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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.
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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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