Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives

This paper studies the existence and computingmethod of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applyingmonotone iterativemethods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.

Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering, and biological sciences.Recently, there have been many papers dealing with the solutions or positive solutions of boundary value problems for nonlinear fractional differential equations.We refer the reader to the papers of Agarwal et al. [1], Ahmad and Sivasundaram [2], Ahmad and Nieto [3], Babakhani and Daftardar-Gejji [4], Bai and Sun [5], Bai et al. [6], Bai and Qiu [7], Caballero et al. [8], Delbosco and Rodino [9], Graef et al. [10], Jiang and Yuan [11], Lakshmikantham and Vatsala [12], Liang and Zhang [13], Qiu and Bai [14], Tian and Liu [15], Wang et al. [16], Yang and Chen [17], Yuan et al. [18], Zhang [19], Zhang and Han [20], and Zhang et al. [21,22] and the references therein.Nonlinear fractional differential equations with two-point boundary conditions (2) have been studied by several authors.For example, in [23], Goodrich studied a fractional differential equation of the form with boundary conditions (2), where  : The author obtained Green's function of the problem and proved that Green's function satisfied a Harnack-like inequality.By using a fixed point theorem due to Krasnosel'skii, the author established the existence results for at least one positive solution.Graef et al. in [24] found sufficient conditions to guarantee that the following fractional differential equation: Mathematical Problems in Engineering with boundary conditions (2) has at least one or two positive solutions when  is small and large, where  is a parameter,  ⩾ 3,  − 1 < ] ⩽ ,  : [0, 1] × [0, ∞) → R, and  : [0, 1] → R are continuous functions.Zhai and Hao [25] discussed the existence and uniqueness of positive solutions for the following fractional differential equation: with boundary conditions (2), where  : are continuous functions and satisfy some monotonicity conditions.The analysis relies on two new fixed point theorems for mixed monotone operators with perturbation.In [26], Su and Feng studied a fractional differential equation with deviating argument of the form with boundary conditions (2), where  : [0,∞) → [0, ∞), ℎ : [0, 1] → (0, ∞), and  : (0, 1) → (0, 1] are continuous functions.The author obtained novel sufficient conditions for the existence of at least one or two positive solutions by using Krasnosel'skii's fixed point theorem, and some other new sufficient conditions for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams, and so forth. Yuan [27] gave sufficient conditions for the existence of multiple positive solution for the semipositone (, )-type boundary value problems of nonlinear fractional differential equations where  is a parameter,  ∈ ( − 1, ] is a real number and  ⩾ 3, 1 ⩽  ⩽  − 1 is fixed and integer, and  : (0, 1) × [0, ∞) → R is a sign-changing continuous function.
The author derived an interval of  such that for any  lying in this interval, the semipositone boundary value problem has multiple positive solutions.The analysis relied on nonlinear alternative of Leray-Schauder type and the Krasnosel'skii fixed point theorem.
We notice that the methods used in the above papers are all fixed point theorems and the derivatives of unknown function are not involved in the nonlinear term explicitly.Different from the works mentioned above, motivated by the works [28][29][30][31][32], we will use monotone iterative techniques to study the existence and iteration of positive solutions for the problem (1)- (2).We not only obtain the existence of positive solutions, but also give two iterative schemes approximating the solutions.Moreover, this method does not demand the existence of upper-lower solutions.To the best of our knowledge, few authors utilize the monotone methods to study the existence of positive solutions for nonlinear fractional boundary value problems.So, it is worthwhile to investigate the problem (1)-(2) by using monotone iterative techniques.
This paper is organized as follows.In Section 2, we recall some definitions and notations from the theory of fractional calculus and give expression and properties of Green's function.The main results will be given in Section 3. Finally, in Section 4, some examples are included to demonstrate the applicability of our results.

Preliminaries
Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [33,34].
where Γ denotes the Euler gamma function and [] denotes the integer part of number  provided that the right side is pointwise defined on (0, ∞).
Definition 2. The Riemann-Liouville fractional integral of order  is defined as Provided that the integral exists.
In [23], the author obtain Green's function associated with the problem (1)-( 2).More precisely, the author proved the following lemma.

Mathematical Problems in Engineering
Obviously, the fixed points of T are solutions of the problem (1)-(2).
Lemma 5. T is completely continuous and T(K) ⊆ K.
Proof.Since (), (, ) are continuous for ,  ∈ [0, 1] and () is integrable on [0, 1], we get that the operator T is well defined on K.By (13), we get T : K → E. Let A ⊆ K be bounded.Then there exists a positive constant  1 > 0 such that ‖‖ ⩽ Then for  ∈ A, by Lemma 4(1) and ( 22), we have Thus, By means of the Arzela-Ascoli theorem, we claim that T is completely continuous.Now, we conclude that T(K) ⊆ K.In fact, for any  ∈ K, it follows from Lemma 4(2) that (T) () = ∫ which together with ( 29) implies In addition, it follows from Lemma 4(1) that Therefore, (31) and (32) show that T ∈ K; that is, T(K) ⊆ K. Then the proof is completed.
For notational convenience, we denote By (2), we know that Λ > 0 is well defined.
The iterative schemes in Theorem 6 start off with the zero function and a known simple function, respectively.
Proof.We divide the proof into four steps.
Step 4. From V (44) By the induction, we have The proof is complete.
Then the problem (1)-( 2) has at least two monotone positive solutions.

Examples
To illustrate the usefulness of the results, we provide two examples.