Recent Progress on Monotone Operator Theory

Abstract

In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions.

Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain

“…We denote As maximally monotone set-valued operators play an important role in this work, it is useful to recall some of basic definitions and some of their properties. More generally, they have frequently shown themselves to be a key class of objects in both modern Optimization and Analysis; see, e.g., [4][5][6]8,28,30].…”

“…The next theorem adds more information on the uniform-like behavior of Lyapunov pairs for (6). V (ȳ)) and letρ > 0 be such that…”

“…Consequently, under condition (19), the pair (V, W ) is an a-Lyapunov pair for (6) provided that Dom A is open and (25) holds on Dom A.…”

“…In [2], we followed a different approach that did not make use of viscosity solutions or contingent derivatives associated to the operator A. We provided general criteria for nonsmooth Lyapunov pairs associated to (6) in terms of proximal and horizon subgradients of the involved function V . Such conditions were written by considering limiting processes required by the fact that the initial condition in (6) was allowed to be any point in the closure of the domain A.…”

Sum Theorems for Maximally Monotone Operators of Type (Fpv)

“…Arguably, the most significant open problem in the theory concerns the maximal monotonicity of the sum of two maximally monotone operators in general Banach spaces; this is called the "sum problem". Some recent developments on the sum problem can be found in Simons' monograph [28] and [4,5,6,8,12,11,36,19,31,37,38,39]. It is known, among other things, that the sum theorem holds under Rockafellar's constraint qualification when both operators are of dense type or when each operator has nonempty domain interior [8,Ch.…”