Uniqueness of the maximal extension of a monotone operator

Abstract

THEOREM [Benilan]. Under the above hypothesis, the graph of A is equal to {(x,x*)EcldomA x E*;vy*~A~y,(x* -y*,x -y)aO}. (1.2) Benilan did not publish his result, but S. Menou quoted it with its proof in [lo]. H. Attouch [9] also proved these two results for the case when A. is the minimal section of A, and applied them to the study of measurable dependence for a family of maximal monotone operators. In Hilbert spaces, these results can be interpreted as saying that a monotone operator can be uniquely extended to a maximal monotone operator within the closure of the convex hull of its domain if the closure of the domain of the original operator is convex and the interior of the convex hull of its domain is nonempty. In section 2, we give a direct proof of such a theorem without using Robert’s theorem. It is interesting to generalize (1.1) to the whole domain of A. The difficulty is the unboundedness of a maximal monotone operator at its boundary points. In section 3, we prove that a monotone operator is bounded on any open segment connecting an interior point and a boundary point belonging to its domain. As a corollary of this result, we also prove that a maximal monotone operator is bounded in a bounded convex set Pin the interior of its domain