Electron. J. Diff. Eqns., Vol. 2005(2005), No. 79, pp. 1-25.

Solutions approaching polynomials at infinity to nonlinear ordinary differential equations

Christos G. Philos, Panagiotis Ch. Tsamatos

Abstract:
This paper concerns the solutions approaching polynomials at $\infty $ to $n$-th order ($n$ greater than 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.

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Christos G. Philos
Department of Mathematics
University of Ioannina
P. O. Box 1186, 451 10 Ioannina, Greece
email: cphilos@cc.uoi.gr
P. Ch. Tsamatos
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: ptsamato@cc.uoi.gr

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