville fractional derivative of order of a function is given byprovided the right-hand side is pointwisely defined on , where , denotes the integer part of the number .

Lemma 3 (see [1, 2]). The unique solution of the following linear Riemann-Liouville fractional differential equation of order iswhere , denotes the integer part of the number .

Lemma 4 (see [1, 2]). If and , thenwhere .

The similar proof of the following three lemmas can be traced to [5, 12]; in order to be convenient for readers to read, we now give the detailed process of proof for Lemma 5; the proofs for other two lemmas are omitted here.

Lemma 5. Let and , then the unique solution of the problemcan be expressed uniquely bywhere

Proof. By Lemma 4, the solution of (13) can be written asIt follows from that , i.e.,thuswhich, together with the boundary value condition , implies thati.e.,thus

Lemma 6. If , then the function satisfies the following conditions:
(1) is a nondecreasing function on ;
(2) there exist , such that , for any , where , .

Remark 7. It is easy to prove that .

Lemma 8. The function defined by (13) has the following properties:(1) for any ;(2) for any ;(3) for any ;(4), where denotes the integer part of the number ,

Set , then (1) can be transformed into the following form:From Lemma 5, we know that the solution of (22) satisfies

Lemma 9 (see [17, 18]). Suppose that is a Banach space and is a bounded convex closed set, the operator is completely continuous, then has one fixed point on .

Lemma 10 (see [20] (Guo-krasnosel’skii fixed point theorem)). Let and be two bounded open sets in Banach space such that and , a completely continuous operator, where denotes the zero element of and a cone of . Suppose that one of the following conditions(i) and ;(ii) and holds. Then has at least one fixed point in .

3. Main Result I: Is Singular with Respect to the Time Variables

Let , , then is a Banach space. Setwhere denotes the integer part of the number . Then is a positive cone of . For convenience, we list some conditions which will be used in this section.

() For any ,where , on , is continuous and increasing on , is continuous and decreasing on .

()where , are the positive part and negative part of , respectively.

() There exists , such thatwhere is the solution of the following linear equationi.e., .

() There exists such thatuniformly holds for , where

For any , letand define operatorFrom condition and , it is easy to know that is well defined.

Lemma 11. is a completely continuous operator.

Proof. For any , it follows from Lemma 8 thatwhich deduce that , i.e., .
Let be a bounded set, i.e., there exists such that for any , thenthereforefor any , which implies that is uniformly bounded.
From , the absolutely continuity of integral and the uniformly continuity of on , we know that for any , , such thatandfor any with .
(37)-(39), together with Lemma 8, imply thatfor any with and any , where , which deduces that is equicontinuous on . Thus, according to Ascoli-Arzela theorem, we know that is a relatively compact set, and that is a completely continuous operator.

Theorem 12. Suppose that hold, then the FVP (1) has at least one positive solution.

Proof. For any , where , by (32), Lemma 8 and condition , one can get thati.e., .
By condition , , such thatfor any and any . Choose such thatwhereFor any , where . Becauseso we haveand then for ,follows from (43) and the definition of cone .
From (42), (43), and (47), one can obtain thati.e., .
It follows from Lemma 10 that has at least fixed point , i.e., satisfiesSet , noticing that are the solutions of BVP (32) and (49), respectively; therefore we can conclude that is a positive solution of (22). Let , then is a positive solution of the nonlinear fractional differential equations (1).

4. Main Result II: Is Singular with Respect to Both the Time Variables and the Space Variable

In this section, we always suppose that the following condition holds.

() is continuous, there exist and , for such thatfor any , where

Setit follows from Lemma 5 that is the solution of the following linear equation DenoteClearly,

Theorem 13. Suppose that the condition (H₅) holds and . Then the FVP (1) has at least one positive solution.

Proof. Because , so we can choose large enough such thatwhere . In fact, sincethere exists such that for any , i.e.,for any . If , then from , one can get that there exists such that for any , i.e.,for any . By (57) (58), we can choose such that satisfies (55). SetFor any , from (23) we haveIt follows from (60), (61), and () thati.e.,And thenwhich deduces that the operator is well defined.
Now, we shall prove that . For , it is easy to see that , and by (55) (63) we can obtain thati.e., .
Next, let us prove that is completely continuous.
For any , and The continuity of deduces thatand it follows from (63) thatBy using the Lebesgue dominated convergence theorem, we obtain thatand this implies that is a continuous operator.
Now, we shall prove that is compact. For any , , which deduces that , for , i.e., is uniformly bounded.
Since , it is also uniformly continuous on , and then for any , , s.t., for any with , we always haveThus, one can obtain by virtue of (63) (70) thatfor any with and any , which shows that is equicontinuous. Thus, Arzela-Ascoli theorem guarantees that is completely continuous. Existence of at least one fixed point follows from Lemma 6, i.e., is a positive solution of differential equation (1), which satisfies

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by Shandong Provincial Natural Science Foundation of China (no. ZR2016FM10) and National Natural Science Foundation of China (no. 11571197).