  The initial-value problem associated with multi-valued operators in Banach
spaces is here reformulated as a minimization principle, extending results of
Brezis-Ekeland, Nayroles and Fitzpatrick. At the focus there is the stability
of these problems w.r.t.\ perturbations not only of data but also of operators;
this is achieved via De Giorgi's $\Gamma$-convergence w.r.t.\ a nonlinear
topology of weak type. A notion of evolutionary $\Gamma$-convergence of weak
type is also introduced. These results are applied to quasilinear PDEs,
including doubly-nonlinear flows.
