Abstract
We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS n =(I+T+…+T n−1 )/n, then lim n ‖S n (x)−TS n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping.
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Partially supported by NSF Grant MCS-7802305A01.
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Bruck, R.E. On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel J. Math. 38, 304–314 (1981). https://doi.org/10.1007/BF02762776
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DOI: https://doi.org/10.1007/BF02762776