Surveys in Mathematics and its Applications
In the present work the use of ternary relations is introduced in fixed point theory to obtain some fixed point results in G-metric spaces. Amongst several generalizations of metric spaces suggested in recent times, G-metric spaces are the ones in which the metric is replaced by a function through which sets of three elements are assigned to non-negative real numbers. A ternary relation is assumed on the space and a generalized contractive condition is assumed for the triplets of elements related by the ternary relation. Fixed point and related results are established for such contractions as generalization of contractive mapping principle. The case without the assumption of ternary relation on the space is also discussed. There are some corollaries and illustrative examples. The illustrations establish the actuality of the generalization. The methodology of the proofs are new in the context of G-metric spaces.
Banach’s contraction mapping principle is a well known result of functional analysis which is established in the general setting of complete metric spaces and has served as the basis of many fundamental results in mathematical studies . It has been extended to several other spaces which are generalizations of metric spaces like cone metric spaces , fuzzy metric spaces , probabenumerateic metric spaces  and the like.
G-metrics spaces, Ternary relation, Fixed point, Orbital point, Contraction
Y. U. Gaba, C. A. Agyingi, Binayak S. Choudhury, P. Maity. (2019). Generalized Banach contraction mapping principle in generalized metric spaces with a ternary relation. Volume 14, 159 – 171.