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

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.