New results of fixed point theory on metric and partial metric spaces

Fixed-point theory has applications in various branches, whether it is applied science or pure science. It gives a unique formula to understand and study a large amount of linear-non-linear mathematical issues. The types of contraction mapping, including the concept of A-contraction and Z- contraction with reference to ζ, and others, are the most important cases which have been studied extensively by many authors to find reliable answers to the existence, and uniqueness of the fixed points in the field of nonlinear analysis. In this present thesis, we introduce a type of contraction mapping by introducing a new class called fφ-contraction with respect to the simulation function ζ and present some findings of the fixed points for mappings concerning in G- metric spaces. The extension and generalization of Banach's contraction principle are greatly supported by the contributions made by fφ- contraction. Furthermore, we have extended the common fixed point results in the setting of partial JS- metric spaces by using g(η; ξ)- contraction. The main goal of the current study is to investigate and continue different aspects of the results for fixed point in G- metric space and common fixed point in partial JS-metric spaces, which will help in providing a generalization to Z-contraction w.r.t ζ.