Fixed point approximation of contractive-like mappings using a stable iterative family and its dynamics via quadratic polynomials

Abstract

This study aims at presenting a novel bi-parametric family of iterative methods for computing the fixed points of a contractive-like mapping. We thoroughly analyze the strong and stable convergence of the proposed technique and explore its applicability across various problem domains. Regarding convergence, it is proven that for several operators, the Mann iteration is analogous to the proposed multi-step class, and vice-versa. Moreover, numerical tests demonstrate the superior performance of the new procedures compared to existing three-step schemes. We further examine the dynamic behavior of several fixed-point iterative techniques when applied to quadratic polynomials. Based on the outcomes of these experiments, it can be concluded that the proposed family demonstrates both validity and effectiveness