  The paper is devoted to proving an existence and uniqueness result for
generalized solutions to semilinear wave equations with a small nonlinearity in
space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of
generalized functions. It is shown that for a nonlinearity of arbitrary growth
and sign, but multiplied with a small parameter, the initial value problem for
the semilinear wave equation has a unique solution in the Colombeau algebra of
generalized functions of bounded type. The proof relies on a fixed point
theorem in the ultra-metric topology on the algebras involved. In classical
terms, the result says that the semilinear wave equations under consideration
have global classical solutions up to a rapidly vanishing error.
