Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation

Motivated by the problem of the existence of a solution of the nonlinear te legraph equation u(tt) + cu(t) - u(xx) + h(t, x, u) = 0, such that u(t, .) satisfies suitable boundary conditions over (0, pi) and p arallel to u(t, .)parallel to is bounded over R for some function space nor m parallel to . parallel to, we prove the existence of bounded solutions ov er R of semilinear evolution equations in a Hilbert space of the form u + cu + Au + g(t, u) = 0, where c > 0, A : D(A) subset of H --> H is self-adjoint, semi-positive defi nite, has compact resolvant and g : R x H --> H, bounded and sufficiently r egular satisfies some Landesman-Lazer type condition. (C) Elsevier, Paris.