A counterexample to Abel-type asymptotics for scaled Volterra equations

TL;DR

This paper demonstrates that Abel-type bounds on kernels do not guarantee convergence of solutions in scaled Volterra equations, providing a counterexample with a positive kernel where solutions diverge.

Contribution

It constructs a specific positive kernel and forcing function showing solutions can diverge, challenging classical assumptions about Abel-type asymptotics.

Findings

A constructed kernel causes solutions to diverge at some point.

Global Abel bounds do not ensure solutions tend to zero.

Resolvants may not form a generalized approximate identity.

Abstract

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.