On diagonal subdifferential operators in nonreflexive Banach spaces
Benar F. Svaiter | Iusem, Alfredo
Equilibrium problem | maximal monotone operator | diagonal subdifferential | convex representation
Consider a real-valued bifunction f defined on C x C, where C is a closed and convex subset of a Banach space X, which is concave in its first argument and convex in the second one. We study its subdifferential with respect to its second argument, evaluated at pairs of the form (x,x), and the subdifferential of -f with respect to its first argument, evaluated at the same pairs. We prove that if f vanishes whenever both arguments coincide, these operators are maximal monotone, under rather undemanding continuity assumptions on f. We also establish similar results under related assumptions on f, e.g. monotonicity and convexity in the second argument. These results were known for the case in which X is reflexive and C = X. Here we use a different approach, based upon a recently established sufficient condition for maximal monotonicity of operators, in order to cover the nonreflexive and constrained case. Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones.