On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh; Taras Banakh; Anatolij Plichko; Anatoliy Prykarpatsky

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2264-2271
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Iryna Banakh, et al. "On local convexity of nonlinear mappings between Banach spaces." Open Mathematics 10.6 (2012): 2264-2271. <http://eudml.org/doc/269784>.

@article{IrynaBanakh2012,
abstract = {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.},
author = {Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky},
journal = {Open Mathematics},
keywords = {Locally convex mapping; Hilbert and Banach spaces; Modulus of convexity; Modulus of smoothness; Lipschitzopen maps; locally convex mapping; modulus of convexity; modulus of smoothness; Lipschitz-open maps},
language = {eng},
number = {6},
pages = {2264-2271},
title = {On local convexity of nonlinear mappings between Banach spaces},
url = {http://eudml.org/doc/269784},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Iryna Banakh
AU - Taras Banakh
AU - Anatolij Plichko
AU - Anatoliy Prykarpatsky
TI - On local convexity of nonlinear mappings between Banach spaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2264
EP - 2271
AB - 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.
LA - eng
KW - Locally convex mapping; Hilbert and Banach spaces; Modulus of convexity; Modulus of smoothness; Lipschitzopen maps; locally convex mapping; modulus of convexity; modulus of smoothness; Lipschitz-open maps
UR - http://eudml.org/doc/269784
ER -

References

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