Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6417117 | Journal of Differential Equations | 2015 | 51 Pages |
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,â ) is smooth for all λ, but, mainly, we do not suppose that F(â ,u) is smooth for all u. We state conditions such that for all λâ0 there exists exactly one solution uâu0, that u is smooth in a certain abstract sense, and that the data-to-solution map λâ¦u is smooth. Then we apply this to time-periodic solutions of first-order hyperbolic systemsâtuj+aj(x,λ)âxuj+bj(t,x,λ,u)=0 and second-order hyperbolic equationsât2uâa(x,λ)2âx2u+b(t,x,λ,u,âtu,âxu)=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.