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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 10, Issue 6


Volume 13 (2015)

On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh
  • Ya. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences, 3b Naukova Str., 79060, Lviv, Ukraine
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/ Taras Banakh
  • Ivan Franko National University of Lviv, 1 Universytetska Str., Lviv, 79000, Ukraine
  • Jan Kochanowski University, Żeromskiego 5, 25-369, Kielce, Poland
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/ Anatolij Plichko / Anatoliy Prykarpatsky
Published Online: 2012-10-12 | DOI: https://doi.org/10.2478/s11533-012-0101-z


We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

MSC: 46B20; 49J50; 52A41; 58C20

Keywords: Locally convex mapping; Hilbert and Banach spaces; Modulus of convexity; Modulus of smoothness; Lipschitzopen maps

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About the article

Published Online: 2012-10-12

Published in Print: 2012-12-01

Citation Information: Open Mathematics, Volume 10, Issue 6, Pages 2264–2271, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0101-z.

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© 2012 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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