ABSTRACT: This paper investigates the applications of binary relations in fixed point theory on complete metric spaces, with particular emphasis on relation-preserving mappings and generalized contraction conditions. By incorporating a binary relation into the structure of a complete metric space, fixed point results can be established even when the underlying mapping does not satisfy standard contraction conditions globally. The study examines how completeness, relation compatibility, and contractive properties work together to guarantee the existence and, under suitable conditions, uniqueness of fixed points. Several generalized fixed point principles are discussed, including relation-theoretic versions of Banach-type contraction results. The approach is applicable to mappings defined on ordered metric spaces, partially ordered structures, and other relational frameworks. These results demonstrate that binary relations provide greater flexibility in modeling and solving fixed point problems arising in nonlinear analysis, optimization, and related mathematical applications. The study highlights the importance of relational structures in extending classical fixed point theory and developing more general and applicable fixed point theorems.
KEYWORDS: Binary relations, Fixed point theory, Complete metric spaces, Relation-preserving mappings, Contractive mappings, Nonlinear analysis, Ordered metric spaces
APPLICATIONS OF BINARY RELATIONS IN FIXED POINT THEORY ON COMPLETE METRIC SPACES
PRIYANKA , DR. NARENDRA SWAMI