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Existence of positive solutions for singular fractional differential equations. (English) Zbl 1172.34313
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 $$2<\alpha\leq 3$$, $$D_{0^+}^\alpha$$ 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:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 26A33 Fractional derivatives and integrals 47N20 Applications of operator theory to differential and integral equations
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