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

• Published in: Applied mathematics and computation Vol. 274; pp. 65 - 82
• Main Authors: Burkotová, Jana, Hubner, Michael, Rachůnková, Irena, Weinmüller, Ewa B.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2016
• Subjects: Asymptotic properties; Kneser solutions; Non-oscillatory solutions; Regular variation; Second order ordinary differential equations; Kneser solutions; Asymptotic properties; 34A12; Second order ordinary differential equations; Non-oscillatory solutions; Regular variation; 34D05
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2015.10.074

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.