The contraction principle for mappings on a metric space with a graph

Jachymski, Jacek

doi:10.1090/S0002-9939-07-09110-1

The contraction principle for mappings on a metric space with a graph
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Proc. Amer. Math. Soc. 136 (2008), 1359-1373 Request permission

Abstract:

We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some linear operators on a Banach space. In particular, the last result easily yields the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space $C[0,1]$.

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