Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings

Domínguez Benavides, T.; Ramírez, P.

doi:10.1090/S0002-9939-01-06141-X

Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings
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by T. Domínguez Benavides and P. Lorenzo Ramírez PDF

Proc. Amer. Math. Soc. 129 (2001), 3549-3557 Request permission

Abstract:

Let $X$ be a Banach space, $C$ a weakly compact convex subset of $X$ and $T:C\to C$ an asymptotically nonexpansive mapping. Under the usual assumptions on $X$ which assure the existence of fixed point for $T$, we prove that the set of fixed points is a nonexpansive retract of $C$. We use this result to prove that all known theorems about existence of fixed point for asymptotically nonexpansive mappings can be extended to obtain a common fixed point for a commuting family of mappings. We also derive some results about convergence of iterates.

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