In this article, by means of fixed point theorem on mixed monotone operator, we establish the uniqueness of positive solution for some nonlocal singular higher-order fractional differential equations involving arbitrary derivatives. We also give iterative schemes for approximating this unique positive solution.

1. Introduction

We are interested in investigating the existence and iterative schemes of the unique positive solution for the following fractional differential equation (FDE): where is the standard Riemann-Liouville derivative, , , is a positive parameter with , , is nonnegative, permits singularities at , and

In recent years, fractional calculus and fractional models play more and more significant role in describing a wide spectrum of nonlinear phenomena in natural sciences, engineering, economics, biology, and signal and image processing; see books and monographs [13] and references [432] to name a few. More and more attention has been paid to nonlocal problem of fractional differential equation because of its wide applications to applied mathematics and physics such as chemical engineering, underground water flow, heat conduction, thermoelasticity, and plasma physics. Under different conjugate type integral conditions such as no parameters, only one or two parameters involved in boundary conditions, [816, 33, 34] investigate the existence, uniqueness, and multiplicity of positive solutions for FDEs when is either continuous or singular. Very recently, in [16], we give two uniqueness results of solution for the following FDE: where , , and is nonnegative. The whole discussion is based on the Banach contraction map principle and the theory of -positive linear operator.

Motivated by the above papers, in this article we aim to obtain the uniqueness result of solution for BVP (1) by means of theory on mixed monotone operator. This article admits some new features. First, compared to [616] the problem considered in this paper performs more general form since another parameter is contained in boundary conditions. Second, the nonlinearity is not only singular on space variable but also relative with -order derivative of an unknown variable . Finally, the method used in this paper is different from that in [16].

2. Preliminaries and Several Lemmas

Definition 1 (see [3]). The Riemann-Liouville fractional integral of order of a function is given by provided the right-hand side is pointwise defined on .

Definition 2 (see [3]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where denotes the integer part of the number , provided that the right-hand side is pointwise defined on .

Lemma 3 (see [23]). (1) If , then (2) If , then

Let . According to the definition of Riemann-Liouville derivative and Lemma 3, we have Let Then, by (7), BVP (1) can reduce to the following modified fractional boundary value problems (MFBVP): In a similar way, we can transform (8) into the form (1). Thus, MFBVP (8) is equivalent to BVP (1).

Lemma 4. Let . Then BVP (1) can transform to (8). In addition, if is a positive solution for (8), then is a positive solution for BVP (1).

Proof. Substituting into (1), we know from Lemma 3 and (7) that Considering this together with the boundary value conditions, we have and Thus, (1) is converted to (8).
Additionally, suppose that is a positive of (8). Let . Then, by Lemma 3, one gets The boundary condition , together with (9) implies that That is to say, is a positive solution of BVP (1).

Remark 5. Direct computation implies that The following two lemmas are isomorphic forms of those in [16].

Lemma 6 (see [16]). Assume that Then for any , the unique solution of the boundary value problems solves where Here, is called the Green function of BVP (15). Obviously, is continuous on

Clearly, is a positive solution of BVP (8) if and only if is a solution of the following nonlinear integral equation:

Lemma 7 (see [16]). The functions and given by (18) and (17), respectively, admit the following properties:
, here ,

Let be a normal cone in a Banach space and nonzero element with A subset of cone is given as follows:

Definition 8 (see [35]). Assume that . is said to be mixed monotone if is nondecreasing in and nonincreasing in , i.e., if implies for any , and implies for any The element is said to be a fixed point of if

Lemma 9 (see [17]). Suppose that is a mixed monotone operator and there exists a constant , , such that Then has a unique fixed point . Moreover, for any , satisfy where where, is a constant, , and dependent on .

3. Main Result

Throughout this paper, we adopt the following assumptions:

, where, is continuous, is nondecreasing on , and is nonincreasing on .

There exists such that for

Remark 10. According to , for any , one has

Theorem 11. Assume that (H1)(H3) hold. Then, the BVP (1) has a unique solution , and there exists a constant such that Moreover, for any , we construct a successive sequence and we have as ; the convergence rate is where is a constant with and is dependent on .

Proof. Let , and we define where where We consider the existence of positive solution for (8). For any , define an operator as follows: It is clear that is a positive solution of BVP (8) if and only if is a fixed point of the operator .
First, we are in position to show that is well defined. By and Remark 5, for any we have and Considering the fact that , we have from (34), , and Remark 5 that and Thus, it follows from (36), (37), Lemma 7, and that This means that is well defined.
On the other hand, we can easily see from (34) and (40) that At the same time, by (38), (39), and Lemma 7, we know that It follows from (40)-(42) that is well defined.
Next, we shall prove that is a mixed monotone operator. To this end, let with . For any , it follows from together with the monotonicity of the operator that which implies that is nondecreasing in for any . In a similar manner, for any with , we have This is to say, is nonincreasing in for any . Thus, is a mixed monotone operator.
Finally, by , one has which means that (23) in Lemma 9 is satisfied. Hence, Lemma 9 guarantees that there exists a unique positive solution for BVP (8). Let ; then is the unique positive solution for BVP (1).
In addition, for any , by Lemma 9, constructing a successive sequence and , then and we have as ; the convergence rate is where is a constant with and is dependent on . Moreover, by Remark 5, we get

4. An Example

Example 1. Consider the following fractional differential equation integral boundary value problems: where , By simple computation, we have . It is easy to know that holds for At the same time, for any and , one has Thus, is valid for . Notice that , and one gets and which implies that is also satisfied. Thus, by Theorem 11 we know that BVP (50) has a unique positive solution.
In addition, for any initial , we construct a successive sequence and ; then and we have as ; the convergence rate is where is a constant with and is dependent on .

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors read and approved the final manuscript.


The project is financially supported by the National Natural Science Foundation of China (11571296, 11571197, 11371221), a Project of Shandong Province Higher Educational Science and Technology Program (J18KA217), the Foundation for NSFC Cultivation Project of Jining Medical University (2016-05), the Natural Science Foundation of Jining Medical University (JY2015BS07, 2017JYQD22), and the Natural Science Foundation of Shandong Province of China (ZR2015AL002).