Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

TL;DR

This paper introduces new closedness-type regularity conditions that ensure surjectivity of sums of maximal monotone operators in reflexive Banach spaces, expanding understanding of operator sum properties.

Contribution

It provides novel regularity conditions based on closedness that guarantee surjectivity of sums of maximal monotone operators, including cases involving shifts and special scenarios.

Findings

Regularity conditions guarantee surjectivity of $S(ullet + p)+T(ullet)$.

Conditions for surjectivity of $S+T$ and when 0 is in the range.

Discussion of special cases with interesting byproducts.

Abstract

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.