  Dirichlet problem in an $n$-dimensional billiard space is investigated. In
particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet
boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$
with elastic impact on the boundary of $K$ is considered. The existence of
multiple solutions having prescribed number of impacts with the boundary is
proved. As a consequence the existence of infinitely many solutions is proved,
too. The problem is solved by reformulation it into non-impulsive problem with
a discontinuous right-hand side. This auxiliary problem is regularized and the
Schauder Fixed Point Theorem is used.
