Generalized Banach contraction mapping principle in generalized metric spaces with a ternary relation

Abstract

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.

Keywords

• G-metric spaces
• Ternary relation
• Fixed point
• Orbital point
• Contractio
• Contraction