Generalizing Kannan Fixed Point Theorem Using Higher-Order Metric Polynomials with Applications to Fractional Differential Equations

In this paper, we propose a new class of self-mappings, referred to as polynomial Kannan contractions, which extend the classical Kannan contractions by incorporating higher-order polynomial distance terms with variable coefficient functions. Unlike polynomial contractions, polynomial Kannan contractions are not necessarily continuous. We establish fixed point results for such mappings under suitable conditions on the coefficient functions, in addition to presenting the error estimates for the associated Picard iteration. Furthermore, we provide some supported numerical examples to show that our extensions are proper and significant. As an application, we show that our results ensure the existence and uniqueness of solutions for a certain class of fractional differential equations.