Periodic solutions of semilinear Duffing equations with impulsive effects

Abstract

In this paper we are concerned with the existence of periodic solutions for semilinear Duffing equations with impulsive effects. Firstly for its autonomous equation, any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time. Under the influence of impulses, these two motions are likely to be quite different. We introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one. Basing on Poincaré–Birkhoff twist theorem, we prove the existence of infinitely many periodic solutions. Secondly, as for the nonautonomous equation where the autonomous case is taken as an auxiliary one, we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by Poincaré–Birkhoff twist theorem. Lastly, an example with special impulses satisfying the above assumptions is given.