Comparison on Contraction Conditions and Conjecture in G-Metric Space

Fixed point theory is an important discipline in Mathematics because of its results which are utilized to investigate the existence of solutions for the problems in applied sciences and engineering. The theorems in fixed point theory assured that a self-map on a metric space has an invariant point that can be noticed as the limit of an iterative scheme obtained by repeated elements under the contraction mapping of an initial point in the metric space. Many fixed-point results have been widely generalized throughout the years in various directions by introducing new metric spaces and setting of new contraction mappings. The results in fixed point theory can be noticed in geometry, computational algorithms, economics, fluid dynamics, micro-structures, nonlinear sciences, medical fields and optimization theory. Recently many Mathematicians have developed various contractions in G-metric spaces. In this paper, brief comparison was done on G-contractions developed by Mohanta, Vats et. al. and Phaneendra et. al. With this comparison, we realize that some more terms need to be added and also need to change the restricted range of constant in the condition, to get the generalized contraction condition. Finally, conjecture was made based on the comparison of these existed G-contractions. It is observed that, conjecture gives the generalization of results established by Mohanta, Vats et. al. and Phaneendra et. al.