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Zbl 1206.34009
Agarwal, Ravi P.; O'Regan, Donal; Staněk, Svatoslav
Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations.
(English)
[J] J. Math. Anal. Appl. 371, No. 1, 57-68 (2010). ISSN 0022-247X

Consider the existence of a positive solution for the singular fractional boundary value problem $$D^\alpha u(t)+ f(t,u(t),D^\mu u(t))=0,\,u(0)=u(1)=0,$$ where $1<\alpha<2$, $\mu>0$ with $\alpha-\mu\geq 1,$ $D^\alpha$ is the standard Riemann-Liouville fractional derivative, the function $f$ is positive, satisfies the Carathéodory conditions on $[0,1]\times (0,\infty)\times {\mathbb R}$ and $f(t,x,y)$ is singular at $x=0$. \par 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 [{\it M. A. Krasnosel'skij}, Positive solutions of operator equations. Groningen: The Netherlands: P.Noordhoff Ltd. (1964; Zbl 0121.10604)].
[Gisèle M. Mophou (Pointe-à-Pitre)]
MSC 2000:
*34A08
34B18 Positive solutions of nonlinear boundary value problems
34B16 Singular nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: Fractional differential equation; Singular Dirichlet problem; Positive solution; Riemann Liouville fractional derivative

Citations: Zbl 0121.10604

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