Weak structural stability of pseudo-monotone equations

The inclusion β(u)∋h in V′ is studied, assuming that V is a reflexive Banach space, and that β: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×V′→R∪{+∞}. A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi's notion of Γ-convergence. The compactness and the convergence of the family of operators β provide the (weak) structural stability of the inclusion β(u)∋h with respect to variations of β and h, under the only assumptions that the βs 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 Dt∂φ(u)+α(u)∋h, ∂φ being the subdifferential of a convex lower semicontinuous mapping φ, and α a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.