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

Published electronically:
September 18, 2006

MathSciNet review:
2262905

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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 has the strong contraction property.

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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:
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.