Existence of Bounded Solutions of a Second-Order System with Dissipation*

In this article, we study the following second-order system of ordinary differential equations with dissipation u00 C cu0 C dAu C kHu‘ D Pt‘; u2 n; t 2 ; where c, d, and k are positive constants, Hx n ! n is a locally Lipschitz function, and Px R ! n is a continuous and bounded function. A is a n n matrix whose eigenvalues are positive. Under these conditions, we prove that for some values of c, d, and k this system has a bounded solution which is exponentially asymptotically stable. Moreover; if Pt‘ is almost periodic, then this bounded solution is also almost periodic. These results are applied to the spatial discretization of very well-known second-order partial differential equations.