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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 $2<\alpha\le 3$, $D_{0^+}^\alpha$ 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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
26A33Fractional derivatives and integrals (real functions)
47N20Applications of operator theory to differential and integral equations
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