Unified Fixed Point Theory in Generalized Metric Structures with Applications to Nonlinear Economic Systems

Abstract

This paper introduces a comprehensive framework unifying recent advancements in fixed point theory through the novel concept of \emph{twisted weighted $\Theta$-$b$-metric spaces}. We establish a framework of fixed point theorems for multi-valued mappings satisfying generalized rational type contractions that incorporate control functions, weight functions, and twisted admissibility conditions. By synthesizing concepts from \v{C}iri\'{c}-type contractions, Berinde's almost contractions, Jleli's $\Theta$-contractions, and weighted $b$-metric spaces, we create a powerful analytical tool with unprecedented theoretical depth. The work provides rigorous proofs, extensive numerical validation, and demonstrates significant applications to economic systems, including production-consumption equilibrium models and fractional economic growth equations. Our results substantially generalize numerous classical theorems while opening new avenues for research in nonlinear analysis and mathematical economics.