Existence and Uniqueness of Positive Solutions for Singular Nonlinear Fractional Differential Equation via Mixed Monotone Operator Method

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result.

In recent years, we can see that differential equations have more and more attention in many fields, like electrical networks, quantum physics, and probability. There are a lot of interesting results from fractional differential equation, Schrodinger equation and k-hessian equations. In [1][2][3][4], the authors studied the solutions of fractional differential equations, and obtained some interesting results. Such as uniqueness of iterative positive solution, existence of multiple positive solutions or maximum and minimum solutions. In [5][6][7][8][9], the authors studied the solutions of Schrödinger equation. From these paper, there have several results about existence of infinitely solutions, existence and nonexistence of blow-up solutions, existence and asymptotic properties of solutions or existence and nonexistence of entire large solutions. In [10][11][12], the authors considered existence of blow-up radial solutions of the k-hessian equations or convergence analysis and uniqueness of blow-up solutions of the k-hessian equations. Inspired by these conclusions,we realize that the questions about differential equations are very interesting and varied. So we focus on fractional differential equation. The authors in [13] studied the following fractional differential equation: where n − 1 < α ≤ n, n > 3, D α 0 + is the Riemann-Liouville fractional derivative, and f : ½0, 1 × ½0, ∞Þ × ½0, ∞Þ ⟶ ½0, ∞Þ is a continuous function.
In [15], Liu et al. investigated the iterative positive solutions for the following singular nonlinear fractional differential equation with integral boundary conditions.
Motivated by the abovementioned papers, the purpose of this article is to study the existence and uniqueness of positive solution for FBVP (1). Obviously, the problem in our article is more general. We not only consider the nonlinear term containing the derivative term, but also study the nonlinear term containing an operator. And we do not need the linearity of operator in nonlinear term. It may be a nonlinear operator.
Comparing with the results in [17], boundary value problem (1) has a more general form. First, we discuss the singularity. It means that p, q are allowed to be singular at t = 0, 1, f may be singular at t = 0, 1 and x = y = 0, and g may be singular at t = 0. Second, we not only consider the derivative term but also consider the operator term, where the operator can be linear or nonlinear. Especially, when pðtÞ = qðtÞ = 1 and Hu = 0, we can see that the problem in [16] is a special case of problem (1). Comparing with [13], firstly, when kðuÞ = 0, β = 0, pðtÞ = qðtÞ = 1, and HuðtÞ = uðtÞ, then our problem (1) reduced compared to the problem in [13]. Third, in [13], the author did not consider the singularity, but in problem (1), we consider the singularity. Compared with [14], first, our method is different from that of [14]. We use the fixed point theorem for the sum of operators rather than the fixed point theorem of cone expansion and compression. Second, f ðt, x, yÞ not only has three variables but also is singular for both time and space variables, and gðt, x, yÞ also has three variables. Compared with [15], we consider the derivative term and the operator term. Our results extended and improved some existing results, such as [13][14][15][16].
The rest of the paper is organized as follows: in Section 2, we give some preliminaries and lemmas that will be used in our main result. In Section 3, we study the existence and uniqueness of a positive solution of the BVP (1). In Section 4, an example is provided to demonstrate our main results.

