Journal of Function Spaces
Volume 2019 (2019), Article ID 5923490, 6 pages
https://doi.org/10.1155/2019/5923490

Research Article

The Unique Positive Solution for Singular Hadamard Fractional Boundary Value Problems

Jinxiu Mao,¹ Zengqin Zhao,¹ and Chenguang Wang²

¹School of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, China
²Department of Mathematics, Jining University, Qufu, Shandong, 273155, China

Correspondence should be addressed to Jinxiu Mao; maojinxiu1982@163.com

Received 21 February 2019; Accepted 31 May 2019; Published 20 June 2019

Guest Editor: Pedro Garrancho

Copyright © 2019 Jinxiu Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

This paper is dedicated to our advisors

1. Introduction

Fractional differential operators play an important role in describing phenomenons in many fields such as physics, chemistry, control, and electromagnetism [1–9]. They have many applications in fractional differential equations and fractional integral equations. In [10], authors investigate a class of fractional integral equations arising from a symmetric transition model

where and are the left and right Riemann-Liouville fractional integrals of order , respectively, and is the gradient of at .

The fractional order diffusion equation

where and , contains fractional differential operators which can describe different diffusion process.

In the past ten years, fractional differential equations have been considered in many papers (see [11–22]). Most of the works on the topic have been based on Riemann-Liouville type and Caputo type fractional differential equations. By means of fixed point theorems and variational methods, authors obtain at least one or multiple positive solutions for boundary value problems of fractional differential equations. Very recently, more studies have been carried out on the boundary value problems of nonlinear Hadamard fractional differential equations. An important characteristic of Hadamard fractional derivative is that it contains logarithmic function of arbitrary exponent. However, there are few results about this topic (see [23–28]).

By using the Krasnoselskii-Zabreiko fixed point theorem, Yang [26] obtained at least one positive solution for the boundary value problem

where , . was the Hadamard fractional derivative of order . , .

In [28], the authors studied the Hadamard fractional differential equation

with boundary conditions

where denoted Hadamard fractional derivative of order and denoted Hadamard fractional integral of order , . By employing the complete continuity of the associated integral operator and the monotone iterative method, the authors obtained twin positive solutions and the unique positive solution.

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:

where denotes Hadamard fractional derivative of order ; is continuous.

In this work, only by using the monotone iterative technique, we aim to establish the unique positive solution for problem (6). The main contributions of this work are as follows: (a) the nonlinear term can be singular at and ; (b) we do not need the continuity and complete continuity of the associated integral operator; (c) we get the unique positive solution.

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

is continuous.

: For , is nondecreasing in and there exists a constant such that, for ,

It is easy to verify that if , then

2. Preliminaries

The way to attack this new problem follows a scheme similar to that used in [21], with the necessary adaptations that Hadamard fractional derivative contains logarithmic function of arbitrary exponent.

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

Definition 1. The Hadamard fractional integral of order of a function is given by

Definition 2. The Hadamard fractional derivative of order is defined by

where , .
Specifically,

Lemma 3. Suppose that , Then the Hadamard type fractional differential equation

has a unique solution

where

Proof. As argued in [9], the solution of the Hadamard differential equation in (10) can be written as the equivalent integral equation

From , we have Thus (47) reduces to

Using we obtain

Substituting (15) into (14), we obtain

Take and for , . We can prove that have the following properties.

Lemma 4 (see [26]). For , Green’s function satisfies the following properties:
(i) is continuous on and ,
(ii) ,
(iii) ,
(iv)
From , and (ii) we get

In this paper, we will work in the Banach space with the norm

Define a set as follows.

there are constants such that Evidently Therefore, is not empty.

3. The Main Results

Theorem 5. Assume hold. And

Then problem (6) has at least one positive solution

Proof. Define the operator by

We can see easily the equivalence between is a solution of (6) and is a fixed point of .
Claim 1. The operator is nondecreasing.
In fact, for , it is obvious that , For any and , from (17),

and

where and are positive constants satisfying

Thus, it follows that there are constants such that, for ,

Therefore, for any , , is the operator From (19), it is easy to see that is nondecreasing with respect to . Hence, Claim 1 holds.
Claim 2. There exist a nondecreasing sequence and a nonincreasing sequence and there exists such that

uniformly on
First, there exist two constants with since Take and to be fixed numbers satisfying

Obviously,
We construct two iterative sequences as follows:

Then

In fact, from (26), we have and Furthermore,

From and nondecreasing, (28) holds. Let ; then It follows from (7) that

And, for any natural number ,

Thus, for any natural number and , we have

which implies that is a cauchy sequence in So there exists such that

and from (32)

From , we have Combining with nondecreasing on ,

Let ,

which implies is a positive solution of problem (6).

