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Existence and exact asymptotic behavior of positive solutions for a fractional boundary value problem. (English) Zbl 1274.34068

Summary: We establish the existence and uniqueness of a positive solution \(u\) for the fractional boundary value problem \(D^\alpha u(x) = -a(x)u^\sigma(x), x \in (0, 1)\) with the condition \(\lim_{x \to 0}D^{\alpha -1} u(x) = 0, u(1) = 0\), where \(1 < \alpha \leq 2, \sigma \in (-1, 1)\) and \(a\) is a nonnegative continuous function on \((0, 1)\) that may be singular at \(x = 0\) or \(x = 1\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
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