Existence of multiple positive solutions for nonlinear fractional boundary value problems on the half-line. (English) Zbl 1358.34016
Summary: In this paper, we deal with the following nonlinear fractional differential problem on the half-line \(\mathbb R^+=(0,+\infty)\)
\[
\begin{cases} D^\alpha u(x)+f(x,u(x),D^pu(x))=0,\quad x\in\mathbb R^+,\\ u(0)=u'(0)=\cdots=u^{(m-2)}(0)=0,\end{cases}
\]
where \(m\in\mathbb N\), \(m\geq 2\), \(m-1<\alpha\leq m\), \(0<p\leq\alpha-1\), the differential operator is taken in the Riemann-Liouville sense and \(f\) is a Borel measurable function in \(\mathbb R^+\times\mathbb R^+\times\mathbb R^+\) satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schauder’s fixed point theorem. Some examples illustrating our main result are also given.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
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