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.


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


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