  Existence of strong solutions of an abstract Cauchy problem for a class of
doubly nonlinear evolution inclusion of second order is established via a
semi-implicit time discretization method. The principal parts of the operators
acting on $u$ and $u'$ are multi-valued subdifferential operators and are
discretized implicitly. A non-variational and non-monotone perturbation acting
nonlineary on $u$ and $u'$ is allowed and discretized explicitly in time. The
convergence of a variational approximation scheme is established using methods
from convex analysis. In addition, it is proven that the solution satisfies an
energy-dissipation equality. Applications of the abstract theory to various
examples, e.g., a model in visco-elastic-plasticity, are provided.
