This is an announcement for the paper “On proximal mappings with Young functions in uniformly convex Banach spaces” by Miroslav Bacakhttps://arxiv.org/find/math/1/au:+Bacak_M/0/1/0/all/0/1, Ulrich Kohlenbachhttps://arxiv.org/find/math/1/au:+Kohlenbach_U/0/1/0/all/0/1.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1709.04700