An Iterative Algorithm for Solving n-Order Fractional Differential Equation with Mixed Integral and Multipoint Boundary Conditions

In this paper, we consider the iterative algorithm for a boundary value problem of n-order fractional differential equation with mixed integral and multipoint boundary conditions. Using an iterative technique, we derive an existence result of the uniqueness of the positive solution, then construct the iterative scheme to approximate the positive solution of the equation, and further establish some numerical results on the estimation of the convergence rate and the approximation error.

Among various techniques of dealing with differential equations, upper and lower solutions' method and fixed-point methods have been verified to be the most efficient approaches. However, the efficiency of those methods depends essentially on the monotonicity and compactness of the operator. So, how to overcome the requirement of the monotonicity and compactness is a huge challenge, since such qualities are not naturally available and difficult to prove which lead to the complexity to solve some BVP, especially for the fractional nonlinear boundary value problems.
Different from [96], in this paper, we develop the new iterative algorithm to overcome the requirement of the compactness for the nonlinear operator. Our work has three major features. Firstly, equation (1) possesses a more general nonlinear term which has two space variables and the boundary condition is a mixed integral and multipoint boundary condition. Secondly, the nonlinear term can be singular in the time variables and the second space variables. In the end, our results are more refined, that is, we not only construct a new iterative process which can perform from any initial value but also obtain the uniform convergence of the iterative sequences; at same time, the estimation of the convergence rate and the approximation error are also given, which imply that more strong results are established under the relatively weaker condition than that of [96]. e paper is organized as follows. In Section 2, we recall some definitions and lemmas. In Section 3, the unique of positive solutions to BVP (1) are obtained. Finally, in Section 4, an illustrative example is also presented.

Preliminaries
In this section, we first list some notations and recall related definitions and lemmas to be used in our proofs later.
Definition 1 (see [5]). Let p > 0, the Riemann-Liouville standard fractional integral derivative of order p > 0 of a function f: (0, ∞) ⟶ R is given by where n � [p] + 1, [p] denotes the integer part of the real number p.
Lemma 1 (see [5]). Suppose that n − 1 < α ≤ n with n ≥ 3 and h ∈ L 1 [0, 1]; then, the boundary value problem is given by where ξ Lemma 2 (see [5]). For all t, s ∈ [0, 1], the functions G(t, s) and H(t, s) in Lemma 1 satisfy the following properties: (1) G(t, s) and H(t, s) are continuous and nonnegative In this paper, we will work in the space E � C[0, 1]. Define a set P in E and an operator T: E × E ⟶ E as follows:

Main Results
Before claiming our main results, we first introduce the following notations: (1) has unique positive solution x * (t) ∈ P, and there exists a constant 0 < l < 1 satisfying Proof. Firstly, it is easy to know that x * is the solution of the BVP (1) if and only if x * satisfies T(x * , x * ) � x * . Next, it follows from (H 1 ) that the operator T: P × P ⟶ P is nondecreasing with respect to x and nonincreasing with respect to y; thus, by (H 1 ), (H 2 ), and (H 3 ), for any (x, y) ∈ P × P and t ∈ (0, 1), there exist two constants 0 < l x < 1, 0 < l y < 1 such that Denote l * x � min l x , l y ; then, we have Consequently, Complexity Let l Tx � min en, there exists a constant 0 < l Tx < 1 such that which implies that the operator T: P × P ⟶ P is well defined. According to the Arzela-Ascoli theorem, it is easy to know that T: P × P ⟶ P is completely continuous. Now, take h(t) � t α− 1 , then (h, h) ∈ P × P, it follows from (12) and (13) that T(h, h) ∈ P. us, by the definition of P, there exists a constant 0 < l Th < 1 such that Take and let x 0 � λh(t), Now, construct the following iterative sequence: x n � T x n− 1 , y n− 1 , y n � T y n− 1 , x n− 1 , n � 1, 2, . . .

