## 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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### References:

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