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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 '' +Φ(t)f(t,y,y ' )=0,t(0,+),
ay(0)-by ' (0)=y 0 0,lim t y ' (t)=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:
34B16Singular nonlinear boundary value problems for ODE
34B40Boundary value problems for ODE on infinite intervals
34C11Qualitative theory of solutions of ODE: growth, boundedness