The topic of interest is the following class of second order differential equations with impulses
where , , , , , is continuous, is of class for every , , for certain and , is continuously differentiable and -periodic, and is -periodic in .
The existence of periodic and homoclinic solutions to this problem is studied via variational methods. In particular, sufficient conditions are given for the existence of at least one non-trivial periodic solution, which is generated by impulses if . An estimate (lower bound) of the number of periodic solutions generated by impulses is also given, showing that this lower bound depends on the number of impulses in a period of the solution. Moreover, under appropriate conditions, the existence of at least a non-trivial homoclinic solution is obtained, that is, a solution satisfying that and . The periodic and homoclinic solutions obtained in the main results are generated by impulses if , due to the non-existence of non-trivial periodic and homoclinic solutions of the problem when and vanish identically.
The main tools for the proofs of the main theorems are the mountain pass theorem and a result on the existence of pairs of critical points by D. C. Clark [Math. J., Indiana Univ. 22, 65–74 (1972; Zbl 0228.58006)] (see also [P. H. Rabinowitz [Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)]), as well as the theory of Sobolev spaces.