Boundedness of solutions for some impulsive pendulum-type equations

Abstract

In the present paper, we study the following impulsive pendulum-type equation $\{\begin{array}{l}\ddot{x}+V^{\mathrm{\prime }}(x)=p(t),t\ne t_j,\\ △x(t_j):=x(t_j^{+})-x(t_j)\equiv 0,\\ △\dot{x}(t_j):=\dot{x}(t_j^{+})-\dot{x}(t_j)=J_j(x(t_j),\dot{x}(t_j)),j\in \mathbf{Z},\end{array}$ where $p(t+1)=p(t)$, $V(x+1)=V(x)$. Under suitable impulses, by Moser's twist theorem, we prove that any solution of the impulsive pendulum-type equation is bounded, and there are infinitely many quasi-periodic solutions for the impulsive pendulum-type equation.

Keywords:

• Lagrange stability
• boundedness
• quasi-periodic solution
• Moser's twist theorem
• impulsive pendulum-type equation

MSC2020:

• 34C27
• 34D15

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is supported by the Doctoral Starting Foundation of Quzhou University (No. BSYJ202115).