  In this paper we show the small data solvability of suitable semilinear wave
and Klein-Gordon equations on geometric classes of spaces, which include
so-called asymptotically de Sitter and Kerr-de Sitter spaces, as well as
asymptotically Minkowski spaces. These spaces allow general infinities, called
conformal infinity in the asymptotically de Sitter setting; the Minkowski type
setting is that of non-trapping Lorentzian scattering metrics introduced by
Baskin, Vasy and Wunsch. Our results are obtained by showing the global
Fredholm property, and indeed invertibility, of the underlying linear operator
on suitable L^2-based function spaces, which also possess appropriate algebra
or more complicated multiplicative properties. The linear framework is based on
the b-analysis, in the sense of Melrose, introduced in this context by Vasy to
describe the asymptotic behavior of solutions of linear equations. An
interesting feature of the analysis is that resonances, namely poles of the
inverse of the Mellin transformed b-normal operator, which are `quantum' (not
purely symbolic) objects, play an important role.
