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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

${D}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right),{D}^{\mu }u\left(t\right)\right)=0,\phantom{\rule{0.166667em}{0ex}}u\left(0\right)=u\left(1\right)=0,$

where $1<\alpha <2$, $\mu >0$ with $\alpha -\mu \ge 1,$ ${D}^{\alpha }$ is the standard Riemann-Liouville fractional derivative, the function $f$ is positive, satisfies the Carathéodory conditions on $\left[0,1\right]×\left(0,\infty \right)×ℝ$ and $f\left(t,x,y\right)$ 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:
 34A08 Fractional differential equations 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
##### References:
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