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

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  • 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.

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

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