On the generalized parallel sum of two maximal monotone operators of Gossez type (D)

TL;DR

This paper investigates conditions under which the generalized parallel sum of two maximal monotone operators of Gossez type (D) remains maximal monotone of the same type, extending the theory to biduals and providing sufficient criteria.

Contribution

It introduces new sufficient conditions for the maximal monotonicity of the generalized parallel sum of Gossez type (D) operators, including extensions to bidual spaces.

Findings

Provided interiority- and closedness-type conditions for maximal monotonicity

Extended the concept to the sum of operators in bidual spaces

Ensured the generalized sum retains Gossez type (D) properties

Abstract

The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more general one concerning the maximal monotonicity of Gossez type (D) of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals.