Some fixed point results for (c)-mappings in Banach spaces

Abstract

In this paper, we solve two fixed point problems associated with the class of (c)-mappings. The first one is devoted to obtain the existence of fixed points for such mappings defined on \(\hbox {weak}^{\star }\) closed bounded convex subsets of duals of separable Banach spaces when the orthogonality relation \(\perp \) is uniformly \(\hbox {weak}^{\star }\) approximately symmetric. For the second problem, using the idea of R. Smarzewski (On firmly nonexpansive mappings, Proc. Am. Math. Soc 113(3):723–725,1991), we prove the existence of fixed points for such mappings which are defined on a finite union of weakly compact convex subsets of UCED (uniformly convex in every direction) Banach spaces.

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Acknowledgements

The authors are very grateful to anonymous referees for their valuable comments and suggestions which helped to improve the quality of the manuscript.

Funding

This work is supported by the research team RPC (Controllability and Perturbation Results) in the Laboratory of Informatics and Mathematics (LIM) at the University of Souk-Ahras (Algeria).

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All authors have contributed equally and significantly in writing this article. All authors read and approved the manuscript.

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Correspondence to Sami Atailia.

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Atailia, S., Redjel, N. & Dehici, A. Some fixed point results for (c)-mappings in Banach spaces. J. Fixed Point Theory Appl. 22, 51 (2020). https://doi.org/10.1007/s11784-020-00787-4

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Keywords

  • Fixed point
  • (c)-mapping
  • asymptotically regular
  • approximate fixed point sequence
  • \(\hbox {weak}^\star \) closed convex subset
  • \(\hbox {weak}^\star \) convergence
  • orthogonality
  • approximately symmetric
  • \(\hbox {weak}^{\star }\) approximately symmetric
  • uniformly \(\hbox {weak}^{\star }\) approximately symmetric
  • dual Banach space
  • uniformly convex Banach space in every direction

Mathematics Subject Classification

  • Primary 47H10
  • Secondary 54H25