Abstract
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.
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With great respect and admiration for Professor Andrzej Granas
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Jachymski, J., Klima, J. Cantor’s intersection theorem for K-metric spaces with a solid cone and a contraction principle. J. Fixed Point Theory Appl. 18, 445–463 (2016). https://doi.org/10.1007/s11784-016-0312-1
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DOI: https://doi.org/10.1007/s11784-016-0312-1