Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials

TL;DR

This paper investigates the boundedness of solutions for a class of impact oscillators with time-dependent polynomial potentials, establishing conditions under which solutions remain bounded despite impacts and variable coefficients.

Contribution

It provides new results on the boundedness of impact oscillator solutions with polynomial potentials and periodic coefficients, extending previous work to more general cases.

Findings

Solutions are proven to be bounded under certain conditions.

Impact impacts are characterized mathematically within the model.

The results apply to oscillators with polynomial potentials of arbitrary degree.

Abstract

In this paper, we consider the boundedness of solutions for a class of impact oscillators $$ \{{array}{ll} \displaystyle \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0,& \quad {\rm for}\quad x(t)> 0, x(t)\geq 0,& \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}),& \quad {\rm if}\quad x(t_{0})=0, {array}. $$ where $n\in\NN^{+}$, $p_{i}(t+1)=p_{i}(t)$ and $p_{i}(t)\in C^5(\RR/\ZZ).$