Abstract
Abdeljawad et al. (2018) introduced a new concept, named double controlled metric type spaces, as a generalization of the notion of extended
b
-metric spaces. In this paper, we extend their concept and introduce the concept of double controlled quasi-metric type spaces with two incomparable functions and prove some unique fixed point results involving new types of contraction conditions. Also, we introduce the concept of
α
−
μ
−
k
double controlled contraction and prove some related fixed point results. We give several examples to show that our results are the proper generalization of the existing works.