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Open Access Research

Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance

Haidong Qu1* and Xuan Liu2

Author Affiliations

1 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

2 Department of Basic Education, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

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Boundary Value Problems 2013, 2013:127  doi:10.1186/1687-2770-2013-127

Published: 16 May 2013

Abstract

We consider the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M1">View MathML</a>

satisfying the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M3">View MathML</a> is the Riemann-Liouville fractional order derivative. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is a Fredholm map of index zero. As a result, the minimal and maximal nonnegative solutions for the problem are obtained by using a fixed point theorem of increasing operators.

MSC: 26A33, 34A08.

Keywords:
fractional order; coincidence degree; at resonance