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In this paper, we investigate singular Hadamard fractional boundary value problems. The existence and uniqueness of the exact iterative solution are established only by using an iterative algorithm. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have also been derived.

Fractional differential operators play an important role in describing phenomenons in many fields such as physics, chemistry, control, and electromagnetism [

The fractional order diffusion equation

In the past ten years, fractional differential equations have been considered in many papers (see [

By using the Krasnoselskii-Zabreiko fixed point theorem, Yang [

In [

Most of the above works required the associated integral operators to be completely continuous because fixed point theorems could be applied. Furthermore, the uniqueness of positive solutions was rarely investigated while the existence and multiplicity of positive solutions were investigated widely.

Inspired by the above results, in this work, we study the existence and uniqueness of positive solutions for the following boundary value problem:

In this work, only by using the monotone iterative technique, we aim to establish the unique positive solution for problem (

Throughout this work, we assume that the following conditions hold without further mention.

It is easy to verify that if

The way to attack this new problem follows a scheme similar to that used in [

In this section, we present some basic concepts and conclusions needed in the proof of our main results.

The Hadamard fractional integral of order

The Hadamard fractional derivative of order

Specifically,

Suppose that

As argued in [

Take

For

(i)

(ii)

(iii)

(iv)

From

In this paper, we will work in the Banach space

Define a set

Assume

Define the operator

In fact, for

First, there exist two constants

We construct two iterative sequences as follows:

From

Assume

(i)Problem (

(ii)For any initial value

Let

(i) It follows from Theorem

(ii) For any

At the same time, (

We just investigate a simple form of boundary value problems for Hadamard differential equations. We can easily apply the monotone iterative technique to multipoint or multistrip boundary value problems.

Suppose that

(1)

(2) If

(3) If

(4) If

The above four facts can be verified directly. This indicates that there are many kinds of functions which satisfy the conditions

Consider the following boundary value problem:

First,

For any

Obviously

Problem (

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

The authors declare that they have no conflicts of interest.

This paper is supported by the Natural Science Foundation of China (11571197) and the Science Foundation of Qufu Normal University of China (XJ201112).