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}(\phi) $. 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.
Citation: Mohammad Sajid, Abhishikta Das, Hemanta Kalita. Metric structure for Riemann-Stieltjes derivable functions on fractals and application[J]. AIMS Mathematics, 2025, 10(7): 15737-15754. doi: 10.3934/math.2025705
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}(\phi) $. 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.
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Mohammad Sajid, Abhishikta Das, Hemanta Kalita. Metric structure for Riemann-Stieltjes derivable functions on fractals and application[J]. AIMS Mathematics, 2025, 10(7): 15737-15754. doi: 10.3934/math.2025705
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