Fixed Point Methodologies for ψ-Contraction Mappings in Cone Metric Spaces over Banach Algebra with Supportive Applications

The explicit aim of this manuscript is to obtain fixed point consequences under novel ψ-contraction mappings in a complete cone metric space over Banach algebra. We connect and relate different fixed point theorems by using the idea of ψ-contraction mappings, providing a thorough viewpoint that deepens our comprehension of this topic. Our theorems generalize and unify many results in the scientific literature. These prospective extensions offer intriguing research directions and have the potential to further advance the study of fixed point theory. The investigation of examples plays an extremely crucial role in verifying the effectiveness and validity of our theoretical results. Moreover, to support the theoretical results, some examples are investigated to emphasize these results. Ultimately, the existence and uniqueness of the solution to the Urysohn integral and nonlinear fractional differential equation are cooperated as applications to provide an authoritative basis for dealing with actual problems that include these equations.