Completeness and the contraction principle

Borwein, J. M.

doi:10.1090/S0002-9939-1983-0681829-1

Completeness and the contraction principle
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by J. M. Borwein

Proc. Amer. Math. Soc. 87 (1983), 246-250

DOI: https://doi.org/10.1090/S0002-9939-1983-0681829-1

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Abstract:

We prove (something more general than) the result that a convex subset of a Banach space is closed if and only if every contraction of the space leaving the convex set invariant has a fixed point in that subset. This implies that a normed space is complete if and only if every contraction on the space has a fixed point. We also show that these results fail if "convex" is replaced by "Lipschitz-connected" or "starshaped".

References

Sur les opérations dans ensembles abstraits et leur application aux équation intégrales

3

James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251. MR 394329, DOI 10.1090/S0002-9947-1976-0394329-4
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
W. A. Kirk, Caristi’s fixed point theorem and metric convexity, Colloq. Math. 36 (1976), no. 1, 81–86. MR 436111, DOI 10.4064/cm-36-1-81-86
G. J. O. Jameson, Topology and normed spaces, Chapman and Hall, London; Halsted Press [John Wiley & Sons, Inc.], New York, 1974. MR 0463890
Ludvík Janoš, A converse of Banach’s contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287–289. MR 208589, DOI 10.1090/S0002-9939-1967-0208589-3

On pseudo-complete spaces

18

Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. MR 254828
D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66, Cambridge University Press, London-New York, 1974. MR 0467717
Francis Sullivan, A characterization of complete metric spaces, Proc. Amer. Math. Soc. 83 (1981), no. 2, 345–346. MR 624927, DOI 10.1090/S0002-9939-1981-0624927-9
Hoàng Tụy, A fixed point theorem involving a hybrid inwardness-contraction condition, Math. Nachr. 102 (1981), 271–275. MR 642157, DOI 10.1002/mana.19811020122