Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property

Yarema A. Prykarpatskyy, Petro Ya. Pukach, Myroslava I. Vovk, Michal Greguš

Abstract

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

Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property

Abstract

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

Paper Structure

This paper contains 9 sections, 9 theorems, 9 equations.

Table of Contents

  1. The local convexity mapping property
  2. Introductory setting
  3. The local convexity property: the Lipschitz smoothness analysis
  4. The local convexity mapping property: Banach spaces case
  5. Banach spaces of the modulus of convexity of degree 2
  6. The Banach space case: main result
  7. The locally convex functions between Banach spaces
  8. Conclusion
  9. Acknowledgements

Key Result

Proposition 1

Let $(X,(\cdot |\cdot ))$ be a Hilbert space and a Frechet differentiable mapping $f:X\rightarrow X\ $ is such that the mapping $h:=f-cI:X\rightarrow X,c\in \mathbb{R}\backslash \{0\},$ is proper, that is the preimage of a compact set is compat too. Assume also, in addition that if $U\subset X$ is o

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Proposition 3
  • ...and 1 more