Maximal Monotone Operators in Nonreflexive Banach Spaces and Nonlinear Integral Equations of Hammerstein Type by Haim Brezis and Felix

Let Y be a Banach space, Y* its conjugate space, X a weak*-dense closed subspace of Y* with the induced norm. We denote the pairing between x in X and ƒ in 7 by (y, x). If T is a mapping from X into 2 F , T is said to be monotone if for each pair of elements [x9 y] and [u, w] of G(T)9 the graph of T, we have (y—w, x—w)^0. Tis said to be maximal monotone from X to 2 F if Tis monotone and maximal among monotone mappings in the sense of inclusion of graphs. The theory of maximal monotone mappings has been intensively developed in the case in which Tis reflexive and X= y*. In this note, we present an extension of this theory to the case in which X and Y are not reflexive, and show that this extended theory can be used to give a new and more conceptual proof of a general existence theorem for solutions of nonlinear integral equations of Hammerstein type established by the writers in [2] by more concrete arguments. An essential tool in our discussion is supplied by the following definitions: