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

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 whose topology is generated by a family of seminorms. Theorem 1. *Let* *be nonempty weakly compact convex with normal structure w.r.t.* *and* *be a* (*not necessarily finite nor commuting*) *family of almost weakly periodic nonexpansive mappings w.r.t.* *on K. Then* *has a common fixed point*. Theorem 2. *Let* *be nonempty weakly compact convex and* *be a semigroup with identity of almost weakly periodic nonexpansive mappings w.r.t.* *on K. If* *has r.o.d. w.r.t.* , *then* *has a common fixed point*. Corollary. *If* *is nonempty weakly compact convex and* *is a finite commuting family of pointwise periodic nonexpansive mappings w.r.t.* *on K, then* *has a common fixed point*.

**[1]**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**0233341**, 10.1090/S0002-9939-1969-0233341-4**[2]**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****[3]**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**0320836****[4]**M. T. Kiang,*Remetrizations and common fixed point theorems for semigroup of mappings*, Ph.D. Thesis, Dalhousie University, 1971.**[5]**Kok Keong Tan,*Fixed point theorems for nonexpansive mappings*, Pacific J. Math.**41**(1972), 829–842. MR**0313902****[6]**-,*Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces*, Ph.D. Thesis, University of British Columbia, 1970.**[7]**C. S. Wong,*Fixed point theorems*(to appear).

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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