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Zbl 1172.34313
Qiu, Tingting; Bai, Zhanbing
Existence of positive solutions for singular fractional differential equations.
(English)
[J] Electron. J. Differ. Equ. 2008, Paper No. 146, 9 p., electronic only (2008). ISSN 1072-6691/e

From the introduction: We discuss the existence of a positive solution to boundary-value problems of the nonlinear fractional differential equation $$D_{0^+}^\alpha u(t)+f(t,u(t))=0, \quad 0<t<1, \qquad u(0)=u'(1)=u''(0)=0,$$ where $2<\alpha\le 3$, $D_{0^+}^\alpha$ is the Caputo's differentiation, and $f:(0,1]\times[0,1)\to [0,1)$ with $\lim_{t\to0^+}f(t,\cdot)=+\infty$ (that is $f$ is singular at $t=0$). We obtain two results about this boundary-value problem, by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone.
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE
34B16 Singular nonlinear boundary value problems
26A33 Fractional derivatives and integrals (real functions)
47N20 Appl. of operator theory to differential and integral equations

Keywords: boundary value problem; positive solution; singular fractional differential equation; fixed-point theorem

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