Sum of two maximal monotone operators in a general Banach space is maximal

TL;DR

This paper proves that the sum of a type (FPV) monotone operator and a maximal monotone operator is maximal in a general Banach space, solving a longstanding problem in monotone operator theory.

Contribution

It establishes the maximality of the sum of specific monotone operators under broad conditions, advancing the understanding of operator sums in Banach spaces.

Findings

Sum of a type (FPV) monotone operator and a maximal monotone operator is maximal.

Provides a solution to the long-standing sum problem in monotone operator theory.

Extends maximality results to general Banach spaces.

Abstract

In a real Banach space, we first prove that the sum of a monotone operator of type (FPV) and maximal monotone operator Rockafellar's constraint qualification is maximal. This prove leads to the solution of most interesting long-time outstanding problem in monotone operator theory is the sum problem.