Abstract
In 1979, Goebel and Kuczumow showed that very large class of non-weak* compact, closed, bounded and convex subsets of \(\ell ^1\) has the fixed point property (FPP) for nonexpansive mappings. Later, in 2008, Lin proved that \(\ell ^1\) can be renormed to have FPP for nonexpansive mappings. \(c_0\)-analogue of Lin’s result is still open. However, renorming \(c_0\), we prove that Goebel and Kuczumow analogy can be proved under affinity condition. That is, we prove that there exist a renorming of \(c_0\) and a very large class of non-weakly compact, closed, bounded and convex subsets of \(c_0\) with FPP for affine nonexpansive mappings whereas Dowling, Lennard and Turett proved in 2004 that weak compactness is equivalent to FPP for nonexpansive mappings when \(c_0\) is considered with its usual norm instead of any equivalent norm.
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Nezir, V., Dutta, H., Oran, S. (2020). A Large Class of Non-weakly Compact Subsets in a Renorming of \(c_0\) with FPP. In: Castillo, O., Jana, D., Giri, D., Ahmed, A. (eds) Recent Advances in Intelligent Information Systems and Applied Mathematics. ICITAM 2019. Studies in Computational Intelligence, vol 863. Springer, Cham. https://doi.org/10.1007/978-3-030-34152-7_64
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DOI: https://doi.org/10.1007/978-3-030-34152-7_64
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