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Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. (English) Zbl 1192.34009
Summary: We give sufficient conditions on the value $\tau \in (0, T]$ such that the nonlinear fractional boundary value problem \align & D_{0^+}^\alpha u(t) + f(t,u(t)) = 0,\quad t \in (0, \tau),\\ & I^\gamma u(0^+) = 0,\quad I^\beta u(\tau) = 0,\endalign where $1 - \alpha < \gamma \leq 2 - \alpha$, $2 - \alpha < \beta < 0$, $D_{0^+}^\alpha$ is the Riemann-Liouville differential operator of order $\alpha$, and $f\in C([0,T] \times \Bbb R)$ is nonnegative, has a positive solution. We also present a nonexistence result.
##### MSC:
 34A08 Fractional differential equations 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE
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