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Existence of positive solution for singular fractional differential equation. (English) Zbl 1185.34004

From the introduction: We discuss the existence of a positive solution to boundary value problem of nonlinear fractional differential equation:

D 0 + α u(t)+f(t,u(t))=0,0<t<1,u(0)=u ' (1)=u '' (0)=0,(1)

where 2<α3 is a real number, D 0 α is the Caputo’s differentiation, and f:(0,1]×[0,)[0,), lim t-0 + f(t,·)=+ (that is f is singular at t=0).

We obtain two results about this boundary value problem by using Krasnoselskii’s fixed point theorem in a cone and nonlinear alternative of Leray-Schauder, respectively.

MSC:
34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
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