Properties of fixed-point sets of nonexpansive mappings in Banach spaces

Bruck, Ronald E.

doi:10.1090/S0002-9947-1973-0324491-8

Properties of fixed-point sets of nonexpansive mappings in Banach spaces
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by Ronald E. Bruck PDF

Trans. Amer. Math. Soc. 179 (1973), 251-262 Request permission

Abstract:

Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either $F = \emptyset$ or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if $T:C \to C$ is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.

References

L. P. Belluce and W. A. Kirk, Fixed-point theorems for families of contraction mappings, Pacific J. Math. 18 (1966), 213–217. MR 199703, DOI 10.2140/pjm.1966.18.213
L. P. Belluce and W. A. Kirk, Nonexpansive mappings and fixed-points in Banach spaces, Illinois J. Math. 11 (1967), 474–479. MR 215145, DOI 10.1215/ijm/1256054569
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
Ronald E. Bruck Jr., Nonexpansive retracts of Banach spaces, Bull. Amer. Math. Soc. 76 (1970), 384–386. MR 256135, DOI 10.1090/S0002-9904-1970-12486-7
Ralph DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139–1141. MR 159229, DOI 10.2140/pjm.1963.13.1139
Michael Edelstein, On nonexpansive mappings, Proc. Amer. Math. Soc. 15 (1964), 689–695. MR 165498, DOI 10.1090/S0002-9939-1964-0165498-3
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