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Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions

Bashir Ahmad1* and Sotiris K Ntouyas2

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece

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Boundary Value Problems 2012, 2012:55  doi:10.1186/1687-2770-2012-55

Published: 9 May 2012

Abstract

This article studies a new class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with strip conditions. We extend the idea of four-point nonlocal boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M1">View MathML</a> to nonlocal strip conditions of the form: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M3">View MathML</a>. These strip conditions may be regarded as six-point boundary conditions. Some new existence and uniqueness results are obtained for this class of nonlocal problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed.

MSC 2000: 26A33; 34A12; 34A40.

Keywords:
fractional differential equations; fractional differential inclusions; nonlocal boundary conditions; fixed point theorems; Leray-Schauder degree