Volume 6, Issue 2 (Vol. 6, No. 2 2020)                   mmr 2020, 6(2): 149-156 | Back to browse issues page


XML Persian Abstract Print


Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

Afshari H, Sajjadmanesh M. Solution of Some Boundary Value Problems Including Nonlinear Fractional Differential Equations. mmr 2020; 6 (2) :149-156
URL: http://mmr.khu.ac.ir/article-1-2735-en.html
1- University of Bonab , hojat.afshari@yahoo.com
2- University of Bonab
Abstract:   (1817 Views)
Introduction
In this paper, by mixed monotone operators and their fixed points, we investigate the existence and uniqueness of positive solution for the following boundary value problems via nonlinear fractional differential equations.
 


and
 

where  is the Riemann–Liouville derivative.          
In recent years, fractional differential equations have been studied by many mathematicians both theoretically and practically, for example, in physics, mechanic, chemistry, engineering, biology, economy, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics and etc.
First time, mixed monotone operators have been introduced by Guo and Lakshmikantham in 1987. Next, many authors studied them in Banach spaces and obtained some results not only in theory fields but also wide applications in chemistry, engineering, biology, technology and other fields.  
In 2009, Xu et. al studied the properties of Green's function for the nonlinear fractional differential equation boundary value problem (2). Here, the existence and uniqueness of its positive solutions are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems.
Material and methods
The content of this paper is organized as follows. First, we present some definitions, lemmas and basic results that will be used in the proofs of our theorems. Then, we consider the existence and uniqueness of positive solution for the operator equation, also we utilize the results obtained to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems.
Results and discussion
In this work, we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary-value problems. Our results guarantee the existence of a unique positive solution, and can be applied for constructing an iterative scheme for obtaining the solution.
Conclusion
The following conclusions were drawn from this research.
Using the properties of cones and the fixed point theorem for mixed monotone operator, the existence and uniqueness of the positive solution are obtained. Our research methods are different from those in the related literature. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems. Our results extend and improve the related conclusions in the literature../files/site1/files/62/1Abstract.pdf
 
 
Full-Text [PDF 600 kb]   (411 Downloads)    
Type of Study: Original Manuscript | Subject: alg
Received: 2018/01/30 | Revised: 2020/09/5 | Accepted: 2019/03/2 | Published: 2020/02/4 | ePublished: 2020/02/4

Add your comments about this article : Your username or Email:
CAPTCHA

Send email to the article author


Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2024 CC BY-NC 4.0 | Mathematical Researches

Designed & Developed by : Yektaweb