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Positive solutions for a class of singular fractional boundary value problems. (English) Zbl 1235.34010
Summary: We investigate the existence and uniqueness of positive solutions for the following singular fractional boundary value problem \[ D^\alpha_{0^+}u(t)+f(t,u(t))=0,\;0<t<1,\quad u(0)=u(1)=0, \] where \(1<\alpha\leq 2, D^\alpha_{0^+}\) is the standard Riemann-Liouville differentiation and \(f:(0,1 \times[0,\infty)\to[0,\infty)\) with \(\lim_{t\to 0^+}f(t,-)= \infty\) (i.e., \(f\) is singular at \(t=0)\). Our analysis relies on a fixed point theorem in partially ordered sets.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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