Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem

Eric R. Kaufmann

doi:10.3934/proc.2009.2009.416

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2009, Volume 2009: 416-423. Doi: 10.3934/proc.2009.2009.416 open access

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Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem

Eric R. Kaufmann^1,

1.
Department of Mathematics & Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204

Received: July 31, 2008

Revised: April 30, 2009

Published: August 2009

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Abstract

We give sufficient conditions on the value $\tau \in (0, T]$ such that the nonlinear fractional boundary value problem

$\D_0^\alpha + u(t) + f(t, u(t)) = 0,$ $t \in (0, \tau),$
$I^\gamma u(0^+) = 0,$ $I^\beta u(\tau) = 0,$

where $1 - \alpha < \gamma \leq 2 - \alpha,$ $2 - \alpha < \beta < 0$, $\D_(0+)^\alpha$ is the Riemann-Liouville differential operator of order $\alpha $, and $f \in C([0,T] \times \mathbb{R})$ is nonnegative, has a positive solution. We also present a nonexistence result.

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Mathematics Subject Classification: Primary: 34B18; Secondary: 26A33, 34B15.

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