A fixed point theorem for asymptotically nonexpansive mappings

Authors:
K. Goebel and W. A. Kirk

Journal:
Proc. Amer. Math. Soc. **35** (1972), 171-174

MSC:
Primary 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298500-3

MathSciNet review:
0298500

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Abstract: Let *K* be a subset of a Banach space *X*. A mapping is said to be asymptotically nonexpansive if there exists a sequence of real numbers with as such that . It is proved that if *K* is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if is asymptotically nonexpansive, then *F* has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.

**[1]**F. E. Browder,*Nonexpansive nonlinear operators in a Banach space*, Proc. Nat. Acad. Sci. U.S.A.**54**(1965), 1041-1044. MR**32**#4574. MR**0187120 (32:4574)****[2]**J. A. Clarkson,*Uniformly convex spaces*, Trans. Amer. Math. Soc.**40**(1936), 396-414. MR**1501880****[3]**K. Goebel,*An elementary proof of the fixed-point theorem of Browder and Kirk*, Michigan Math. J.**16**(1969), 381-383. MR**40**#4831. MR**0251604 (40:4831)****[4]**D. Göhde,*Zum prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251-258. MR**32**#8129. MR**0190718 (32:8129)****[5]**W. A. Kirk,*A fixed point theorem for mappings which do not increase distances*, Amer. Math. Monthly**72**(1965), 1004-1006. MR**32**#6436. MR**0189009 (32:6436)****[6]**-,*On nonlinear mappings of strongly semicontractive type*, J. Math. Anal. Appl.**27**(1969), 409-412. MR**39**#6128. MR**0244814 (39:6128)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298500-3

Keywords:
Fixed point theorem,
nonexpansive mapping,
asymptotically nonexpansive mapping,
uniformly convex Banach space,
lipschitzian mapping

Article copyright:
© Copyright 1972
American Mathematical Society