In this paper, some fixed point results of self mapping which is defined on orthogonal CONE METRIC SPACEs are given by using extensions of orthogonal contractions. And by taking advantage of these results, the necessary conditions for self mappings on orthogonal CONE METRIC SPACE to have P property are investigated. Also an example is given to illustrate the main results.

In this paper we introduce the CONE bounded linear mapping and show that the CONE norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-CONE normed SPACEs. We also show that under some restrictions on the CONE, two TVS-CONE norms are equivalent if and only if they induce equivalent topologies. In the sequel, the notion of algebraic CONE METRIC is introduced and it is shown that every algebraic CONE METRIC SPACE has a topology and the Banach fixed point theorem for contraction mappings on algebraic CONE METRIC SPACEs is proved.

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.

In this paper, we prove some fixed point theorems for self-mappings on an algebraic CONE METRIC SPACE. These results are related to the product of the CONE, and improve some well-known results by inserting an algebraic CONE $\PP$ instead of $\mathbb{R}^{+}$.