Refinements to Relation-Theoretic Contraction Principle

Abstract

After the appearance of relation-theoretic contraction principle proved in a metric space equipped with an amorphous binary relation (often termed as relational metric space), various core fixed point results have been proved in the setting of different relational distance spaces by varying underlying contraction conditions. In proving such results, the notions of completeness of ambient space, continuity of involved mapping and d-self-closedness of underlying binary relation are of paramount importance. The aim of this paper is to further refine the relation-theoretic contraction principle by relaxing the conditions of completeness and continuity by replacing their respective relation-theoretic analogues. Moreover, we observe that the notion of d-self-closedness utilized in relation-theoretic contraction principle is more general than the concepts of regularity and strong regularity utilized by earlier authors.

Keywords

• R -continuous mappings%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm200"> <mml:semantics> <mml:mi mathvariant="script">R</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous mappings
• <i>T</i>-closed binary relations
• R -connected sets%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm201"> <mml:semantics> <mml:mi mathvariant="script">R</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-connected sets