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.

In this paper, we discuss the existence and computing method of positive solutions of the

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. [

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 [

This paper is organized as follows. In Section

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

The Riemann-Liouville fractional derivative of order

The Riemann-Liouville fractional integral of order

In [

Let

Obviously, for

The following properties of Green’s function

Green’s function

Firstly, we prove that (1) is true. In fact, for all

Next, we show part (2). In fact, on the one hand, from (1), we know that

In this section, we discuss the existence and iteration of positive solutions for the problem (

Since

Now, we conclude that

For notational convenience, we denote

Suppose that

The iterative schemes in Theorem

We divide the proof into four steps.

In fact, if

Obviously,

Obviously,

Of course,

Assume that

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

Consider the fractional boundary value problem

Obviously, the problem (

Moreover, the two iterative schemes are

After direct calculations, we get

Consider the problem

In this problem,

Moreover, the two iterative schemes are

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

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).