A counterexample to Abel-type asymptotics for scaled Volterra equations

We consider scaled Volterra equations of the form $f_n+nk\ast f_n=g$ 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 $n\to +\mathrm{\infty }$. More precisely, we construct a continuous strictly positive kernel globally comparable with the Abel kernel $x^{-1\mathrm{/}2}$, and a continuous strictly positive $g$, for which a subsequence of $(f_n{)}_{n\in \mathbb{N}}$ diverges to $+\mathrm{\infty }$ 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.