On the variational representation of monotone operators

Let V be a Banach space, z' is an element of V', and alpha : V' -> P(V') be a maximal monotone operator. A large number of phenomena can be modelled by inclusions of the form alpha(u) (sic) z', or by the associated flow D(t)u + alpha(u) (sic) z'. Fitzpatrick proved that there exists a lower semicontinuous, convex representative function f(alpha) : V x V' -> RU {+infinity} such that f(alpha) (v,v') >= < v',v > for all (v,v'), f(alpha) (v,v') = < v',v > double left right arrow v' is an element of alpha(v). (0.1) This provides a variational formulation for the above inclusions. Here we use this approach to prove two results of existence of a solution, without using the classical theory of maximal monotone operators. This is based on a minimax theorem, and on the duality theory of convex optimization.