ABSTRACT: This study presents a comprehensive analysis of fixed point theorems in metric spaces equipped with arbitrary binary relations. Classical fixed point theory, particularly the Banach contraction principle, generally relies on the global structure of the metric space. However, many applications involve additional relational structures that allow contraction conditions to be imposed only on related elements. In this framework, we investigate fixed point results for self-mappings satisfying relation-dependent contractive conditions in a metric space. Appropriate assumptions concerning relation preservation, completeness, and the existence of suitable initial points are considered to establish the existence and, where applicable, uniqueness of fixed points. The approach extends several classical fixed point principles by incorporating arbitrary binary relations, thereby providing a flexible framework for studying mappings that do not satisfy conventional contraction conditions throughout the entire space. The results demonstrate that binary relations can effectively replace global assumptions with more localized and structurally meaningful conditions. This framework has potential applications in nonlinear analysis, optimization, and other areas where relational constraints naturally arise.
KEYWORDS: Fixed point theorem, metric space, binary relation, relational contraction, Banach contraction principle, complete metric space.
FIXED POINT THEOREMS IN METRIC SPACES WITH ARBITRARY BINARY RELATIONS
PRIYANKA, DR. NARENDRA SWAMI