Abstract
Let X be a strictly convex Banach space, whose predual space is \( Y (X=Y^{\prime })\), having the weak star sequentially compact unit ball for the topology \( \sigma (X, Y) \) and the weak star Kadec–Klee property. Furthermore, we suppose that the unit ball of the dual space \(X^{\prime }\) is weak star sequentially compact for the topology \(\sigma (X^{\prime }, X)\). Let C be a nonempty convex bounded closed subset of X; then every nonexpansive map** \( T:C \rightarrow C \) has a fixed point. As consequences of this result, we generalize the Browder (Proc Natl Acad Sci USA 54:1041–1044, 1965) and Göhde (Math Nachr 301:251–258, 1965) theorems, where X is a uniformly convex Banach space and the Lin’s theorem by Lin (Nonlinear Anal 68:2303–2308, 2008) and Lin (J Math Anal Appl 362:534–541, 2010), where \( X=l^{1} \).
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The author would like to thank the referees for their positive remarks on this work and their instructive critical readings.
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Hanebaly, E. The fixed point property in Banach spaces via the strict convexity and the Kadec–Klee property. J. Fixed Point Theory Appl. 22, 30 (2020). https://doi.org/10.1007/s11784-020-0768-x
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DOI: https://doi.org/10.1007/s11784-020-0768-x