Positive solutions for a class of singular fractional boundary value problems. (English) Zbl 1235.34010

Caballero, J.; Harjani, J.; Sadarangani, K.

Comput. Math. Appl. 62, No. 3, 1325-1332 (2011).

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
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

Keywords:

fractional boundary value problem; fixed point theorem; positive solution

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