Neutrosophy deals with neutrosophic logic, probability, and sets. Actually, the neutrosophic set is a generalization of the classical set, fuzzy set, and intuitionistic fuzzy set. A neutrosophic set is a mathematical notion serving issues containing inconsistent, indeterminate, and imprecise data. The notion of intuitionistic fuzzy metric space is useful in modelling some phenomena, where it is necessary to study the relationship between two probability functions. In this study, the concept of an orthogonal neutrosophic metric space is initiated. It is a generalization of the neutrosophic metric space. Some fixed point results are investigated in this setting. For the validity of the obtained results, some nontrivial examples are given.

In this article, we are generalizing the concept of control fuzzy metric spaces by introducing orthogonal control fuzzy metric spaces. We prove some fixed point results in this setting. We provide nontrivial examples to show the validity of our main results and the introduced concepts. An application to fuzzy integral equations is also included. Our results generalize and improve several developments from the existing literature.

In this manuscript, we introduce the concept of intuitionistic fuzzy controlled metric-like spaces via continuous t-norms and continuous t-conorms. This new metric space is an extension to intuitionistic fuzzy controlled metric-like spaces, controlled metric-like spaces and controlled fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We prove some fixed-point theorems and we present non-trivial examples to illustrate our results. We used different techniques based on the properties of the considered spaces notably the symmetry of the metric. Moreover, we present an application to non-linear fractional differential equations.

In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result.

In this manuscript, owing to the concept of orthogonal coupled contraction mappings type I and II, we prove coupled fixed point theorem in orthogonal metric spaces. In order to strengthen our main results, a suitable example is presented. Moreover, the results we obtained supplement and improve previous research findings. A fruitful application is also supplied to endorse our outcomes.

In this article, we introduce the concept of intuitionistic fuzzy double controlled metric spaces that generalizes the concept of intuitionistic fuzzy b-metric spaces. For this purpose, two noncomparable functions are used in triangle inequalities. We generalize the concepts of the Banach contraction principle and fuzzy contractive mappings by giving authentic proof of these mappings in the sense of intuitionistic fuzzy double controlled metric spaces. To validate the superiority of these results, examples are imparted. A possible application to solving integral equations is also set forth towards the end of this work to support the proposed results.