We propose a set-valued version of the implicit Euler scheme for
relaxed one-sided Lipschitz differential inclusions and prove that
the defining implicit inclusions have a well-defined
solution. Furthermore, we give a convergence analysis based on stability theorems, which shows that the
set-valued implicit Euler method inherits all favourable stability properties from the
single-valued scheme. The impact of spatial discretization is discussed, a fully discretized
version of the scheme is analyzed, and a numerical example is given.