Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

where D0+ is the standard fractional derivative of order p satisfying 2 < p ≤ 3, and f ðt, x, yÞ may be singular at y = 0, t = 0, 1. Fractional calculus differential equations are an important branch of differential equations. In recent years, it has attracted the interest of many researchers and has become a hot-button issue [1–14]. Compared with the integer order, it has a wider range of applications as it can be used to describespecific problems more precisely, such as the problem in complex analysis, polymer rheology, physical chemistry, electrical networks, and many other branches of science, For specific applications, see [15, 16, 20–28]. In [3], the authors study the following BVP:


Introduction
In this work, we present the uniqueness of a positive solution for the following boundary value problem (BVP for short): where D p 0+ is the standard fractional derivative of order p satisfying 2 < p ≤ 3, and f ðt, x, yÞ may be singular at y = 0, t = 0, 1.
Fractional calculus differential equations are an important branch of differential equations. In recent years, it has attracted the interest of many researchers and has become a hot-button issue [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Compared with the integer order, it has a wider range of applications as it can be used to describespecific problems more precisely, such as the problem in complex analysis, polymer rheology, physical chemistry, electrical networks, and many other branches of science, For specific applications, see [15,16,[20][21][22][23][24][25][26][27][28]. In [3], the authors study the following BVP: They obtained the existence of multiple positive solutions by means of the Gou-Krasnoselskii fixed point theorem. In [4], the authors also study the same BVP. By constructing a special u 0 -positive operator and using its properties, they obtained a unique solution for BVP (2). BVP (1) is more general than the problem in papers [3] and [4] in four aspects. First, the nonlinear term has two space variables and can be singular with respect to the second space variable. Second, the method we used is different from papers [3] and [4]. Through constructing an iterative process which can be from any initial value, only by the iterative algorithm, we can prove that it converges uniformly to the unique positive solution. Third, we calculate the estimation of the convergence rate and the approximation error. Finally, we compare with it [3] and [4]. We lower the conditional constraint on the nonlinear term since we do not need the monotone of the nonlinear term. So the result of this paper is most general, not only did it weaken the restrictions but it also strengthened the conclusions in [3] and [4]. Also, we can show that the main result in [4] is a corollary of our work.
The rest of our presentation is as follows. In Section 2, we recall some definitions and lemmas. In Section 3, we establish the result of the uniqueness of the positive solutions to BVP (1).
Finally, in Section 4, an illustrative example is also presented.

Preliminaries
We first list some definitions and lemmas which will be used later.
Definition 1 (see [5]). Let p > 0, and the Riemann-Liouvill standard fractional integral and the Riemann-Liouvill standard fractional derivative of order p > 0 of a function f : ð0,∞Þ ⟶ R are given by where n = ½p + 1 and ½p denotes the integer part of the real number p and provides that the right side integral is pointwise defined in ½0, ∞Þ.
H 3 . The third condition is as follows: The basic space used in this paper is E = C½0, 1. Define a set P in E as follows: Denote hðtÞ = ð1 − tÞt p−1 . Evidently, ðhðtÞ, hðtÞÞ ∈ P × P, i.e., P × P is not empty.
Let the operator T::

Main Results
In this section, we will prove the existence of the positive solution to BVP (1) and calculate the approximation error and the convergence rate.
where l is a constant that belongs to ð0, 1Þ.
Proof. The solution of BVP (1) coincides with the fixed point of operator T. So our goal is to show that operator T has a unique fixed point in P. We divide our proof in four steps.
Step 1. We verify that operator T is well defined. From H 1 , H 2 , and H 3 , for any ðx, yÞ ∈ P × P and t ∈ ð0, 1Þ, we know that So, operator T is well defined.

