Some Results of Fixed Points in Generalized Metric Space by Methods of Suzuki and Samet

Abstract

In 1992 Dhage introduced the notion of generalized metric or D-metric spaces and claimed that D-metric convergence define a Hausdorff topology and that $D$-metric is sequentially continuous in all the three variables. Many authors have taken these claims for granted and used them in proving fixed point theorems in $D$-metric spaces. In 1996 Rhoades generalized Dhages contractive condition by increasing the number of factors and proved the existence of unique fixed point of a self map in $D$-metric space. Recently motivated by the concept of compatibility for metric space. In 2002 Sing and Sharma introduced the concept of $D$-compatibility of maps in $D$-metric space and proved some fixed point theorems using a contractive condition. In this paper ,we prove some fixed point theorems and common fixed point theorems in $D^*$-complete metric spaces under particular conditions among weak compatibility. Also by Using method of Suzuki and Samet we prove some theorems in generalised metric spaces.