Abstract
Suppose that Q is a family of seminorms on a locally convex space E which determines the topology of E. We study the existence of Q-nonexpansive retractions for families of Q-nonexpansive mappings and prove that a separated and sequentially complete locally convex space E has the weak fixed point property for commuting separable semitopological semigroups of Q-nonexpansive mappings. This proves the Bruck’s problem (Pacific J Math 53:59–71, 1974) for locally convex spaces. Moreover, we prove the existence of Q-nonexpansive retractions for the right amenable Q-nonexpansive semigroups.
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Acknowledgements
We would like to thank the anonymous referees for the careful readings and suggestions, which led to improvements in the presentation of this paper. The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. Moreover, the third author is grateful to the University of Lorestan for their support.
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Dhompongsa, S., Kumam, P. & Soori, E. Fixed Point Properties and \({\varvec{Q}}\)-Nonexpansive Retractions in Locally Convex Spaces. Results Math 73, 66 (2018). https://doi.org/10.1007/s00025-018-0821-x
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DOI: https://doi.org/10.1007/s00025-018-0821-x