| Summary: | Since 1964, when I.A. Perov introduced the so-called generalized metric space where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an element of the vector space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></semantics></math></inline-formula>, many researchers have considered various contractive conditions in this type of space. In this paper, we generalize, extend and unify some of those established results. We are primarily concerned with examining the existence of a fixed point of some mapping from <i>X</i> to itself, but if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> belongs to some relation <i>R</i> on the set <i>X</i>, then the binary relation <i>R</i> and some F contraction defined on the space cone <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></semantics></math></inline-formula> are combined. We start our consideration with the recently announced results and give them strict, critical remarks. In addition, we improve several announced results by weakening some of the given conditions.
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