Gossez’s skew linear map and its pathological maximally monotone multifunctions

Maximally monotone linear subspace extensions of monotone subspaces: explicit constructions and characterizations

Monotone linear relations play important roles in variational inequality problems and quadratic optimizations. In this paper, we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional spaces. Examples are provided to illustrate our extensions.Our results generalize a recent result by Crouzeix and Ocaña-Anaya.

Maximal Monotone Multifunctions of BrndstedRockafellar Type

We consider whether the “inequality-splitting” property established in the Brøndsted– Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions...

Maximal Monotone Multifunctions of Brrndsted{rockafellar Type

Construction of pathological maximally monotone operators on non-reflexive Banach spaces

In this paper, we construct maximally monotone operators that are not of Gossez’s dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC–functions will not always be a BC–function. This provides a negative answer to a challenging question posed by S...

The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying domA ∩ int dom...