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

We establish an extension of Cantor's intersection theorem for a -metric space (), where is a generalized metric taking values in a solid cone in a Banach space . This generalizes a recent result of Alnafei, RadenoviAc and Shahzad (2011) obtained for a -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 CvetkoviAc and RakoeviAc (2014): we assume that a mapping satisfies the condition "" for , where is a partial order induced by , and 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 -metric space.