Some Fixed Point Theorems in Generalized Metric Spaces and its Applications

Abstract

There has been broad research work in the arena of fixed point theory and the area still ensues to attract researchers. But the foremost investigation of fixed points focuses has been centered around metric spaces, spaces with higher designs. Still, there is a wide extension to apply this idea in pretty much every part of Mathematics that has arisen as of not long ago. With a similar perspective on featuring the relevance of Fixed Point Theory in assorted parts of Mathematics, in our work, we have touched lot of structures for pursuit of fixed points. newlineLet and#119879; be a self map of a nonempty set and#119883;. A point and#119901;and#8712;and#119883; such that and#119879;and#119901;=and#119901; is called a fixed point of the map and#119879;. If and#119879; is a multivalued map, i.e., from and#119883; to the collection of nonempty subsets of and#119883;, then a point and#119901; in and#119883; is termed as a fixed point of and#119879; if and#119901;and#8712; and#119879;and#119901;. The significance of the fixed point theory lies principally in the way that the vast majority of the situations emerging in the different actual plans might be changed to fixed point conditions or considerations. The theorems concerning the properties and existence of fixed points are identified as Fixed Point Theorems. The roots of fixed point theory lie in the process of successive estimates for proving existence of solutions of differential equations announced autonomously in 1837 by J. Liouville [52] and in 1890 by C. E. Picard [69] and related results developed by Granas and Dugundji [34], Kirk and Sims [47], Zeidler [92]. Be that as it may, officially its starting point returns to the start of twentieth century as an imperative part of nonlinear analysis. The generalization of this old style hypothesis is the spearheading work of the incredible Polish Mathematician S. Banach [2] put out in 1922 which gives a productive strategy to find the fixed points of a map. Fixed point theory has a delightful combination of analysis, topology, and geometry. Since from 1922 the theory of fixed points has been uncovered as an extremely newlineincredible and significant method for tackling an assortment of applied iss