Equivalence of completeness and contraction property
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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 $X$ has the strong contraction property.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2262905