Debnath and De La Sen introduced the notion of set valued interpolative Hardy-Rogers type contraction mappings on b-metric spaces and proved that on a complete b-metric space, whose all closed and bounded subsets are compact, the set valued interpolative Hardy-Rogers type contraction mapping has a fixed point. This article presents generalizations of above results by omitting the assumption that all closed and bounded subsets are compact.

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 introduce a few new generalizations of the classical Perov-fixed point theorem for single-valued and multi-valued mappings in a complete generalized metric space endowed with a binary relation. We have furnished our work with examples to show that several metrical-fixed point theorems can be obtained from an arbitrary binary relation.

In this paper, we investigate the existence of a fixed point for modified multivalued ?*-?-contractive type mapping in the context of complete metric space. We also construct some examples to illustrate the main result. Our results extend, improve and generalize the results on the topic in the literature.