A counterexample to Abel-type asymptotics for scaled Volterra equations
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913058203369472 |
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| author | Gregosiewicz, Adam |
| author_facet | Gregosiewicz, Adam |
| contents | We consider scaled Volterra equations of the form $f_n + n k*f_n = g$ for $n \in \mathbb{N}$, where $g$ is given and $f_n$ is sought. We show that global two-sided Abel-type bounds on a positive kernel $k$ do not force the solutions $f_n$ to converge to zero as $n \to +\infty$. More precisely, we construct a continuous strictly positive kernel globally comparable with the Abel kernel $x^{-1/2}$, and a continuous strictly positive $g$, for which a subsequence of $(f_n)_{n \in \mathbb{N}}$ diverges to $+\infty$ at some point $x_0 > 0$. Consequently, the resolvents associated with the scaled kernels $nk$ need not form a generalized approximate identity, in contrast to a couple of classical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21944 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A counterexample to Abel-type asymptotics for scaled Volterra equations Gregosiewicz, Adam Classical Analysis and ODEs Functional Analysis 45D05 (Primary) 45M05, 45E10 (Secondary) We consider scaled Volterra equations of the form $f_n + n k*f_n = g$ for $n \in \mathbb{N}$, where $g$ is given and $f_n$ is sought. We show that global two-sided Abel-type bounds on a positive kernel $k$ do not force the solutions $f_n$ to converge to zero as $n \to +\infty$. More precisely, we construct a continuous strictly positive kernel globally comparable with the Abel kernel $x^{-1/2}$, and a continuous strictly positive $g$, for which a subsequence of $(f_n)_{n \in \mathbb{N}}$ diverges to $+\infty$ at some point $x_0 > 0$. Consequently, the resolvents associated with the scaled kernels $nk$ need not form a generalized approximate identity, in contrast to a couple of classical results. |
| title | A counterexample to Abel-type asymptotics for scaled Volterra equations |
| topic | Classical Analysis and ODEs Functional Analysis 45D05 (Primary) 45M05, 45E10 (Secondary) |
| url | https://arxiv.org/abs/2604.21944 |