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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References
Alspach, D.E.: A fixed point free nonexpansive map. Proc. Amer. Math. Soc. 82, 423–424 (1981)
Brouwer, L.E.J.: Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen 71, 97–115 (1912)
Dowling, P.N., Lennard, C.J., Turett, B.: Asymptotically isometric copies of \(c_0\) in Banach spaces. J. Math. Anal. Appl. 219, 377–391 (1998)
Dowling, P.N., Lennard, C.J., Turett, B.: Renormings of \(l^1\) and \(c_0\) and fixed point properties. In: Kirk, W., Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp. 269–297. Springer (2001)
Dowling, P.N., Lennard, C.J., Turett, B.: Weak compactness is equivalent to the fixed point property in \(c_0\). Proc. Am. Math. Soc. 132(6), 1659–1666 (2004)
Everest, T.: Fixed points of nonexpansive maps on closed, bounded, convex sets in \(l^1\). Ph.D. thesis, University of Pittsburgh (2013)
Gallagher, T., Lennard, C., Popescu, R.: Weak compactness is not equivalent to the fixed point property in \(c\). J. Math. Anal. Appl. 431(1), 471–481 (2015)
Goebel, K., Kuczumow, T.: Irregular convex sets with fixed-point property for nonexpansive mappings. Colloquium Mathematicum 40(2), 259–264 (1979)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)
Lennard, C., Nezir, V.: The closed, convex hull of every AI \(c_0\)-summing basic sequence fails the FPP for affine nonexpansive mappings. J. Math. Anal. Appl. 381, 678–688 (2011)
Lin, P.K.: There is an equivalent norm on \(\ell _1\) that has the fixed point property. Nonlinear Anal. 68, 2303–2308 (2008)
Maurey, B.: Points fixes contractions de certains faiblement compacts de \({L}^1\). Seminaire d’Analyse Fonctionelle, 1980–1981, Centre de Mathématiques. École Polytech., Palaiseau, Exp. No. VIII, 19 pp. (1981)
Nezir, V.: Fixed point properties for \(c_0\)-like spaces. Ph.D. thesis, University of Pittsburgh (2012)
Nezir, V., Mustafa, N., Dutta, H.: Renorming \(c_0\) and the fixed point property. In: Ruzhansky, M., Dutta, H. (eds.) Advanced Topics in Mathematical Analysis, pp. 481–540. CRC Press, Boca Raton (2018)
Nuñez, C.: Characterization of Banach spaces of continuous vector valued functions with the weak Banach-Saks property. Illinois J. Math. 33(1), 27–41 (1989)
Oran, S.: Fixed point property for affine nonexpansive mappings on a very large class of nonweakly compact subsets in \(c_0\) respect to an equivalent norm. M.Sc. thesis, Kafkas University Graduate School of Natural and Applied Sciences (2017)
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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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