Uniformly contractive fixed points in compact metric spaces

Leader, Solomon

doi:10.1090/S0002-9939-1982-0663887-2

Uniformly contractive fixed points in compact metric spaces
HTML articles powered by AMS MathViewer

by Solomon Leader PDF

Proc. Amer. Math. Soc. 86 (1982), 153-158 Request permission

Abstract:

Let $f$ be a continuous self-map on a compact metric space. Equivalent conditions are given for the existence of a uniformly contractive fixed point $w({f^n}x \to w$ uniformly for all $x$). Corollaries give sufficient conditions for uniformly contractive fixed points subsuming some known results.

References

Josef Daneš, Two fixed point theorems in topological and metric spaces, Bull. Austral. Math. Soc. 14 (1976), no. 2, 259–265. MR 418066, DOI 10.1017/S0004972700025077
M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 133102, DOI 10.1112/jlms/s1-37.1.74
B. Fisher, A fixed-point theorem for compact metric spaces, Publ. Math. Debrecen 25 (1978), no. 3-4, 193–194. MR 516999
M. Hegedűs, New generalizations of Banach’s contraction principle, Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 87–89. MR 576939
Ludvik Janos, Hwei Mei Ko, and Kok Keong Tan, Edelstein’s contractivity and attractors, Proc. Amer. Math. Soc. 76 (1979), no. 2, 339–344. MR 537101, DOI 10.1090/S0002-9939-1979-0537101-2
Gerald Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261–263. MR 400196, DOI 10.2307/2318216
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
M. S. Khan and S. P. Singh, Some fixed point theorems for densifying mappings, Kyungpook Math. J. 20 (1980), no. 1, 105–110. MR 584339
Solomon Leader and Stephen L. Hoyle, Contractive fixed points, Fund. Math. 87 (1975), 93–108. MR 405397, DOI 10.4064/fm-87-2-93-108
Philip R. Meyers, A converse to Banach’s contraction theorem, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 73–76. MR 221469
Pedro Morales, Topological contraction principle, Fund. Math. 110 (1980), no. 2, 135–144. MR 600587, DOI 10.4064/fm-110-2-135-144
Roger D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375. MR 310719, DOI 10.1090/S0002-9947-1972-0310719-6
J. L. Solomon and Ludvik Janos, Even continuity and the Banach contraction principle, Proc. Amer. Math. Soc. 69 (1978), no. 1, 166–168. MR 500891, DOI 10.1090/S0002-9939-1978-0500891-8
Wolfgang Walter, Remarks on a paper by F. Browder about contraction: “Remarks on fixed point theorems of contractive type” [Nonlinear Anal. 3 (1979), no. 5, 657–661; MR 81a:54046], Nonlinear Anal. 5 (1981), no. 1, 21–25. MR 597277, DOI 10.1016/0362-546X(81)90066-3