 Solution

Let $y$ be any continuous function solving the integral equation on $[0,\infty )$, and retain $B=|b|>0$. A first-exit argument avoids assuming in advance that its supremum norm is finite. If $|y|$ ever reached $2B$, continuity would give a first such time $T>0$. On $[0,T]$ we would have $|y|\le 2B$, so the kernel estimate would implya contradiction. Therefore $|y(t)|<2B$ for every finite $t$.

That pointwise strict inequality alone would not give a strict bound on the supremum norm. Apply the integral equation once more, now using $|y|\le 2B$ globally: