Generalized monotone operators on dense sets

TL;DR

This paper demonstrates that local generalized monotonicity on dense sets implies global generalized monotonicity for lower semicontinuous set-valued operators, extending results from real functions to Banach spaces and linking to generalized convexity.

Contribution

It establishes the connection between local and global generalized monotonicity for set-valued operators on dense sets, extending to Banach spaces and exploring implications for generalized convexity.

Findings

Local generalized monotonicity on dense sets implies global generalized monotonicity.

Results extended from real functions to operators on Banach spaces.

Links between generalized monotonicity and convexity are established.

Abstract

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.