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Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems. (English) Zbl 1182.34005
From the introduction: We discuss the boundary-value problem
\[ \begin{aligned} & D_{0^+}^\alpha,u(t)+f(t,u(t))=0,\quad 0<t<1,\\ & u(0)=u'(1)=u''(0)=0,\end{aligned}\tag{*} \]
where \(2<\alpha\leq 3\), \(D_{0^+}^\alpha\) is the Caputo’s differentiation and \(f:(0,1]\times [0,\infty)\to [0,\infty)\) with \(\lim_{t\to 0^+}f(t,-)=\infty\) (i.e., \(f\) is singular at \(t=0\)).
We prove the existence and uniqueness of a positive and nondecreasing solution for the problem (*) by using a fixed point theorem in partially ordered sets.

MSC:
34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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