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

Pages: 416 - 423, Issue Special, September 2009

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Eric R. Kaufmann - Department of Mathematics & Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, United States (email)

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.

Keywords:  Fractional derivative, nonlinear dynamic equation, positive solution
Mathematics Subject Classification:  Primary: 34B18; Secondary: 26A33, 34B15

Received: August 2008;      Revised: May 2009;      Published: September 2009.