A fixed point theorem for mappings with a nonexpansive iterate
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- by W. A. Kirk PDF
- Proc. Amer. Math. Soc. 29 (1971), 294-298 Request permission
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 $T:K \to K$ has the property that, for some positive integer $N,{T^N}$ is nonexpansive ($\left \| {{T^N}x - {T^N}y} \right \| \leqq \left \| {x - y} \right \|$ for all $x,y \in K$). A function $\gamma (N)$ is determined, $\gamma (N) > 1$, such that if $\left \| {{T^j}x - {T^j}y} \right \| \leqq k\left \| {x - y} \right \|$ for all $x,y \in K,1 \leqq j \leqq N - 1$, where $k < \gamma (N)$, then T has a fixed point in K.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
- Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
- K. Goebel, Convexivity of balls and fixed-point theorems for mappings with nonexpansive square, Compositio Math. 22 (1970), 269–274. MR 273477
- Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
- W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
- Victor L. Klee Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43. MR 54850, DOI 10.1090/S0002-9947-1953-0054850-X
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 294-298
- MSC: Primary 47.85
- DOI: https://doi.org/10.1090/S0002-9939-1971-0284887-3
- MathSciNet review: 0284887