Are fixed point theorems in G-metric spaces an authentic generalization of their classical counterparts?

Abstract

Many G-metric fixed point results can be retrieved from classical ones given in the (quasi-)metric framework. Indeed, many G-contractive conditions can be reduced to a quasi-metric counterpart assumed in the statement of celebrated fixed point results. In this paper, we show that the existence of fixed points for the most part in the aforesaid G-metric fixed point results is guaranteed by a very general celebrated result by Park, even when the G-contractive condition is reduced to a quasi-metric one which is not considered as a contractive condition in any celebrated fixed point result. Moreover, in all those cases in which a quasi-metric contractivity can be raised, we show that the uniqueness of the fixed point is also derived from it. Despite our finding, we also show that there are a few G-metric fixed point results in the literature whose contractive condition cannot be reduced to any (quasi-)metric counterpart. In these cases, however, we have seen that Park’s result applies again for all cases analyzed, and the G-metric techniques become essential to yield the uniqueness of the fixed point, and, even, to check that the self-mapping under study satisfies some requirements of the Park result. Therefore, it seems natural to encourage the researchers in this field to only study new results in this direction, i.e., in which the aforesaid Park’s result cannot be applied.