Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Equivalence of completeness and contraction property


Author: Shu-wen Xiang
Journal: Proc. Amer. Math. Soc. 135 (2007), 1051-1058
MSC (2000): Primary 47H10, 54H25; Secondary 54E50, 54E35
DOI: https://doi.org/10.1090/S0002-9939-06-08684-9
Published electronically: September 18, 2006
MathSciNet review: 2262905
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ X$ has the strong contraction property.


References [Enhancements On Off] (What's this?)

  • 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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H10, 54H25, 54E50, 54E35

Retrieve articles in all journals with MSC (2000): 47H10, 54H25, 54E50, 54E35


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: https://doi.org/10.1090/S0002-9939-06-08684-9
Keywords: Completeness, contraction property, Lipschitz-completeness, arcwise-completeness
Received by editor(s): October 22, 2005
Published electronically: September 18, 2006
Additional Notes: This work was completed with the support of NSF of China (No: 10561003)
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society