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Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

Chengbo Zhai* and Mengru Hao

Author Affiliations

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, P.R. China

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Boundary Value Problems 2013, 2013:85  doi:10.1186/1687-2770-2013-85

Published: 10 April 2013

Abstract

This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order ν, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4">View MathML</a>. Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.

MSC: 26A33, 34B18, 34B27.

Keywords:
Riemann-Liouville fractional derivative; fractional differential equation; positive solution; existence and uniqueness; fixed point theorem for mixed monotone operator