Exploring Fixed Point Theorems in \(C^\ast\)-Algebra Valued Fuzzy Soft Metric Spaces Using Cyclic \((\alpha, \beta)\)-\(\mathscr{F}_{\mathit{G}}(\varphi, \wp, \varpi)\)-Rational Contractions via \(\mathcal{C}_{\mathit{G}^\star}\)-Class Functions

Abstract

This study primarily aims to develop generalized \(\mathcal{C}_{\mathit{G}^\star}\)-class functions and establish fixed point results for cyclic \((\alpha, \beta)\)-\(\mathscr{F}_{\mathit{G}}(\varphi, \wp, \varpi)\)-rational contraction mappings within the framework of \(C^*\)-algebra-valued fuzzy soft metric spaces (\(\mathscr{C}^\ast\)-\(\mathscr{A}\mathbb{V}\mathbb{F}\mathbb{S}\mathbb{M}\mathbb{S}\)). The analysis incorporates key properties of various control functions to support the theoretical development. The findings not only extend previous work but also align with and enhance existing results in the literature. To illustrate the validity of our theorems, we present concrete examples. Additionally, we explore applications to homotopy theory and integral equations, highlighting the broader significance of the results.