On hybrid self-correcting contractions and the convergence of picard iterations in metric spaces

In this paper we present a new class of contractive mappings, called accelerated hybrid self-correcting contractions, in metric spaces. The proposed contractive condition combines Banach, Kannan and Chatterjea type distance terms with a nonlinear geometric mean component, thereby providing a unified hybrid structure for generalized contractions. We establish existence and uniqueness results for fixed points of such mappings in complete metric spaces and prove the geometric convergence of the associated Picard iteration together with explicit error estimates. An example is presented to show that the proposed class non-trivially extends the classical Banach contraction principle. Finally, an application to a nonlinear integral equation is provided to illustrate the applicability of our results.