Journal of Convex Analysis 25 (2018), No. 4, 1291--1318
Copyright Heldermann Verlag 2018
On Proximal Mappings with Young Functions in Uniformly Convex Banach Spaces
Miroslav Bacák
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
bacak@mis.mpg.de
Ulrich Kohlenbach
Department of Mathematics, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany
kohlenbach@mathematik.tu-darmstadt.de
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.
Keywords: Convex function, Duality mapping, modulus of uniform convexity, proximal mapping, uniformly convex Banach space, uniformly convex function, Young function.
MSC: 46T20; 46B20
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