Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem
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
$\D_0^\alpha + u(t) + f(t, u(t)) = 0,$ $t \in (0, \tau),$
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
Received: August 2008; Revised: May 2009; Published: September 2009. |