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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:
\[ \begin{cases} D^\alpha_{0^+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ u(0) = u'(1) = u''(0) = 0,\end{cases}\tag{1} \]
where \(2 <\alpha\leq 3\) 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 ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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