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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.

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