Double controlled metric-like spaces

In this paper, we introduce a new extension of the double controlled metric-type spaces, called double controlled metric-like spaces, by assuming that the “self-distance” may not be zero On the other hand, if the value of the metric is zero, then it has to be a “self-distance” (i.e., we replace $[\varsigma(g,h)=0 \Leftrightarrow g=h]$ [ ς ( g , h ) = 0 ⇔ g = h ] by $[\varsigma(g,h)=0 \Rightarrow g=h]$ [ ς ( g , h ) = 0 ⇒ g = h ] ). Using this new type of metric spaces, we generalize many results in the literature. We prove fixed point results along with examples illustrating our theorems. Also, we present double controlled metric-like spaces endowed with a graph along with an open question.