Theorem 6. Assume hold. Then, we have the following:
(i)Problem (6) has unique positive solution in and there exist constants with such that

(ii)For any initial value , there exists a sequence that uniformly converges to the unique positive solution , and we have the error estimation

where is a constant with and determined by ; represents the same order infinitesimal of

Proof. Let be defined in (26) and (27).
(i) It follows from Theorem 5 that problem (6) has a positive solution , which implies that there exists constants and with such that

Let be another positive solution of problem (6). Then there exist constants and with such that

Let defined in (25) be small enough so that and defined in (25) be large enough so that Then

Note that and is nondecreasing; we have

Letting , we obtain that Hence, the positive solution of problem (6) is unique.
(ii) For any , there exist constants and with such that

Similar to (i), take and to be defined by (25) satisfying and Then

Let

Note that is nondecreasing,

Letting , it follows from (33) and (34) that uniformly on .
At the same time, (38) follows from (32).

Remark 7. 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.

Remark 8. Suppose that are nonnegative continuous functions on , which may be unbounded at the end points of . is the set of functions which satisfy the conditions and . Then we have the following conclusions:
(1) , where .
(2) If and , then .
(3) If , then .
(4) If , then .
The above four facts can be verified directly. This indicates that there are many kinds of functions which satisfy the conditions and .

4. An Example

Consider the following boundary value problem:

where , ,

Analysis 1. First, and so holds.
For any , we take and have

Then holds.
Obviously

where . Hence all conditions of Theorem 5 are satisfied, and consequently we have the following corollary.

Corollary 9. Problem (47) has unique positive solution . For any initial value , the successive iterative sequence generated by

uniformly converges to the unique positive solution on . We have the error estimation

where is a constant with and determined by the initial value And there are constants with such that

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

References

K. Diethelm and A. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity,” in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer-Verlag, Heidelberg, Germany, 1999.

B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008.

W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

A. Arafa, S. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, pp. 1–7, 2012.

J. W. Kirchner, X. Feng, and C. Neal, “Frail chemistry and its implications for contaminant transport in catchments,” Nature, vol. 403, no. 6769, pp. 524–527, 2000.

D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.

D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Lévy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.

X. Zhang, L. Liu, and Y. Wu, “Variational structure and multiple solutions for a fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 68, no. 12, part A, pp. 1794–1805, 2014.

Y. Wang and L. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,” Advances in Difference Equations, vol. 2017, no. 13, 2017.

Z. Yue and Y. Zou, “New uniqueness results for fractional differential equation with dependence on the first order derivative,” Advances in Difference Equations, vol. 2019, no. 1, p. 38, 2019.

K. M. Zhang, “On a sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 2017, no. 59, 2017.

Y. Guan, Z. Zhao, and X. Lin, “On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques,” Boundary Value Problems, vol. 2016, no. 141, pp. 1–18, 2016.

Y. L. Guan, Z. Q. Zhao, and X. Lin, “On the existence of solutions for impulsive fractional differential equations,” Advances in Mathematical Physics, vol. 2017, Article ID 1207456, 12 pages, 2017.

J. Jiang, W. Liu, and H. Wang, “Positive solutions for higher order nonlocal fractional differential equation with integral boundary conditions,” Journal of Function Spaces, vol. 2018, Article ID 6598351, 12 pages, 2018.

J. Q. Jiang, W. W. Liu, and H. C. Wang, “Positive solutions to singular dirichlet-type boundary value problems of nonlinear fractional differential equations,” Advances in Difference Equations, vol. 2018, no. 169, 2018.

X. S. Du and A. M. Mao, “Existence and multiplicity of nontrivial solutions for a class of semilinear fractional Schrödinger equations,” Journal of Function Spaces, vol. 2017, Article ID 3793872, 7 pages, 2017.

J. X. Mao and Z. Q. Zhao, “The existence and uniqueness of positive solutions for integral boundary value problems,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 34, no. 1, pp. 153–164, 2011.

J. X. Mao and Z. Q. Zhao, “On existence and uniqueness of positive solutions for integral boundary boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 16, pp. 1–8, 2010.

J. X. Mao, Z. Q. Zhao, and C. G. Wang, “The exact iterative solution of fractional differential equation with nonlocal boundary value conditions,” Journal of Function Spaces, vol. 2018, Article ID 8346398, 6 pages, 2018.

J. X. Mao, Z. Q. Zhao, and A. X. Qian, “Laplace's equation with concave and convex boundary nonlinearities on an exterior region,” Boundary Value Problems, vol. 51, pp. 1–12, 2019.

K. Y. Zhang and Z. Q. Fu, “Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity,” Journal of Function Spaces, vol. 2019, Article ID 9046472, 7 pages, 2019.

J. Tariboon, S. K. Ntouyas, S. Asawasamrit, and C. Promsakon, “Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain,” Open Mathematics, vol. 15, no. 1, pp. 645–666, 2017.

B. Ahmad and S. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 2, pp. 348–360, 2014.

W. G. Yang and Y. P. Qin, “Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions,” ScienceAsia, vol. 43, no. 3, pp. 201–206, 2017.

Q. Ma, R. Wang, J. Wang, and Y. Ma, “Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative,” Applied Mathematics and Computation, vol. 257, pp. 436–445, 2015.

G. Wang, K. Pei, R. P. Agarwal, L. Zhang, and B. Ahmad, “Nonlocal hadamard fractional boundary value problem with hadamard integral and discrete boundary conditions on a half-line,” Journal of Computational and Applied Mathematics, vol. 343, pp. 230–239, 2018.