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Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. (English) Zbl 1116.34016

The authors show the existence of unbounded solutions to the singular boundary value problem

${y}^{\text{'}\text{'}}+{\Phi }\left(t\right)f\left(t,y,{y}^{\text{'}}\right)=0,\phantom{\rule{0.166667em}{0ex}}t\in \left(0,+\infty \right),$
$ay\left(0\right)-b{y}^{\text{'}}\left(0\right)={y}_{0}\ge 0,\underset{t\to \infty }{lim}{y}^{\text{'}}\left(t\right)=k>0$

using two different techniques. In section 3, the authors use the upper and lower solution technique to establish necessary and sufficient conditions for the existence of a positive solution to the boundary value problem. Under the additional assumption that $f$ is nondecreasing in the second and third variables, the authors show that the boundary value problem has a unique solution. In section 4, the authors use index theory to show the existence of at least one and at least two positive solutions to the boundary value problem.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B40 Boundary value problems for ODE on infinite intervals 34C11 Qualitative theory of solutions of ODE: growth, boundedness
##### Keywords:
lower and upper solutions; fixed point index