Abstract
LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→x f(T n x/n)=limn→x‖T n x/n ‖=α, where α≡inf y∈c ‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT n x/n converges weakly for allx (infz∈f g(T n x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT.
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Supported by National Science Foundation Grant MCS-79-066.
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Kohlberg, E., Neyman, A. Asymptotic behavior of nonexpansive mappings in normed linear spaces. Israel J. Math. 38, 269–275 (1981). https://doi.org/10.1007/BF02762772
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DOI: https://doi.org/10.1007/BF02762772