= 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 $|y|\leq2B$, so the kernel estimate would imply
$$
2B=|y(T)|\leq B+\frac{4B^2}{C_1}=B(1+\gamma)<2B,
$$
a 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 $|y|\leq2B$ globally:
$$
\boxed{\|y\|_\infty\leq B(1+\gamma)<2B.}
$$
Back to article page
