On weakly contractions via $w$-distances

In this article, we will check whether the known results remain valid if the metric $d$ is replaced by the $w$ -distance $p$. we show that in some contractive conditions where $w$-distance $p$ participates instead of metric $d$, symmetry of $w$-distance $p$ can be assumed and the proofs can be shorter. We are talking about results such as Banach's contraction principle, Kannan's theorem, Boyd–Wong, Meir–Keeler, Chatterje's, Reich's, Hardy–Rogers', Karapinars' and Wardowskis' theorems and many others.
By doing so, we would obtain generalizations of the above results.