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Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. (English) Zbl 1206.34009

Consider the existence of a positive solution for the singular fractional boundary value problem

D α u(t)+f(t,u(t),D μ u(t))=0,u(0)=u(1)=0,

where 1<α<2, μ>0 with α-μ1, D α is the standard Riemann-Liouville fractional derivative, the function f is positive, satisfies the Carathéodory conditions on [0,1]×(0,)× and f(t,x,y) is singular at x=0.

The proofs are based on regularization and sequential techniques and the results are obtained by means of fixed point theorem of cone compression type due to [M. A. Krasnosel’skij, Positive solutions of operator equations. Groningen: The Netherlands: P.Noordhoff Ltd. (1964; Zbl 0121.10604)].

MSC:
34A08Fractional differential equations
34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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