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Zbl 1116.34016
Yan, Baoqiang; O'Regan, Donal; Agarwal, Ravi P.
Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity.
(English)
[J] J. Comput. Appl. Math. 197, No. 2, 365-386 (2006). ISSN 0377-0427

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.
[Eric R. Kaufmann (Little Rock)]
MSC 2000:
*34B16 Singular nonlinear boundary value problems
34B40 Boundary value problems on infinite intervals
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: lower and upper solutions; fixed point index

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