Portal de Periódicos da CAPES

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

2007; American Mathematical Society; Volume: 136; Issue: 04 Linguagem: Inglês

10.1090/s0002-9939-07-09110-1

1088-6826

Jacek Jachymski,

Advanced Differential Geometry Research

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 $C[0,1]$.