Cantor’s intersection theorem for K-metric spaces with a solid cone and a contraction principle

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• 作者：Jacek Jachymski ; Jakub Klima
• 关键词：K ; metric space ; cone metric space ; solid cone ; Cantor’s intersection theorem ; fixed point ; spectral radius ; contraction principle
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：18
• 期：3
• 页码：445-463
• 全文大小：776 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746
• 卷排序：18

文摘

We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenović and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone. Next we show that our Cantor’s theorem yields a special case of a generalization of Banach’s contraction principle given very recently by Cvetković and Rakočević (2014): we assume that a mapping \({T}\) satisfies the condition “\({d(Tx, Ty) \preceq \Lambda (d(x, y))}\)” for \({x, y \in X}\), where \({\preceq}\) is a partial order induced by \({K}\), and \({\Lambda : E \rightarrow E}\) is a linear positive operator with the spectral radius less than one. We also obtain new characterizations of convergence in the sense of Huang and Zhang in a \({K}\)-metric space.