## Fredholmness and smooth dependence for linear hyperbolic periodic-Dirichlet problems

Please always quote using this URN: urn:nbn:de:0296-matheon-6911

- This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$ such that the system has a Fredholm like solvability behavior. Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those sufficient conditions imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general.