Extended Sehgal-Guseman Contractions in Generalized Metric Spaces with Applications to Fractional and Elastic Systems

This paper introduces and analyzes a novel class of Sehgal--Guseman-type contractions in the framework of extended $b$-metric spaces. By incorporating functional parameters that depend on iterates of the mapping, we establish generalized fixed-point theorems that significantly extend classical results. The proposed contraction conditions offer enhanced flexibility and applicability, particularly in nonlinear analysis. We demonstrate the practical relevance of our theoretical findings through applications to nonlinear fractional differential equations and boundary value problems, supported by numerical examples and comparative analysis. Our results contribute to the advancement of fixed-point theory in generalized metric settings and open new avenues for solving complex functional equations.