Preliminaries and Lemmas
In this section, we present some notations, definitions and lemmas that will be used in the paper.
Let ðE, k⋅kÞ be a Banach space, and θ is defined as the Putting P ∘ = fx ∈ Pjx is an interior point of Pg, a cone P is said to be solid if P ∘ is nonempty. In this paper, suppose that ðE, k⋅kÞ is a Banach space partially ordered by a cone P ⊂ E, that is, x ≤ y if and only if y − x ∈ P. A cone P called normal if there exists a constant N > 0 such that for all x, y ∈ E, θ ≤ x ≤ y implies kxk ≤ Nkyk, and the smallest such N is called the normality constant of P.
For all x, y ∈ P, we denote the notation x~y if there have constants λ, μ > 0 such that λy ≤ x ≤ μy. Clearly,~is an equivalence relation. Given h > θ, we denote by P h the set P h = fx ∈ Ejx~hg. And it is easy to know that P h ⊂ P.
Definition 1 (see [18]). An operator A : P × P ⟶ P is said to be a mixed monotone operator if Aðx, yÞ is increasing in the first component and decreasing in the second Journal of Function Spaces Definition 2 (see [19]). Let D = P or D = P ∘ and α be a real Definition 3 (see [19]). An operator B : P ⟶ P is said to be subhomogeneous if it satisfies Definition 4 (see [20]). The Riemann-Liouville fractional integral of order α > 0 of a function y : ð0, ∞Þ ⟶ ℝ is given by provided that the right-hand side is pointwise defined on ð0, ∞Þ.
Definition 5 (see [20]). The Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : ð0, ∞Þ ⟶ ℝ is given by where n = ½α + 1, where ½α denotes the integer part of the number α, provided that the right-hand side is pointwise defined on ð0, ∞Þ.
From the definition of the Riemann-Liouville derivative, we can obtain the statement.
In the following, we present the Green's function of FBVP (1).
Lemma 8 (see [16]). Let h ∈ C½0, 1, then the unique solution of the linear problem is given by where Lemma 9 (see [16]). The Green's function Gðt, sÞ defined by (15) satisfies the following properties: (3) For any t, s ∈ ½0, 1, there is In [17], the operator equation Aðx, xÞ + Bðx, xÞ = x was studied, where A, B are mixed monotone operators, and for all t ∈ ð0, 1Þ, ψðtÞ ∈ ðt, 1, there have Aðtx, t −1 yÞ ≥ ψðtÞ Aðx, yÞ, Bðtx, t −1 yÞ ≥ tBðx, yÞ. Based on the result in [17], we get the following lemma: Lemma 10. Let P be a normal cone in E, and let A, B : P × P ⟶ P be two mixed monotone operators and C : P ⟶ P is a decreasing operator, suppose that (A1) for all t ∈ ð0, 1Þ, there have ψðtÞ ∈ ðt, 1 such that Then (2) There exist u 0 , v 0 ∈ P h and r ∈ ð0, 1Þ such that we have x n ⟶ x * and y n ⟶ x * as n ⟶ ∞ Proof. Now we define the operator T : P × P ⟶ P by So it is easy to know that T is a mixed monotone operator. Next, we will prove that Tðh, hÞ ∈ P h and Tðtx, t −1 yÞ ≥ φðtÞTðx, yÞ, where φðtÞ ∈ ðt, 1, Then according to the results of [21], the conclusions of Lemma 10 holds.
Thus, by the results of Zhai and Zhang in [21], we know what the conclusions of Lemma 10 hold. 4 Journal of Function Spaces

Main Results
For convenience, in this section, we set Clearly, E is a Banach space with the norm Now, we let the cone P be defined by There have P ⊂ E, and P is a normal cone. And, the order relation in E is given by x ≺ y if xðtÞ ≤ yðtÞ, D β 0+ xðtÞ ≤ D β 0+ yðtÞ. From Lemma 8, we get that the unique solution of the problem (1) satisfies the following equation: where Gðt, sÞ is the Green's function given in (15).  Then problem (1) has a unique positive solution u * ∈ P h , where hðtÞ = t α−1 . For any initial value u 0 , v 0 ∈ P h , there are two iterative sequences fu n g, fv n g for approximating u * , v * , that is, u n ⟶ u * , v n ⟶ v * , as n ⟶ ∞, where Proof. We define operators A, B, C as follows: Cv Furthermore, for u, v ∈ P, we obtain that Clearly, u is the solution of problem (1) if and only if u = Aðu, uÞ + Bðu, uÞ + Cu. So, if we can prove that the operators A, B, C satisfy all conditions in Lemma 10, then we will obtain the conclusions in Theorem 11.
First, we will prove that A, B are well defined. Evidently, C is well defined, so we omit it.