On the relation of Kannan contraction and Banach contraction
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Marija Cvetković
Abstract
Kannan contraction is broadly investigated topic in Metric Fixed Point Theory due to its importance in omitting the continuity presumption. Of a great significance is also its role in characterizing completeness of a metric space through existence and uniqueness of a fixed point of arbitrary Kannan contraction in the observed setting. The concept of Kannan contraction has been adapted, extended and transferred to various types of spaces including cone metric spaces, quasi metric spaces, b-metric spaces, partial metric spaces, among others. The main aim of this article is to prove that for any Kannan contraction T on a complete metric space (X, d) there exists another metric on the set X in relation to which T is a Banach contraction while the completeness is preserved. In that way, all results on Kannan contraction may be derived as corollaries of the Banach contraction principle. The converse also holds since, by altering the metric, Banach contraction becomes a Kannan contraction. The obtained theoretical results are substantiated with adequate examples.
The work is supported by Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 451-03-68/2022-14/200124.
Communicated by Marcus Waurick
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