Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces

During the sixties, the notion of 2-metric space introduced by Gähler see 1, 2 as a generalization of usual notion ofmetric space X, d . But different authors proved that there is no relation between these two functions, for instance, Ha et al. in 3 show that 2-metric need not be continuous function, further there is no easy relationship between results obtained in the two settings. In 1992, Bapure Dhage in his Ph.D. thesis introduce a new class of generalized metric space called D-metric spaces 4, 5 . In a subsequent series of papers, Dhage attempted to develop topological structures in such spaces see 5–7 . He claimed that D-metrics provide a generalization of ordinary metric functions and went on to present several fixed point results. But in 2003 in collaboration with Brailey Sims, we demonstrated in 8 that most of the claims concerning the fundamental topological structure of D-metric space are incorrect, so, we introduced more appropriate notion of generalized metric space as follows.