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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Common fixed point theorems for almost weakly periodic nonexpansive mappings


Author: Kok Keong Tan
Journal: Proc. Amer. Math. Soc. 33 (1972), 355-360
MSC: Primary 47H10; Secondary 46N05
DOI: https://doi.org/10.1090/S0002-9939-1972-0295169-9
MathSciNet review: 0295169
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Abstract: The notions of normal structure, (convex) diminishing orbital diameters, regular orbital diameters (r.o.d.) have been generalized into a Hausdorff locally convex space $ (X,\tau )$ whose topology $ \tau $ is generated by a family $ \mathcal{P}$ of seminorms. Theorem 1. Let $ K \subseteq X$ be nonempty weakly compact convex with normal structure w.r.t. $ \mathcal{P}$ and $ \mathcal{F}$ be a (not necessarily finite nor commuting) family of almost weakly periodic nonexpansive mappings w.r.t. $ \mathcal{P}$ on K. Then $ \mathcal{F}$ has a common fixed point. Theorem 2. Let $ K \subseteq X$ be nonempty weakly compact convex and $ \mathcal{F}$ be a semigroup with identity of almost weakly periodic nonexpansive mappings w.r.t. $ \mathcal{P}$ on K. If $ \mathcal{F}$ has r.o.d. w.r.t. $ \mathcal{P}$, then $ \mathcal{F}$ has a common fixed point. Corollary. If $ K \subseteq X$ is nonempty weakly compact convex and $ \mathcal{F} = \{ {f_1}, \cdots ,{f_n}\} $ is a finite commuting family of pointwise periodic nonexpansive mappings w.r.t. $ \mathcal{P}$ on K, then $ \mathcal{F}$ has a common fixed point.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0295169-9
Keywords: Almost weakly periodic, weakly periodic, pointwise periodic, periodic, nonexpansive mappings, diminishing orbital diameters, convex diminishing orbital diameters, regular orbital diameters, normal structure
Article copyright: © Copyright 1972 American Mathematical Society