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2012 | 10 | 6 | 2264-2271

Tytuł artykułu

On local convexity of nonlinear mappings between Banach spaces

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Języki publikacji

EN

Abstrakty

EN
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.

Twórcy

autor
  • Ya. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences, 3b Naukova Str., 79060, Lviv, Ukraine
autor
  • Cracow University of Technology, Warszawska 24, 31-155, Kraków, Poland
  • AGH University of Science and Technology, Mickiewicza 30, 30-059, Kraków, Poland

Bibliografia

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  • [2] Borwein J., Guirao A.J., Hájek P., Vanderwerff J., Uniformly convex functions on Banach spaces, Proc. Amer. Math. Soc., 2009, 137(3), 1081–1091 http://dx.doi.org/10.1090/S0002-9939-08-09630-5[Crossref]
  • [3] Deimling K., Nonlinear Functional Analysis, Springer, Berlin, 1985 http://dx.doi.org/10.1007/978-3-662-00547-7[Crossref]
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  • [13] Lajara S., Pallarés A.J., Troyanski S., Moduli of convexity and smoothness of reflexive subspaces of L 1, J. Funct. Anal., 2011, 261(11), 3211–3225 http://dx.doi.org/10.1016/j.jfa.2011.07.024[Crossref]
  • [14] Lindenstrauss J., Tzafriri L., Classical Banach Spaces II. Function Spaces, Ergeb. Math. Grenzgeb., 97, Springer, Berlin-New York, 1979
  • [15] Linke Yu.É., Application of Michaelś theorem and its converse to sublinear operators, Math. Notes, 1992, 52(1), 680–686 http://dx.doi.org/10.1007/BF01247650[Crossref]
  • [16] Nirenberg L., Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York, 1974
  • [17] Prykarpatska N.K., Blackmore D.L., Prykarpatsky A.K., Pytel-Kudela M., On the inf-type extremality solutions to Hamilton-Jacobi equations, their regularity properties, and some generalizations, Miskolc Math. Notes, 2003, 4(2), 153–180
  • [18] Prykarpatsky A.K., A Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem, preprint available at http://arxiv.org/abs/0902.4416
  • [19] Prykarpatsky A.K., Blackmore D., A solution set analysis of a nonlinear operator equation using a Leray-Schauder type fixed point approach, Topology, 2009, 48(2–4) 182–185 http://dx.doi.org/10.1016/j.top.2009.11.017[WoS][Crossref]
  • [20] Prykarpats’kyi A.K., An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications, Ukrainian Math. J., 2008, 60(1), 114–120 http://dx.doi.org/10.1007/s11253-008-0046-3[Crossref]
  • [21] Samoilenko A.M., Prykarpats’kyi A.K., Samoilenko V.H., Lyapunov-Schmidt approach to studying homoclinics splitting in weakly perturbed Lagrangian and Hamiltonian systems, Ukrainian Math. J., 2003, 55(1), 82–92 http://dx.doi.org/10.1023/A:1025072619144[Crossref]
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Bibliografia

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