Fractional Integral Contractions in Symmetric S-Metric Spaces with Lucas Polynomial Spectral Collocation

This study introduces a novel class of fractional integral contractions within the framework of symmetric S-metric spaces. By embedding a Riemann–Liouville fractional integral (RLFI) operator into an integral-type contractive condition, we offer a versatile extension of classical fixed point principles. Assuming suitable properties for the associated locally bounded and locally integrable function, we prove the existence and uniqueness of a fixed point for self-mappings on symmetric complete S-metric spaces, demonstrating that the corresponding Picard iterative sequence converges to this unique point. Notably, setting the fractional order to unity recovers the classical integral-type contractive condition. To showcase the practical utility of this theoretical framework, we develop a numerical spectral collocation method for solving related fractional differential equations. Utilizing Lucas polynomials as basis functions alongside Riemann equidistant collocation nodes, we transform continuous equations into discrete algebraic systems. The established fixed point theorems provide the analytical foundation for the convergence of the iterative algorithms used to solve these nonlinear systems. We present a comprehensive algorithm, derive rigorous error bounds, and provide numerical examples with tabular and graphical illustrations that confirm the spectral accuracy and exponential convergence rates of the proposed methodology.