Morawo Common Fixed Point under Commutative Contraction Mappings in Digital Metric Spaces

Abstract

The contraction mapping principle is a fundamental tool for establishing solutions to linear and nonlinear systems of ordinary differential equations. It guarantees that a contraction self-mapping on a complete metric space has a unique fixed point that can be approximated through an iterative sequence generated from an arbitrary initial point. This study examines the existence, uniqueness, and related analytical properties of fixed points for contraction mappings in digital metric spaces, including the uniform continuity of such mappings. Using a theoretical and analytical approach, existing results on dual-commutative contraction mappings were reviewed to determine the conditions under which unique fixed points exist in digital metric spaces. These results were subsequently extended from dual-commutative to triple-commutative contraction mappings. The analysis establishes that the proposed triple-commutative mappings satisfy the stated contraction conditions required for the existence and uniqueness of fixed points within the digital metric framework. Illustrative examples are provided to demonstrate the applicability of the established results. This extension broadens the theoretical framework of fixed-point analysis in digital metric spaces and provides a foundation for further investigations of higher-order commutative contraction mappings.