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A fixed point theorem for conical shells

  • A. Domokos
  • M. M. Marsh
Article
  • 98 Downloads

Abstract

We prove a fixed point theorem for mappings f defined on conical shells F in \(\mathbb R^n\), where the image of f need not be a subset of F, nor even a subset of the cone that contains F. In this sense, our results extend, in \(\mathbb R^n\), Krasnosel’skiĭ’s well-known fixed point result on cones in Banach spaces (Krasnosel’skiĭ, Soviet Math Dokl 1:1285–1288, 1960). Sufficiency for fixed points of f is dependent only on the behavior of f on the boundary of F. This behavior is related to notions of compressing or extending the conical shell F. We also discuss possible extensions of our theorem to infinite dimensional Banach spaces.

Keywords

Convex cone conical shell fixed point 

Mathematics Subject Classification

Primary 54H25 Secondary 55M20 54F15 

Notes

Acknowledgements

The authors wish to thank the referees for suggestions that improved the paper. In particular, we thank the referee who noticed that Theorem 3.1(2) could be strengthened to its present form.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsCalifornia State University, SacramentoSacramentoUSA

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