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

34B16Singular nonlinear boundary value problems for ODE
34B40Boundary value problems for ODE on infinite intervals
34C11Qualitative theory of solutions of ODE: growth, boundedness
Full Text: DOI
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