The aim of this manuscript is to initiate the study of the Banach contraction in R-fuzzy b-metric spaces and discuss some related fixed point results to ensure the existence and uniqueness of a fixed point. A nontrivial example is imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the provided results, an application is presented to solve the first kind of a Fredholm-type integral equation.

The aim of this manuscript is to introduce the concept of fuzzy b-metric-like spaces and discuss some related fixed point results. Some examples are imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the obtained results, an application is provided to solve a first kind of Fredholm type integral equations.

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 paper, we initiate the concept of orthogonal partial b -metric spaces. We ensure the existence of a unique fixed point for some orthogonal contractive type mappings. Some useful examples are given, and an application is also provided in support of the obtained results.

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.