A fixed point theorem for mappings with a nonexpansive iterate

Author:
W. A. Kirk

Journal:
Proc. Amer. Math. Soc. **29** (1971), 294-298

MSC:
Primary 47.85

MathSciNet review:
0284887

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Abstract: Let *X* be a reflexive Banach space which has strictly convex norm and suppose *K* is a nonempty, bounded, closed and convex subset of *X*. Suppose has the property that, for some positive integer is nonexpansive ( for all ). A function is determined, , such that if for all , where , then *T* has a fixed point in *K*.

**[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]**Felix E. Browder,*Nonexpansive nonlinear operators in a Banach space*, Proc. Nat. Acad. Sci. U.S.A.**54**(1965), 1041–1044. MR**0187120****[4]**K. Goebel,*Convexivity of balls and fixed-point theorems for mappings with nonexpansive square*, Compositio Math.**22**(1970), 269–274. MR**0273477****[5]**Dietrich Göhde,*Zum Prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251–258 (German). MR**0190718****[6]**W. A. Kirk,*A fixed point theorem for mappings which do not increase distances*, Amer. Math. Monthly**72**(1965), 1004–1006. MR**0189009****[7]**Victor L. Klee Jr.,*Convex bodies and periodic homeomorphisms in Hilbert space*, Trans. Amer. Math. Soc.**74**(1953), 10–43. MR**0054850**, 10.1090/S0002-9947-1953-0054850-X

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0284887-3

Keywords:
Fixed point theory,
nonexpansive mappings,
uniformly convex Banach spaces,
normal structure,
nonexpansive iterate

Article copyright:
© Copyright 1971
American Mathematical Society