New fixed point theorems for double generalized contraction in Banach spaces

Abstract

In this work, we establish new fixed point theorems for a novel class of mappings, called double generalized contractions, in the setting of Banach spaces. We prove the existence and uniqueness of fixed points and analyze the convergence of the associated iterative schemes. Our approach extends and unifies several known results, including Banach contraction, enriched contraction, and weak enriched contractions. To illustrate the applicability of the theory, we present numerical examples together with applications to both linear systems and nonlinear functional equation. These findings provide a constructive answer to recent open questions concerning the extension of weak enriched contractions and the development of more flexible iterative processes. The proposed framework significantly broadens the scope of fixed point theory in Banach spaces and offers new tools for solving problems that cannot be treated by classical or single-averaging techniques.