  In the first part we present a generalized implicit function theorem for
abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a
solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all
$\lambda$, but, mainly, we do not suppose that $F(\cdot,u)$ is smooth for all
$u$. Even so, we state conditions such that for all $\lambda \approx 0$ there
exists exactly one solution $u \approx u_0$, that $u$ is smooth in a certain
abstract sense, and that the data-to-solution map $\lambda \mapsto u$ is
smooth.
  In the second part we apply the results of the first part to time-periodic
solutions of first-order hyperbolic systems of the type $$ \partial_tu_j +
a_j(x,\lambda)\partial_xu_j + b_j(t,x,\lambda,u) = 0, \; x\in(0,1),
\;j=1,\dots,n $$ with reflection boundary conditions and of second-order
hyperbolic equations of the type $$
\partial_t^2u-a(x,\lambda)^2\partial^2_xu+b(t,x,\lambda,u,\partial_tu,\partial_xu)=0,
\; x\in(0,1) $$ with mixed boundary conditions (one Dirichlet and one Neumann).
There are at least two distinguishing features of these results in comparison
with the corresponding ones for parabolic PDEs: First, one has to prevent small
divisors from coming up, and we present explicit sufficient conditions for that
in terms of $u_0$ and of the data of the PDEs and of the boundary conditions.
And second, in general smooth dependence of the coefficient functions $b_j$ and
$b$ on $t$ is needed in order to get smooth dependence of the solution on
$\lambda$, this is completely different to what is known for parabolic PDEs.
