This study aims to provide some new classes of (α,β,F*)-weak Geraghty contraction and (α,β,F**)-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we find that the mappings satisfying the definition of such contractions have a unique fixed point if the underlying space is complete. In addition, we provide an application showing the uniqueness of the solution of the two-point boundary value problem.

This study aims to provide new classes of (α,β,F∗)-weak Geraghty contraction and (α,β,F∗∗)-weak Geraghty contraction, a self generalized contractions on any metric space. Further we find that the mappings satisfying the definition of such contractions have unique fixed point if the base space is complete. Also there is an application for existing unique solution of two-point boundary value problem.

Two new types of metric spaces namely C∗−algebra valued Branciari Sb−metric space and C∗−algebra valued extended Branciari Sb−metric space are introduced in this work, which are a generalization of all known related metric spaces.We explained the generalizations with two examples respectively. There are examples of self-mappings having a unique fixed point, which are concluded using our main theorems. As an application, we used one of our main results to show the existence of a unique solution of an algebraic system of linear equations with a numerical example. Mathematics Subject Classification (2010). Primary 54H25; Secondary 47H10.

This study aims to provide a new class of Ćirić α+(θ, ϕ)-proximal contraction, a non-self generalized poximal contraction mapping on a non-empty closed subset of any metric space. Also we have proved that such contractions satisfying some conditions must have unique best proximity point if we take the base space as complete. For some particular values of the constants, that we have used to generalize the proximal contraction, we conclude different types of proximal contractions with proof, that all such type of contractions must have unique best proximity point. Mathematics Subject Classification (2010). Primary 47H10, 54H25; Secondary 41A65, 46B20, 47H05, 47H09.

A new variety of non-self generalised poximal contraction,called Hardy-Rogers α+F-proximal contraction, is shown in this work.Also we have proved that such contractions satisfying some conditionsmust have unique best proximity point. For some particular values of theconstants, that we have used to generalize the proximal contraction, weconclude different α+F-proximal contraction results of the types ´Ciri´c,Chatterjea, Reich, Kannan and Banach with proof, that all such typeof contractions must have unique best proximity point.