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

Abstract

This paper introduces a novel framework that unifies Istrătescu-type contractions with simulation functions in the context of $b$ metric spaces. We define a new class of mappings, termed Istrătescu type $\Xi$-contractions, which generalize and extend several well-known contraction types from the literature. Our main result establishes the existence and uniqueness of fixed points for such mappings under mild continuity conditions, providing a unified approach to various fixed point theorems. The flexibility of our framework is demonstrated through several corollaries that recover important classical results as special cases. To illustrate the practical utility of our theoretical developments, we apply our main theorems to prove the existence and uniqueness of solutions for nonlinear fractional differential equations and nonlinear Volterra integral equations. The results presented herein not only advance fixed point theory in generalized metric spaces but also offer powerful tools for analyzing nonlinear problems in applied mathematics and related fields.