Finer Properties of Monotone Operators

Abstract

In this chapter, we deepen our study of monotone operators. The main results are Minty’s theorem, which conveniently characterizes maximal monotonicity, and the Debrunner–Flor theorem, which concerns the existence of a maximally monotone extension with a prescribed domain localization. Another highlight is the fact that the closures of the range and of the domain of a maximally monotone operator are convex, which yields the classical Bunt–Kritikos–Motzkin result on the convexity of Chebyshev sets in Euclidean spaces. Results on local boundedness, surjectivity, and single-valuedness are also presented.