Some Remarks on Smooth Mappings of Hilbert and Banach Spaces and Their Local Convexity Property

Author:

Prykarpatskyy Yarema A.12ORCID,Pukach Petro Ya.3ORCID,Vovk Myroslava I.4,Greguš Michal5ORCID

Affiliation:

1. Department of Applied Mathematics, University of Agriculture in Krakow, Al. Mickiewicza 21, 31-120 Kraków, Poland

2. Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Str., Kyiv 01024, Ukraine

3. Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine

4. Department of Mathematics, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine

5. Department of Information Systems, Faculty of Management, Comenius University in Bratislava, Odbojárov 10, 82005 Bratislava, Slovakia

Abstract

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radii of the balls are small enough. We focus on the study of new and mildly sufficient conditions for the nonlinear mapping of Hilbert and Banach spaces to be locally convex, and address a suitably reformulated local convexity problem analyzed within the Leray–Schauder homotopy method approach for Hilbert spaces, and within the Lipschitz smoothness condition for both Hilbert and Banach spaces. Some of the results presented in this work prove to be interesting and novel, even for finite-dimensional problems. Open problems related to the local convexity property for nonlinear mappings of Banach spaces are also formulated.

Funder

Ministry of Education and Science of Ukraine

Publisher

MDPI AG

Reference36 articles.

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