Open Access Research

Three-point boundary value problems of fractional functional differential equations with delay

Yanan Li, Shurong Sun*, Dianwu Yang and Zhenlai Han

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School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, P.R. China

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Citation and License

Boundary Value Problems 2013, 2013:38  doi:10.1186/1687-2770-2013-38

Published: 22 February 2013

Abstract

In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M3">View MathML</a> denote Caputo fractional derivatives, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/38/mathml/M7">View MathML</a>. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.

MSC: 34A08, 34K37.

Keywords:
fractional functional differential equation; delay; three-point boundary value problems; fixed point theorem; existence of solutions