  We consider generalized gradient systems in Banach spaces whose evolutions
are generated by the interplay between an energy functional and a dissipation
potential. We focus on the case in which the dual dissipation potential is
given by a sum of two functionals and show that solutions of the associated
gradient-flow evolution equation with combined dissipation can be constructed
by a split-step method, i.e. by solving alternately the gradient systems
featuring only one of the dissipation potentials and concatenating the
corresponding trajectories. Thereby the construction of solutions is provided
either by semiflows, on the time-continuous level, or by using Alternating
Minimizing Movements in the time-discrete setting. In both cases the
convergence analysis relies on the energy-dissipation principle for gradient
systems.
