On the kernel of the Stieltjes derivative and the space of bounded Stieltjes-differentiable functions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917972473282560 |
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| author | Fernández, Francisco J. Albés, Ignacio Márquez Tojo, F. Adrián F. Mariz, Carlos Villanueva |
| author_facet | Fernández, Francisco J. Albés, Ignacio Márquez Tojo, F. Adrián F. Mariz, Carlos Villanueva |
| contents | We investigate the existence and uniqueness of solutions to first-order Stieltjes differential problems, focusing on the role of the Stieltjes derivative and its kernel. Unlike the classical case, the kernel of the Stieltjes derivative operator is nontrivial, leading to non-uniqueness issues in Cauchy problems. We characterize this kernel by providing necessary and sufficient conditions for a function to have a zero Stieltjes derivative. To address the implications of this nontrivial kernel, we introduce a function space which serves as a suitable framework for studying Stieltjes differential problems. We explore its topological structure and propose a metric that facilitates the formulation of existence and uniqueness results. Our findings demonstrate that solutions to first-order Stieltjes differential equations are, in general, not unique, underscoring the need for a refined analytical approach to such problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the kernel of the Stieltjes derivative and the space of bounded Stieltjes-differentiable functions Fernández, Francisco J. Albés, Ignacio Márquez Tojo, F. Adrián F. Mariz, Carlos Villanueva Classical Analysis and ODEs 34A12, 34A30, 34A36, 26A24, 46E30 We investigate the existence and uniqueness of solutions to first-order Stieltjes differential problems, focusing on the role of the Stieltjes derivative and its kernel. Unlike the classical case, the kernel of the Stieltjes derivative operator is nontrivial, leading to non-uniqueness issues in Cauchy problems. We characterize this kernel by providing necessary and sufficient conditions for a function to have a zero Stieltjes derivative. To address the implications of this nontrivial kernel, we introduce a function space which serves as a suitable framework for studying Stieltjes differential problems. We explore its topological structure and propose a metric that facilitates the formulation of existence and uniqueness results. Our findings demonstrate that solutions to first-order Stieltjes differential equations are, in general, not unique, underscoring the need for a refined analytical approach to such problems. |
| title | On the kernel of the Stieltjes derivative and the space of bounded Stieltjes-differentiable functions |
| topic | Classical Analysis and ODEs 34A12, 34A30, 34A36, 26A24, 46E30 |
| url | https://arxiv.org/abs/2503.23962 |