Translation Mappings of Quasimonotonicity Beyond Smoothness

TL;DR

This paper investigates conditions under which quasimonotonicity implies monotonicity in multi-valued mappings, generalizing existing results without requiring differentiability, but assuming continuity.

Contribution

It extends the understanding of the relationship between quasimonotonicity and monotonicity for multi-valued mappings using translation maps in topological spaces.

Findings

Quasimonotonicity can imply monotonicity under certain conditions.

Continuity, but not differentiability, is sufficient for the implications.

Generalizes previous results in the literature on monotonicity relations.

Abstract

Monotonicity of a mapping implies its pseudomonotonicity and hence quasimonotonocity, the converse is not true. In this note we intend to study the situations under which quasimono tonicity of a mapping implies its monotonicity. Thus we generalize some results in the literature related to the connection between monotonocity and its generalized classes for multi-valued mappings via translation maps in real topological spaces. No differentiability assumption is required but continuity assumption is imposed.