Remarks on “Perov Fixed-Point Results on F-contraction Mappings Equipped with Binary Relation”

Abstract

Since 1964, when I.A. Perov introduced the so-called generalized metric space where \(d(x,y)\) is an element of the vector space \(\mathbb{R}^m\). Since then, many researchers have considered various contractive conditions on this type of spaces. In this paper, we generalize, extend and unify some of those established results. It is primarily about examining the existence of a fixed point of some mapping from \(X\) to itself, but if \((x,y)\) belong to some relation \(R\) on the set \(X\). Then the binary relation \(R\) and some \(F\) contraction defined on the space cone \(\mathbb{R}^m\) are combined. We start our consideration on the paper [1] and give strict critical remarks on the results published in there. Also, we improve their result by weakening one condition.