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,⋅) is smooth for all λ, but, mainly, we do not suppose that F(⋅,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∂tuj+aj(x,λ)∂xuj+bj(t,x,λ,u)=0 and second-order hyperbolic equations∂t2u−a(x,λ)2∂x2u+b(t,x,λ,u,∂tu,∂xu)=0. Here we have to prevent small divisors from coming up. Moreover, we need smooth dependence of bj and b on t to get smooth dependence of the solution on λ. This is completely different to the case of parabolic PDEs.