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

By Jacek R. Jachymski. 2007

647 citations recorded by the source corpus.

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].

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9484fe1e1de92bed8d1767c48b7315e3ea88f5ba

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

FIXED POINTS FOR MULTIVALUED CONTRACTIONS IN $b$-METRIC SPACES WITH APPLICATIONS TO FRACTALS (2014, result): “Our results extend and complement some previous theorems given in [1, 2, 4, 5, 6], etc.”
Fixed point theorems for Reich type contractions on metric spaces with a graph (2012, methodology): “The first result in this direction was given by Jachymski [3] who also presented its applications to the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space C [0, 1].”
Generalized contractions in metric spaces endowed with a graph (2012, methodology): “Denote also Xf WD ¹x 2 X W .x; f .x// 2 E.G/º: One of the main results given by Jachymski [12] is Theorem 3.2.”
Various Suzuki type theorems in b -metric spaces (2015, result): “The first result in this direction was given by Jachymski [15].”
Fixed Point Theory in -Complete Metric Spaces with Applications (2014, methodology): “8 Abstract and Applied Analysis Definition 26 (see [11]).”
Common fixed-point results for nonlinear contractions in ordered partial metric spaces (2011, background): “There exist a vast literature concerning its various generalizations and extensions (see [1-45]).”
Fixed point results for multivalued contractions on a metric space with a graph (2013, background): “13 (Jachymski [5]).”
A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations (2011, neutral): “Many authors have studied this problem, for a small sample of such work, we refer the reader to works [4,6,9,21,22,27,30,38,42].”
Fixed point of φ-contraction in metric spaces endowed with a graph (2010, methodology): “By using the idea of Jakhymski [3], we will say that:”
Fixed points of Kannan mappings in metric spaces endowed with a graph (2012, methodology): “In this section, by using the idea of Jakhymski [5], we will consider the following concept: Definition 4.”
Caristi Fixed Point Theorem in Metric Spaces with a Graph (2014, methodology): “This work was motivated by some recent works on the extension of Banach contraction principle to metric spaces with a partial order [1] or a graph [2].”
Nonlinear contractions involving simulation functions in a metric space with a partial order (2015, result): “Further, several results appeared in this direction, we mention [1, 2, 3, 4, 5, 7, 8, 9, 10, 13] and the references therein.”
Some fixed point results on a metric space with a graph (2012, background): “In 2008 and 2009, Jachymski continued this idea by using different view (see [8] and [7]).”