Abstract
LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup i→∞{sup y∈K ‖t ix−Tiy∼−‖x−y‖}≦0. IfT N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k i‖x−y∼,x,y∈K, wherek i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ0 (X), be less than one.
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Research supported by National Science Foundation Grant GP 18045.
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Kirk, W.A. Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel J. Math. 17, 339–346 (1974). https://doi.org/10.1007/BF02757136
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DOI: https://doi.org/10.1007/BF02757136