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

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 $p_{i}(t)\in C^5(\RR/\ZZ).$