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Positive solutions for mixed problems of singular fractional differential equations. (English) Zbl 1232.26005
Summary: We investigate the existence of positive solutions to the singular fractional boundary value problem: $^cD^{\alpha}u+ f(t,u,u',^cD^{\mu}u)=0, u^{\prime}(0) = 0, u(1) = 0$, where $1 < \alpha < 2$, $0 < \mu < 1$, $f$ is a $L^{q}$-Carathéodory function, $q> \frac{1}{\alpha-1}$, and $f(t, x, y, z)$ may be singular at the value 0 of its space variables $x, y, z$. Here $^cD$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.

26A33Fractional derivatives and integrals (real functions)
34B16Singular nonlinear boundary value problems for ODE
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