The concept of a $w$-distance on a‎ ‎metric space has been introduced by Kada et al‎. ‎\cite{Kst}‎. ‎They generalized Caristi‎ ‎fixed point theorem‎, ‎Ekeland variational principle and the‎ ‎nonconvex minimization theorem according to Takahashi‎. ‎In the present paper‎, ‎we first introduce the notion of quasi $w$-distances in quasi-metric spaces and then we will prove some fixed point theorems for $\mathcal{L}$-contractive mappings in the class of quasi-metric spaces with $w$-distances via a control function introduced by Jleli and Samet \cite{JL}‎. ‎These results generalize many fixed point theorems by Kada et al‎. ‎\cite{Kst}‎, ‎Suzuki \cite{S}‎, ‎Ciri\'{c} \cite{ciric}‎, ‎Aydi et al‎. ‎\cite{Aydbarlak}‎, ‎Abbas and Rhoades \cite{Ar}‎, ‎Kannan \cite{Kannan}‎, ‎Hicks and Rhoades \cite{H}‎, ‎Du \cite{D}‎, ‎Lakzian et al‎. ‎\cite{LAR}‎, ‎Lakzian and Rhoades \cite{LR} and others‎. ‎Some examples in support of the given concepts and presented results‎.