Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 79, pp. 1-25.
Title: Solutions approaching polynomials at infinity to nonlinear
ordinary differential equations
Author: Christos G. Philos (Univ. of Ioannina, Greece)
Panagiotis Ch. Tsamatos (Univ. of Ioannina, Greece)
Abstract:
This paper concerns the solutions approaching polynomials at
$\infty $ to $n$-th order ($n>1$) nonlinear ordinary differential
equations, in which the nonlinear term depends on time $t$ and
on $x,x',\dots ,x^{(N)}$, where $x$ is the unknown function and
$N$ is an integer with $0\leq N\leq n-1$. For each given integer
$m$ with $\max \{1,N\}\leq m\leq n-1$, conditions are given which
guarantee that, for any real polynomial of degree at most $m$,
there exists a solution that is asymptotic at $\infty $ to this
polynomial. Sufficient conditions are also presented for every
solution to be asymptotic at $\infty $ to a real polynomial of
degree at most $n-1$. The results obtained extend those by the
authors and by Purnaras [25] concerning the particular case
$N=0$.
Submitted February 13, 2005. Published July 11, 2005.
Math Subject Classifications: 34E05, 34E10, 34D05
Key Words: Nonlinear differential equation; asymptotic properties;
asymptotic expansions; asymptotic to polynomials solutions.