Double Controlled Metric Type Spaces and Some Fixed Point Results

Abstract

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions $$\alpha (x,y)$$ and $$\mu (x,y)$$ on the right-hand side of the $$b-$$triangle inequality, that is,$$d(x,y)\le \alpha (x,z)d(x,z)+\mu (z,y)d(z,y),\mathrm{for}\mathrm{all}x,y,z\in X.$$Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and $$\varphi$$-nonlinear type contractions in the setting of double controlled metric type spaces.