The coexistence of quasi-periodic and blow-up solutions for a class of impact oscillators without the twist condition

The purpose of this paper was the study of the dynamic behavior of impact oscillators:

$$\begin{array}{r}\begin{array}{r}\{\begin{array}{l}{x}^{″}+a(x)x^{2n+1}+\sum _{l=0}^m{p}_l(t)x^l=0,\text{for}x(t)>0,\\ x(t)\ge 0,\\ x^{\prime }(t_0^{+})=-x^{\prime }(t_0^{-}),\text{if}x(t_0)=0,\end{array}\end{array}\end{array}$$

where the positive function $a(x)$ is a smooth $T$-periodic oscillator violating the monotone twist condition. We have proved that the above equation has an infinite number of bounded solutions as well as a solution that escapes to infinity in a finite amount of time.