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Research Article | Open Access

V-Moreau envelope of nonconvex functions on smooth Banach spaces

Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
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Abstract

We continue the study of the properties of the V-Moreau envelope and generalized (f,λ)-projection that we started in [5]. In this paper, we study the differentiability and the regularity of the V-Moreau envelope and the Hölder continuity of the generalized (f,λ)-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.

Keywords

V-Moreau envelopegeneralized (f, λ)-projectionp-uniformly convex Banach spacesq-uniformly smooth Banach spacesuniformly generalized prox-regular setsV-prox-regular functions
CLC number: 34A60, 49J53

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AIMS Mathematics
Volume 9 Issue 10,
October 2024
Pages 28589-28610
DOI: 10.3934/math.20241387

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Bounkhel M. V-Moreau envelope of nonconvex functions on smooth Banach spaces. AIMS Mathematics, 2024, 9(10): 28589-28610. https://doi.org/10.3934/math.20241387

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Received: 24 August 2024
Revised: 15 September 2024
Accepted: 19 September 2024
Published: 15 October 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)