Existence Result for Difference Equations on Non-Uniform Grids via Upper and Lower Solution Method
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| Format: | Preprint |
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2025
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| _version_ | 1866918116371464192 |
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| author | Bandyopadhyay, Shalmali Lor, Kimser |
| author_facet | Bandyopadhyay, Shalmali Lor, Kimser |
| contents | This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form $u^{ΔΔ}(t_{i-1}) + f(t_i, u(t_i), u^Δ(t_{i-1})) = 0$ on a non-uniform time grid ${t_0, t_1, \ldots, t_{n+2}}$ with mixed boundary conditions $u^Δ(t_0) = 0$ and $u(t_{n+2}) = g(t_{n+2})$. This extends previous work on homogeneous boundary conditions to the non-homogeneous case, requiring a sophisticated functional analytic framework to handle the resulting affine function spaces. Our approach employs a decomposition strategy that separates boundary effects from the differential structure, enabling the application of Brouwer's Fixed Point Theorem to establish existence with solutions bounded between upper and lower functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04706 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence Result for Difference Equations on Non-Uniform Grids via Upper and Lower Solution Method Bandyopadhyay, Shalmali Lor, Kimser General Mathematics This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form $u^{ΔΔ}(t_{i-1}) + f(t_i, u(t_i), u^Δ(t_{i-1})) = 0$ on a non-uniform time grid ${t_0, t_1, \ldots, t_{n+2}}$ with mixed boundary conditions $u^Δ(t_0) = 0$ and $u(t_{n+2}) = g(t_{n+2})$. This extends previous work on homogeneous boundary conditions to the non-homogeneous case, requiring a sophisticated functional analytic framework to handle the resulting affine function spaces. Our approach employs a decomposition strategy that separates boundary effects from the differential structure, enabling the application of Brouwer's Fixed Point Theorem to establish existence with solutions bounded between upper and lower functions. |
| title | Existence Result for Difference Equations on Non-Uniform Grids via Upper and Lower Solution Method |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2508.04706 |