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Boundary Point Method and the Mann–Dotson Algorithm for Non-self Mappings in Banach Spaces

  • Giuseppe MarinoEmail author
  • Luigi Muglia
Article

Abstract

Let C be a closed, convex and nonempty subset of a Banach space X. Let \({T : C \rightarrow X}\) be a nonexpansive inward mapping. We consider the boundary point map \({h_{C,T } : C \rightarrow \mathbb{R}}\) depending on C and T defined by \({h_{C,T} = {\rm max}\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}}\), for all \({x \in C}\). Then for a suitable step-by-step construction of the control coefficients by using the function \({h_{C,T }}\), we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} < \infty}\) and weak convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} = \infty}\).

Keywords

Boundary point method nonexpansive mappings inward condition non-self mappings 

Mathematics Subject Classification (2010)

Primary 47H05 Secondary 47H10 65J05 

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Notes

Acknowledgements

The authors thank the anonymous reviewers for for their valuable suggestions regarding the improvement of the paper. This paper is funded by Ministero dell’Istruzione, Universitá e Ricerca (MIUR) and Gruppo Nazionale di Analisi Matemarica e Probabilitá e Applicazioni (GNAMPA).

The first author declares that his contribution would not have been possible without the help of Maria Grazia Atzeri, Carla Mazzone and of the co-author Luigi Muglia.

Author’s contribution

Both authors contributed equally and significantly in writing the paper. Both authors read and approved the final manuscript.

Competing interest

The author declare that they have no competing interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá della CalabriaRende (CS)Italy

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