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
\[ y'' + \Phi(t) f(t, y, y') = 0, \, t \in (0, +\infty), \]
\[ a y(0) - b y'(0) = y_0 \geq 0, \lim_{t \to \infty} y'(t) = k > 0 \]
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.


34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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