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SOLVABILITY FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Part of: Boundary value problems

Published online by Cambridge University Press:  19 June 2009

YINGXIN GUO
YINGXIN GUO*
Affiliation:
Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, PR China (email: [email protected])
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Abstract

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In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) where 1<α≤2, η∈(0,1),β∈ℝ=(−∞,+∞), βηα−1≠1, Dα is the Riemann–Liouville differential operator of order α, and f:[0,1]×ℝ→ℝ is continuous, q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

Keywords

Caputo’s fractional derivativefractional differential equationboundary-value problemnontrivial solutionLeray–Schauder nonlinear alternative

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 125 - 138
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The authors were supported financially by the NNSF of China (10726004) and the NSF of Shandong Province of China (Q2007A02).

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