Controlled metric type spaces and the related contraction principle

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.