An abstract fixed point theorem for nonexpansive mappings

Author:
W. A. Kirk

Journal:
Proc. Amer. Math. Soc. **82** (1981), 640-642

MSC:
Primary 54H25

MathSciNet review:
614894

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Abstract: A class of subsets of a bounded metric space is said to be normal if each member of contains a nondiametral point. An induction proof is given for the following. Suppose is a nonempty bounded metric space which contains a class of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in . Then every nonexpansive self-mapping of has a fixed point.

**[1]**Benno Fuchssteiner,*Iterations and fixpoints*, Pacific J. Math.**68**(1977), no. 1, 73–80. MR**0513055****[2]**A. A. Gillespie and B. B. Williams,*Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure*, Applicable Anal.**9**(1979), no. 2, 121–124. MR**539537**, 10.1080/00036817908839259**[3]**W. A. Kirk,*A fixed point theorem for mappings which do not increase distances*, Amer. Math. Monthly**72**(1965), 1004–1006. MR**0189009****[4]**W. A. Kirk,*Mappings of generalized contractive type*, J. Math. Anal. Appl.**32**(1970), 567–572. MR**0271794****[5]**Teck Cheong Lim,*A constructive proof of the infinite version of the Belluce-Kirk theorem*, Pacific J. Math.**81**(1979), no. 2, 467–469. MR**547612****[6]**-,*Asymptotic centers and nonexpansive mappings in some conjugate spaces*, Pacific J. Math. (to appear).**[7]**Jean-Paul Penot,*Fixed point theorems without convexity*, Bull. Soc. Math. France Mém.**60**(1979), 129–152. Analyse non convexe (Proc. Colloq., Pau, 1977). MR**562261**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1981-0614894-6

Keywords:
Fixed point theorem,
nonexpansive mappings

Article copyright:
© Copyright 1981
American Mathematical Society