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On local convexity of nonlinear mappings between Banach spaces |
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| Authors: | Iryna Banakh Taras Banakh Anatolij Plichko Anatoliy Prykarpatsky |
| Institution: | 1. Ya. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences, 3b Naukova Str., 79060, Lviv, Ukraine 2. Ivan Franko National University of Lviv, 1 Universytetska Str., Lviv, 79000, Ukraine 3. Jan Kochanowski University, ?eromskiego 5, 25-369, Kielce, Poland 4. Cracow University of Technology, Warszawska 24, 31-155, Kraków, Poland 5. AGH University of Science and Technology, Mickiewicza 30, 30-059, Kraków, Poland |
| 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 ? < c the image f(B ?(x)) of each ?-ball B ?(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. |
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