Further investigation on fixed point theorems via C-class functions in extended b-metric spaces

Abstract

The purpose of this research project is to develop new theories, discuss, and extend some recent common fixed point results established when the underlying ambient space is an extended b-metric space and the contraction condition involves a new class of ψ-φ-C-contraction type mappings where ψ is the altering distance function and φ is the ultra-altering distance function. The unique fixed point theorems for such mappings in the setting of ψ-φ-complete metric spaces are proven. We also prove the fixed point theorem in partially ordered metric spaces. Moreover, some examples supporting the main results are given. Our results extend and generalize corresponding results in the literature. The start of the development of the theory of fixed points is tied to the end of the 19^th century. The method of successive approximations is used in order to prove the solution's existence and uniqueness at the beginning of differential and integral equations. This branch of nonlinear analysis has been developed through various classes of spaces, such as metric spaces, topological spaces, probabilistic metric spaces, fuzzy metric spaces, and others. In developing the theory of fixed points, achievements are applied in various sciences, such as optimization, economics, and approximation theory. A very important step in the development of fixed point theory was taken by A.H. Ansari through the introduction of a C-class function. Using C-class functions, we generalize some known fixed point results, and Kamran et al. introduced a new intuitive concept of distance measure to extend the notion of b-metric space by further weakening the triangle inequality.