This article presents a metric structure of the space of all Riemann-Stieltjes derivable functions defined over a fractal subset of the real line. Within this framework, we formulate, and analyze a measure of non-compactness tailored to fractal domains. Building upon this foundation, we develop a fixed point theorem in normed linear spaces under a generalized contraction condition, thereby extending Darbo's classical results. To illustrate the applicability of our theoretical findings, we apply this framework to the analysis of a class of fractal $\alpha$-linear differential equations, particularly focusing on models that exhibit oscillatory behaviors in fractal media settings where traditional methods often fail due to the irregularity of the domain. A numerical simulation using MATLAB supports the theoretical assertions, thus demonstrating that the noncompactness-based conditions imposed on the operator $\mathcal{L}$ are sufficient to ensure convergence within the function space $\mathcal{D}(\varphi )$. This approach offers novel insights into solving differential equations in non-Euclidean geometries, thereby emphasizing the interplay between metric structures, the fixed point theory, and mechanical systems in complex media.