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

Abstract

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, 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.

1. Introduction

In this paper, we discuss the existence and computing method of positive solutions of the th-order fractional boundary value problem consisting of the nonlinear fractional differential equation and the two-point boundary conditions where is an integer, is a real number, is the Riemann-Liouville fractional derivative of order , denotes the integer part of real number , and in boundary conditions (2) represents the th (ordinary) derivative of .

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 is continuous. 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: with boundary conditions (2) has at least one or two positive solutions when is small and large, where is a parameter, , , , and 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 and 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 , and 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, is a real number and , is fixed and integer, and 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 [28–32], 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.

2. Preliminaries

Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [33, 34].

Definition 1. The Riemann-Liouville fractional derivative of order of a continuous function is defined to be where denotes the Euler gamma function and denotes the integer part of number provided that the right side is pointwise defined on .

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.

Lemma 3 (see [23]). Let , then the differential equation with boundary conditions (2) has a unique solution where

Obviously, for , is continuous on .

The following properties of Green’s function defined by (12) will be used later.

Lemma 4. Green’s function defined by (12) has the following properties: (1) on , for ,(2) on .

Proof. Firstly, we prove that (1) is true. In fact, for all , if , it is obvious that for . If , from (13), we obtain that On the other hand, by (13), we find From (14) and (15) we get part (1).
Next, we show part (2). In fact, on the one hand, from (1), we know that for any . Thus, is increasing in , so On the other hand, if , then from (13), we have If , then from (13), we have Thus, (17) and (18) show From (16) and (19), we get part (2). Then the proof is completed.

3. Main Results

In this section, we discuss the existence and iteration of positive solutions for the problem (1)-(2). In the sequel, the following conditions hold:(H1) is continuous and .(H2) is nonnegative and .For any , we define . Let the Banach space be equipped with the norm We define a cone by and an integral operator by where Obviously, the fixed points of are solutions of the problem (1)-(2).

Lemma 5. is completely continuous and .

Proof. Since , are continuous for and is integrable on , we get that the operator is well defined on . By (13), we get . Let be bounded. Then there exists a positive constant such that . Denote Then for , by Lemma 4(1) and (22), we have Hence, is bounded. For , one has Thus, By means of the Arzela-Ascoli theorem, we claim that is completely continuous.
Now, we conclude that . In fact, for any , it follows from Lemma 4(2) that which implies that On the other hand, which together with (29) implies In addition, it follows from Lemma 4(1) that Therefore, (31) and (32) show that ; that is, . Then the proof is completed.

For notational convenience, we denote By , we know that is well defined.

Theorem 6. Suppose that and hold. In addition, assume that there exists such that (H3) for , , ;(H4).Then, the problem (1)-(2) has two positive solutions and satisfying . Moreover, there exist monotone increasing sequence and monotone decreasing sequence in such that , , and and , where , and .

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 1. Let . Then .
In fact, if , then ; thus, By the conditions and , we have Thus, by the definition of and Lemma 4(2), for , we get Then (36) shows that ; thus, .
Step 2. Let . Then is increasing; there exists such that , and is a positive solution of the problem (1)-(2).
Obviously, . Since , we have . Since is completely continuous, we assert that is a sequentially compact set. Since , we have It follows from that is increasing; then Thus, by the induction, we have Hence, there exists such that . Applying the continuity of and equation , we get . Moreover, because the zero function is not a solution of the problem (1)-(2), thus, . It follows from the definition of the cone that we have . That is, is a positive solution of the problem (1)-(2).
Step 3. Let . Then is decreasing; there exists such that , and is a positive solution of the problem (1)-(2).
Obviously, . Since , we have . Since is completely continuous, we assert that is a sequentially compact set. Since , by Lemma 4(2), , and , for , we have Thus, we obtain that So by , we have By the induction, we have Hence, there exists such that . Applying the continuity of and equation , we get . Thus, is a nonnegative solution of the problem (1)-(2). Moreover, the zero function is not a solution of the problem (1)-(2). Thus, , it follows from the definition of the cone that we have ; that is, is a positive solution of the problem (1)-(2).
Step 4. From , we have By the induction, we have The proof is complete.

Remark 7. Of course, may happen and then the problem (1)-(2) has only one solution in .

Corollary 8. Assume that and hold. Suppose that is increasing in . Moreover, Then the problem (1)-(2) has at least two monotone positive solutions.

4. Examples

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

Example 1. Consider the fractional boundary value problem

Obviously, the problem (47) fits the framework of problem (1)-(2) with , . In addition, we have set , . Obviously, and satisfy the conditions and . Moreover, It is easy to see that is increasing in and , and Let ; then for any , we have Then conditions and hold. Consequently, applying Theorem 6, the problem (47) has at least two positive solutions and satisfying .

Moreover, the two iterative schemes are

After direct calculations, we get

Example 2. Consider the problem

In this problem, , , , and . Obviously, and satisfy the conditions and . In addition, is increasing with regard to , and and Let ; then for any , by simple computation, we obtain that Therefore, all assumptions of Theorem 6 are satisfied. Thus, Theorem 6 ensures that the problem (52) has two monotone positive solutions and satisfying and and .

Moreover, the two iterative schemes are

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their careful reading of the paper and a lot of valuable suggestions and comments, which greatly improved this paper. This work was supported financially by the Natural Science Foundation of Zhejiang Province of China (LY12A01012).