Periodic and homoclinic solutions generated by impulses

Abstract

In this paper we study the existence of periodic and homoclinic solutions for a class of second order differential equations of the form q ̈ + V q ( t , q ) = f ( t ) with impulsive conditions Δ q ̇ ( s k ) = g k ( q ( s k ) ) via variational methods. Our results show that under appropriate conditions such a system possesses at least one non-zero periodic solution and at least one non-zero homoclinic solution and these solutions are generated by impulses when f ≡ 0 . Furthermore, one of the results gives us a lower bound of the number of periodic solutions generated by impulses and this lower bound is determined by the number of impulses of the system in a period of the solution.