  Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locally
prescribed data: $u$ along a non-characteristic curve $M$ and $u_x$ along a
non-characteristic curve $N$. We assume that $M$ and $N$ are graphs of
one-to-one functions, intersecting only at the origin, and located in the first
quadrant of the $(x,y)$-plane. It is known that if $M$ is located above $N$,
then there is a unique local solution, obtainable by successive approximation.
We show that in the opposite case, when $M$ lies below $N$, the uniqueness can
fail in the following strong sense: for the same boundary data, there are two
solutions that differ at points arbitrarily close to the origin. In the latter
case, we also establish existence of a local solution (under a Lipschitz
condition on the function $f$). The construction, via Picard iteration, makes
use of a careful choice of additional $u$-data which are updated in each
iteration step.
