Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfills a certain growth and a Hölder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfills a certain growth condition and is (up to some shift) monotone and coercive.