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Equivalence of completeness and contraction property
Author(s):
Shu-wen
Xiang
Journal:
Proc. Amer. Math. Soc.
135
(2007),
1051-1058.
MSC (2000):
Primary 47H10, 54H25;
Secondary 54E50, 54E35
Posted:
September 18, 2006
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Abstract:
In this paper, we consider the completeness and the contraction property in metric spaces and show that the contraction property implies Lipschitz-completeness or arcwise-completeness in a metric space. However, in a metric space, the contraction property does not imply the usual completeness. We prove that a locally Lipschitz-connected metric space has the contraction property if and only if it is Lipschitz-complete and that a locally arcwise-connected metric space is arcwise-complete if and only if  has the strong contraction property.
References:
-
- 1.
- J. M. Borwein, Completeness and contraction principle, Proc. Amer. Math. Soc. 87 (1983), 246-250. MR 0681829 (84a:54080)
- 2.
- J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125(1997),2327-2335. MR 1389524 (97j:54047)
- 3.
- J. R. Jachymski, Short proof of the converse to the contraction principle and some related results, Topol. Methods Nonlinear Anal. 15(2000),179-186. MR 1786260 (2001h:54071)
- 4.
- L. Janos, A converse of Banach's contraction theorem, Proc. Amer. Math. Soc. 18(1968),287-289. MR 0208589 (34:8398)
- 5.
- L. Janos, On pseudo-complete spaces, Notices Amer. Math. Soc. 18(1971), 97-163.
- 6.
- L. Janos, The Banach contraction mapping principle and cohomology, Comment. Math. Univ. Carolinae, 43:3(2000),605-610. MR 1795089 (2001m:54039)
- 7.
- W. A. Kirk, Caristi's fixed point theorem and metric convexity, Colloq. Math. 36(1976), 81-86. MR 0436111 (55:9061)
- 8.
- S. Leader, A topological characterization of Banach contractions, Pacific J. Math. 69:2(1977),461-466. MR 0436093 (55:9044)
- 9.
- F. Sullivan, A characterization of complete metric spaces, Proc. Amer. Math. Soc. 83(1981),345-346. MR 0624927 (83b:54036)
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Additional Information:
Shu-wen
Xiang
Affiliation:
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, People's Republic of China
Email:
shwxiang@vip.163.com
DOI:
10.1090/S0002-9939-06-08684-9
PII:
S 0002-9939(06)08684-9
Keywords:
Completeness,
contraction property,
Lipschitz-completeness,
arcwise-completeness
Received by editor(s):
October 22, 2005
Posted:
September 18, 2006
Additional Notes:
This work was completed with the support of NSF of China (No: 10561003)
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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