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 u " + cu' + dAu + kH(u) = P(t), u is an element of R-n, t is an element of R, where c, d, and k are positive constants, H: R-n --> R-n is a locally Lipschitz function, and P: R --> R-n is a continuous and bounded function. A is a n x 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. (C) 1999 Academic Press.