The inclusion beta(u) there exists h in V' is studied, assuming that V is a reflexive Banach space, and that beta : V -> P (V') is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242]. A notion of strict generalized pseudo-monotonicity is also introduced. The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional V x V' -> R boolean OR {+infinity}. A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi's notion of Gamma-convergence. The compactness and the convergence of the family of operators beta provide the (weak) structural stability of the inclusion beta(u) there exists h with respect to variations of beta and h, under the only assumptions that the beta s are equi-coercive and the hs are equi-bounded. These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form D-t partial derivative phi(u) + alpha(u) there exists h, partial derivative phi being the subdifferential of a convex lower semicontinuous mapping phi, and alpha a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.
