A Unified Framework for Generalized Contractions via Simulation Functions in Sb- Metric Spaces with Applications to Nonlinear Analysis

Abstract

This paper establishes a comprehensive theory of multi-coupled fixed points for nonlinear operators in two significant generalizations of metric spaces: G-metric spaces and Sb-metric spaces. We introduce several new existence and uniqueness theorems under diverse contractive conditions, substantially extending previous results in the literature. The main contributions include: (1) a unified treatment of n-coupled fixed points under ϕ-contraction conditions in complete G-metric spaces; (2) theorems for asymmetric linear contractions with coefficients summing to less than one; (3) results for mixed monotone operators in partially ordered G-metric spaces; (4) extensions to Sb-metric spaces with contraction constants respecting the geometric constraints imposed by the coefficient b ≥ 1; and (5) common multi-coupled fixed point theorems for compatible mappings. Each theorem is accompanied by a complete proof employing novel iterative techniques and careful analytical estimates adapted to the ternary structures of these generalized metric spaces. We provide illustrative examples demonstrating the applicability and advantages of our approach, including cases with discontinuous operators and asymmetric dependencies. Furthermore, we develop substantial applications for systems of nonlinear integral equations, boundary value problems with coupled boundary conditions, and economic equilibrium models with interdependent markets. These applications validate the practical utility of our theoretical framework and establish meaningful connections to problems in applied mathematics. The results presented herein provide both theoretical advances and practical tools for analyzing complex nonlinear systems with multiple interdependent variables,contributing significantly to fixed point theory and its applications.