= Solution
Existence was proved by <Picard iteration>. If $y$ and $v$ are any two continuous solutions with the same $b\ne0$, the previous part makes both bounded by $2B$. Subtracting their <Volterra integral equations> gives
$$
\|y-v\|_\infty\leq\frac{\|y\|_\infty+\|v\|_\infty}{C_1}\|y-v\|_\infty
\leq\gamma\|y-v\|_\infty.
$$
Since $\gamma<1$, this forces $y=v$.
For $b=0$, zero is a solution. On any finite interval $[0,T]$, another continuous solution has a finite bound $K_T=\max_{[0,T]}|y|$. Since $K(t,s)\leq1$, its <integral equation> implies
$$
|y(t)|\leq K_T\int_0^t|y(s)|\,ds.
$$
The <Gronwall inequality> gives $y=0$ on every such interval. Hence
$$
\boxed{|b|<C_1/4\quad\Longrightarrow\quad\text{exactly one continuous solution on }[0,\infty).}
$$
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