Journal of Function Spaces
Step 2. We verify the properties of T. Let and Consequently, we can prove that there exists constants 0 < l Tx < 1 such that Therefore, the operator T : P × P ⟶ P is well defined. It follows from H1 that T is nondecreasing with respect to x and nonincreasing with respect to y.
Step 3. We will establish the existence of a positive solution to BVP (1). Since ðh, hÞ ∈ P × P, from the above steps, we have Tðh, hÞ ∈ P. According to the definition of P, there exists a constant 0 < l Th < 1 such that Taking x 0 = ρh t ð Þ, x n = T x n−1 , y n−1 ð Þ , where ρ is a fixed number satisfying In fact, 0 < ρ < 1. From (19), we get x 0 , y 0 ∈ P and x 0 ≤ y 0 . Moreover, and Combining x 0 ≤ y 0 with T is nondecreasing with respect to x and nonincreasing with respect to y, so we have On the other hand, for any nature number n, denote ρ 2 by c and we have Therefore, for any nature number n and n * , we obtain So, there exists x * ∈ P such that uniformly on ð0, 1Þ. By the same argument, we can also prove that 3 Journal of Function Spaces uniformly on ð0, 1Þ. Since T is continuous, we can take the limits in x n = Tðx n , y n Þ and we get x * = Tðx * , x * Þ. Therefore, x * is a positive solution of BVP (1). Owing to x * ∈ P, for any t ∈ ð0, 1Þ, there exists a constant l ∈ ð0, 1Þ such that holds.
Step 4. We further show its uniqueness. Let y * ðtÞ be another positive solution of BVP (1), then for any t ∈ ð0, 1Þ, there exists a constant r ∈ ð0, 1Þ such that Taking ρ defined in (21) as being small enough such that ρ < r. So In view of Tðy * , y * Þ = y * , by means of the nondecreasing T, we obtain Taking limits to the above inequality, we get x * = y * . Therefore, the solutions of BVP (1) are unique which completes the proof of Theorem 4. Now, we are in a position to construct the successive sequence which converges to the unique solution.
Theorem 5. Suppose conditions H1, H2, and H3 are satisfied. Then, for any initial value u 0 ∈ P, the successive sequence uniformly converges to the unique positive solution x * ðtÞ where the error estimation is the same order infinitesimal of ð1 − c k n Þ, where c ∈ ð0, 1Þ and determined by u 0 .
Proof. According to Theorem 4, notice that the positive solution x * is unique; for any u 0 ∈ P, there exists a constant l ∈ ð0, 1Þ, such that We shall adopt the similar argument as in the proof of Theorem 4; take ρ < l to be a fixed number 0 < ρ ≤ l 1/1−k Th , thus ð36Þ then x n t ð Þ ≤ u n t ð Þ ≤ y n t ð Þ, t ∈ 0, 1 ð Þ: ð37Þ Taking limits to inequality (37), by the same method of the proof to (26), we can show that fu n ðtÞg uniformly converges to the unique positive solution x * of BVP (1), and which means the error estimation is the same order infinitesimal of ð1 − c k n Þ, where c = ρ 2 and determined by u 0 .
The proof is completed.

Example
Let us illustrate the main results with an example.
Example. Take p = ð5/2Þ, pðtÞ = ð1 − tÞ 2 , qðtÞ = t 2/3 , and f ðt, x, yÞ = aðtÞx 1/4 + bðtÞy −1/3 . We consider the following BVP: For any λ ∈ ð0, 1Þ, take k = 1/2, we can prove that f t, λx, λ −1 y À Á ≥ λ k f t, x, y ð Þ: ð40Þ Combined with the expression of f , p, and q, it is easy to see that H1 and H2 are holding. In addition, So all of the assumptions in Theorem 4 are satisfied. Then, BVP (39) has a unique positive solution x * . For any initial value x 0 ∈ P, we can construct the successive iterative sequence fxnðtÞg as follows which uniformly converges to the unique positive solution x * on ð0, 1Þ. The error estimation is the same order infinitesimal of ð1 − c k n Þ, i.e.,