Fifty years of maximal monotonicity

Abstract

Maximal monotone operator theory is about to turn (or just has turned) 50. I intend to briefly survey the history of the subject. I shall try to explain why maximal monotone operators are both interesting and important-culminating with a description of the remarkable progress made during the past decade.

A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalities

“…for all x, y C; see [2,[6][7][8][9][10]. If A is an a-inverse strongly monotone mapping of C into H, then it is obvious that A is 1 α -Lipschitz continuous, that is, Ax − Ay ≤ 1 α x − y for all x, y C. Clearly, the class of monotone mappings include the class of a-inverse strongly monotone mappings.…”

Recent Progress on Monotone Operator Theory

“…As discussed in [12,13,15,21], the two most central open questions in monotone operator theory in a general real Banach space are almost certainly the following: Let A, B : X ⇒ X * be maximally monotone.…”

“…While monotone operator theory is rather complete in reflexive space -and for type (D) operators in general space -the general situation is less clear [21,15]. Hence our continuing interest in operators which are not of type (D).…”

“…We say a Banach space X is of type (D) [15] if every maximally monotone operator on X is of type (D). At present the only known type (D) spaces are the reflexive spaces; and our work here suggests that there are no non-reflexive type (D) spaces.…”

Monotone Operators Without Enlargements

“…Their study in the context of Banach spaces, and in particular nonreflexive ones, arises naturally in the theory of partial differential equations, equilibrium problems, and variational inequalities. For a detailed study of these operators, see, e.g., [12,13,14], or the books [6,15,19,25,31,32,30,41,42].…”