An affirmative answer to quasi-contraction's open problem under some local constraints in JS-metric spaces

In this work, we first present JS-Pompeiu-Hausdorff metric in JS metric spaces and then introduce well-behaved quasi-contraction in order to find an affirmative answer to quasi-contractions’ open problem under some local constraints in JS-metric spaces. In the literature, this problem solved when the constant modules α ∈ [0,1/2] and when α ∈ (1/2,1], finding conditions by which the set of all fixed points be non-empty, has remained open yet. Moreover, we support our result by a notable example. Finally, by taking into account the approximate strict fixed point property we present some worthwhile open problems in these spaces