@article{Sajid2025, 
author = {Mohammad Sajid and Abhishikta Das and Hemanta Kalita},
title = {Metric structure for Riemann-Stieltjes derivable functions on fractals and application},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {15737-15754},
keywords = {fractal set, fractal functions, fractal calculus, α-linear differential equations, measure of non-compactness, Riemann-Stieltjes integration, metric space, fixed point, Darbo's theorem},
url = {https://www.sciopen.com/article/10.3934/math.2025705},
doi = {10.3934/math.2025705},
abstract = {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    α-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        L   are sufficient to ensure convergence within the function space        D    (  ϕ  ). 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.}
}