Controlled Metric Type Spaces and the Related Contraction Principle

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Abstract

In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function α ( x , y ) of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + α ( z , y ) d ( z , y ) , for all x , y , z ∈ X . Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension.

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