Common fixed point theorems for almost weakly periodic nonexpansive mappings
HTML articles powered by AMS MathViewer
- by Kok Keong Tan PDF
- Proc. Amer. Math. Soc. 33 (1972), 355-360 Request permission
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.References
- L. P. Belluce and W. A. Kirk, Fixed-point theorems for certain classes of nonexpansive mappings, Proc. Amer. Math. Soc. 20 (1969), 141–146. MR 233341, DOI 10.1090/S0002-9939-1969-0233341-4
- M. S. Brodskiĭ and D. P. Mil′man, On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837–840 (Russian). MR 0024073
- R. D. Holmes and Anthony T. Lau, Asymptotically non-expansive actions of topological semigroups and fixed points, Bull. London Math. Soc. 3 (1971), 343–347. MR 320836, DOI 10.1112/blms/3.3.343 M. T. Kiang, Remetrizations and common fixed point theorems for semigroup of mappings, Ph.D. Thesis, Dalhousie University, 1971.
- Kok Keong Tan, Fixed point theorems for nonexpansive mappings, Pacific J. Math. 41 (1972), 829–842. MR 313902 —, Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces, Ph.D. Thesis, University of British Columbia, 1970. C. S. Wong, Fixed point theorems (to appear).
Additional Information
- © Copyright 1972 American Mathematical Society
- 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