G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications

The primary aim of this manuscript is to establish unique fixed point results for a class of Ψ-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of Ψ-contraction operators, we introduce a novel function, denoted as ψ, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions.