Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs

First we present a generalized implicit function theorem for abstract equations of the type F (λ , u) = 0. We suppose that u₀ is a solution for λ = 0 and that F (λ , ṡ) is smooth for all λ, but, mainly, we do not suppose that F (ṡ , u) is smooth for all u. We state conditions such that for all λ ≈ 0 there exists exactly one solution u ≈u₀, that u is smooth in a certain abstract sense, and that the data-to-solution map λ ↦ u is smooth. Then we apply this to time-periodic solutions of first-order hyperbolic systems