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Existence of positive solution for singular fractional differential equation. (English) Zbl 1185.34004
From the introduction: We discuss the existence of a positive solution to boundary value problem of nonlinear fractional differential equation: $$\cases D^\alpha_{0^+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ u(0) = u'(1) = u''(0) = 0,\endcases\tag1$$ where $2 <\alpha\le 3$ is a real number, $D^\alpha_0$ is the Caputo’s differentiation, and $f : (0,1]\times [0,\infty)\to [0,\infty)$, $\lim_{t-0^+}f(t,\cdot) = +\infty$ (that is $f$ is singular at $t=0$). We obtain two results about this boundary value problem by using Krasnoselskii’s fixed point theorem in a cone and nonlinear alternative of Leray-Schauder, respectively.

MSC:
 34A08 Fractional differential equations 34B15 Nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE
Full Text:
References:
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