Existence and Uniqueness Theorem of Fixed Points in Generalized Metric Spaces in Generalized Metrics Spaces

Abstract

The study of fixed points is one of the most important areas of mathematical analysis, and it has many uses in different fields. In recent years, the study of generalized metric spaces has expanded the field of research, which has helped us learn more about fixed point theorems. In this paper, the uniqueness theorem of fixed points is looked at in detail in the setting of generalized metric spaces. Generalized metric spaces are more broad than traditional metric spaces because they loosen some or all of the axioms. They are useful in situations where the basic structure is more complex and where the traditional idea of distance is replaced by a more flexible idea. This paper presents the uniqueness theorem, which is one of the most important ideas in the study of fixed points in generalized metric spaces. It gives the conditions for a single fixed point to exist for a certain self-mapping. This makes mathematical analysis in these complex spaces clearer and more reliable. Our research includes an overview of the basic ideas behind generalized metric spaces, which helps us understand the features and structures that are at play. We investigate the uniqueness theorem's importance and what it means in real-world situations by carefully looking at related theorems and proofs. The wide range of applications of this theorem is shown by examples from different areas of mathematics. The uniqueness theorem of fixed points in generalized metric spaces not only makes fixed point theory better, but it also opens the door to new areas of study and uses in many different areas of science. We end by encouraging more research into how generalized measure spaces can help solve difficult problems and broaden the scope of mathematical analysis.