Asymptotic properties of Kneser solutions to nonlinear second order ODEs with regularly varying coefficients

Jana Burkotová, Michael Hubner, Irena Rachůnková and Ewa B. Weinmüller

Applied Mathematics and Computation, 2016, vol. 274, issue C, 65-82

Abstract: In this work, we investigate properties of a class of solutions to the second order ODE, (p(t)u′(t))′+q(t)f(u(t))=0on the interval [a, ∞), a ≥ 0, where p and q are functions regularly varying at infinity, and f satisfies f(L0)=f(0)=f(L)=0, with L0 < 0 < L. Our aim is to describe the asymptotic behaviour of the non-oscillatory solutions satisfying one of the following conditions: u(a)=u0∈(0,L),0≤u(t)≤L,t∈[a,∞),u(a)=u0∈(L0,0),L0≤u(t)≤0,t∈[a,∞).The existence of Kneser solutions on [a, ∞) is investigated and asymptotic properties of such solutions and their first derivatives are derived. The analytical findings are illustrated by numerical simulations using the collocation method.

Keywords: Second order ordinary differential equations; Regular variation; Asymptotic properties; Non-oscillatory solutions; Kneser solutions (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:274:y:2016:i:c:p:65-82

DOI: 10.1016/j.amc.2015.10.074

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