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

作者：Burkotova Jana^*; Hubner Michael; Rachunkova Irena; Weinmueller Ewa B

来源：Applied Mathematics and Computation, 2016, 274: 65-82.

DOI：10.1016/j.amc.2015.10.074

• Second order ordinary differential equations
• Regular variation
• Asymptotic properties
• Non-oscillatory solutions
• Kneser solutions

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)) = 0
on the interval [a,infinity), a >= 0, where p and q are functions regularly varying at infinity, and f satisfies f(L-0) = f(0) = f(L) = 0, with L-0 < 0 < L. Our aim is to describe the asymptotic behaviour of the non-oscillatory solutions satisfying one of the following conditions:
u(a)=u(0) is an element of(0, L), 0 <= u(t)<= L, t is an element of[a,infinity),
u(a)=u(0) is an element of(L-0,0), L-0 <= u(t)<= 0, t is an element of[a,infinity).
The existence of Kneser solutions on [a,infinity) 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.

• 出版日期2016-2-1