Electronic Journal of Differential Equations,
Conference 01 (1998), pp. 97-108.
Title: On Properties of Nonlinear Second Order Systems under
Nonlinear Impulse Perturbations
Authors: John R. Graef (Mississippi State Univ., MS USA)
Janos Karsai (Albert Szent-Gyorgyi Medical Univ., Szeged, Hungary)
Abstract:
In this paper, we consider the impulsive second order system
\[
\ddot{x}+f(x)=0\quad (t\neq t_{n});\quad \dot{x}(t_{n}+0)=b_{n}\dot{x}(t_{n})
\quad (t=t_{n})
\]
where $t_n=t_0+n\,p$ $(p>0, n=1,2\dots )$. In a previous paper, the authors
proved that if $f(x)$ is strictly nonlinear, then this system has
infinitely many
periodic solutions. The impulses account for the main differences in
the attractivity properties of the zero solution. Here,
we prove that these periodic
solutions are attractive in some sense, and we give good estimates for
the attractivity region.
Published November 12, 1998.
Math Subject Classifications: 34D05, 34D20, 34C15.
Key Words: Asymptotic stability; attractrivity of periodic
solutions; impulsive systems; nonlinear equations; second order systems.