The goal of this research article is to study some fixed point results in dislocated Quasi metric spaces. These results are proved for certain generalized contraction conditions involving linear and rational expressions. Our results extend and generalize some of the well-known fixed point results of the literature available in fixed point theory. Suitable examples of the established results are also given.

This paper, considers the fuzzy topological subsets, fuzzy topological spaces and introduces a novel concept of fuzzy Hausdorff space and fuzzy manifold space in this regards. Based on these concepts, we present a concept of fuzzy metric Hausdorff spaces and fuzzy metric manifold spaces via the notations of KM-fuzzy metric spaces. This study, generalises the concept of fuzzy metric space to union and product of fuzzy metric spaces in classical logic and in this regard investigates the some product of fuzzy metric fuzzy manifold spaces. We apply the notation of valued-level subsets and make a relation between of topological space, Husdorff space, manifold space and fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space. In final, we extended the fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space to fuzzy metric topological space, fuzzy metric Husdorff space and fuzzy metric manifold space Indeed, this study analyses the notation of fuzzy metric manifold based on valued-level subset.

We introduce the notion of Quasi-cyclic-noncyclic pair and its relevant new notion of coincidence Quasi-best proximity points in a con-vex metric space. In this way we generalize the notion of coincidence-best proximity point already introduced by M. Gabeleh et al [14]. It turns out that under some circumstances this new class of mappings contains the class of cyclic-noncyclic mappings as a subclass. The existence and con-vergence of coincidence-best and coincidence Quasi-best proximity points in the setting of convex metric spaces are investigated.

The main aim of this research paper is to introduce concept of Quasi-partial Branciari b-metric space. Such spaces are an extension of Quasi-partial metric spaces, Quasi-partial b-metric spaces and Quasi-partial Branciari metric spaces. In this article, firstly, Conditions for the existence and uniqueness of fixed points in underlying spaces are discussed and related theorems are proved. After that various consequences of these theorems are given and specific examples are presented.