Complexity
On the other hand, it follows from x 0 ≤ y 0 and the fact of T being nondecreasing with respect to second variable and nonincreasing with respect to third variable that x 1 ≤ y 1 .
us, according to the above fact, we have (20) holds. Notice that, for any nature number n, where c � λ 2 . So, for any nature numbers n and n * , we have which implies that there exists x * ∈ P such that uniformly on (0, 1). By the same method, we can also prove that uniformly on (0, 1). In view of the continuous of T, take the limits in x n � T(x n , y n ), we have x * � T(x * , x * ). So, x * is a positive solution of BVP (1). Since x * ∈ P, for any t ∈ (0, 1), there exists a constant l ∈ (0, 1) such that holds. Finally, we show that the uniqueness of the positive solution. Let y * (t) be another positive solution of BVP (1); then, for any t ∈ (0, 1), there exists a constant m ∈ (0, 1) such that Taking λ defined in (17) be small enough such that λ < m. So, According to T(y * , y * ) � y * , using the nondecreasing of T, we can show that x n (t) ≤ y * (t) ≤ y n (t), t ∈ (0, 1).
Taking limits to the both sides of (29), we have x * � y * . It follows that the solution of BVP (1) is unique. e proof of eorem 1 is completed. In the following, we consider the error estimation between unique solution and iterative value. en, for any initial value z 0 ∈ P, there exists a sequence z n (t) that uniformly converges to the unique positive solution x * (t) with the following error estimation: where c ∈ (0, 1) is determined by z 0 .
Proof. By eorem 1, we know that the positive solution x * is unique. For any z 0 ∈ P, there exists a constant l z 0 ∈ (0, 1), such that Similar to eorem 1, we can take λ < min l z 0 , l (1/1− μ) Th as a fixed number. It follows that Let z n � T(z n− 1 , z n− 1 ), n � 1, 2, . . .; then, by monotonicity of the operator T, we have us, it follows from mathematical induction that Take limits for the above inequality, we get that z n (t) uniformly converges to the unique positive solution x * of BVP (1). Using (23), we can now derive the error estimation (30), which implies that the error estimation is the same order infinitesimal of (1 − c μ n ), where c � λ 2 and determined by z 0 . is completes the proof of eorem 2.

Example
Let us illustrate the main results with an example. Example 1. Let α � (7/2), m � 4, η 1 � (1/3), η 2 � (2/3), β 1 � (3/2), β 2 � 4, c 1 � (5/2), and c 2 � 2. We consider the following BVP: x(s)ds where For any k ∈ (0, 1), take μ � (1/4), and it is easy to verify that f t, kx, k − 1 y ≥ k μ f(t, x, y). (36) According to the expression of f and the above inequality, it follows that (H 1 ) and (H 2 ) are held. In addition, So, all of the assumptions of eorem 1 are satisfied. As a result, BVP (35) has a unique positive solution x * and for any initial value x 0 ∈ P, and the successive iterative sequence x n (t) is generated by and uniformly converges to the unique positive solution x * on (0, 1). We also obtain the error estimation where c ∈ (0, 1) is a constant and determined by the initial value x 0 . Moreover, for any t ∈ (0, 1), there exists a constant l ∈ (0, 1) which satisfies that Remark 1. According to the definition of f(t, x, y), let f(t, x, x) � a(t)x (1/8) + b(t)x − (1/5) ; then, f does not have monotonicity with respect to x. us, the iterative process in some previous work such as [31,33,35] cannot be performed, which implies our developed iterative technique in this paper can be suitable for a wider range of functions; in particular, even if f(t, x, y) is reduced to only have one space variable f(t, x), our results is more general than those of [96].

Result and Discussion
In this paper, we obtain the results of the existence solutions of the n-order fractional equation involving mixed integral and multipoint boundary conditions by using a new iterative algorithm. e efficiency of those methods depends essentially on the monotonicity and compactness of the operator. Different from [96], the iterative process does not need to start with the fixed upper and lower solution. We can not only construct a new iterative process which can perform from any initial value but also obtain the uniform convergence of the iterative sequences; at the same time, the estimation of the convergence rate and the approximation error are also given, which imply that the more strong results are established under the relatively weaker condition than that of [96].

Data Availability
e calculating data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions
e study was carried out in collaboration with all authors. All authors read and approved the final manuscript.