Double controlled $ M $-metric spaces and some fixed point results

Abstract

<abstract><p>In this article, we introduce the idea of double controlled $ M $-metric space by employing two control functions $ a(u, w) $ and $ \beta (w, v) $ on the right-hand side of the triangle inequality of $ M $-metric space. We provide some examples of double controlled $ M $-metric spaces. We also provide some fixed point results under new type of contractions in the setting of double controlled $ M $-metric spaces. Moreover, we give an example to highlight the importance of one of our main results.</p></abstract>