PERIODIC SOLUTIONS TO SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

In this paper, we consider a non-instantaneous impulsive system represented by the second order nonlinear differential equations in a Banach space. We use the strongly continuous cosine family of linear operators along with Schauder and Banach fixed point theorems to study the existence and uniqueness of the periodic solutions of the non-instantaneous impulsive system. Moreover, we construct a Poincar´e operator, which is a composition of the maps and we apply the techniques of a priori estimate for this operator. Finally, we give an example to illustrate the application of these obtained abstract results.