On proximal mappings with Young functions in uniformly convex Banach spaces

TL;DR

This paper extends the understanding of proximal mappings from Hilbert spaces to uniformly convex Banach spaces, introducing a generalized proximal mapping involving Young functions and providing explicit continuity moduli.

Contribution

It introduces a new generalized proximal mapping with Young functions in Banach spaces and derives explicit uniform continuity moduli based on convexity and Young function properties.

Findings

Proximal mappings in uniformly convex Banach spaces are uniformly continuous on bounded sets.

A new general proximal mapping involving Young functions is formulated.

Explicit modulus of uniform continuity is obtained in terms of convexity and Young functions.

Abstract

It is well known in convex analysis that proximal mappings on Hilbert spaces are $1$-Lipschitz. In the present paper we show that proximal mappings on uniformly convex Banach spaces are uniformly continuous on bounded sets. Moreover, we introduce a new general proximal mapping whose regularization term is given as a composition of a Young function and the norm, and formulate our results at this level of generality. It is our aim to obtain the corresponding modulus of uniform continuity explicitly in terms of a modulus of uniform convexity of the norm and of moduli witnessing properties of the Young function. We also derive several quantitative results on uniform convexity, which may be of interest on their own.