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

Abstract:

In this article, we study the following second-order system of ordinary differential equations with dissipationu″+cu′+dAu+kHu=Pt,u∈Rn,t∈R,where c, d, and k are positive constants, H: Rn→Rn is a locally Lipschitz function, and P: R→Rn 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. © 1999 Academic Press.