Solutions approaching polynomials at infinity to nonlinear ordinary differential equations

Philos, Christos G. ; Tsamatos, Panagiotis Ch.

Electronic Journal of Differential Equations, Tome 2005 (2005).

Résumé

Summary: This paper concerns the solutions approaching polynomials at $\infty $ to $n$-th order ($n$) 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 $N=0$.

Détail

Classification : 34E05, 34E10, 34D05
Keywords: nonlinear differential equation, asymptotic properties, asymptotic expansions, asymptotic to polynomials solutions