= Solution
The difference of two successive <Picard iterates> factors as
$$
y_{n+1}(t)-y_n(t)=\int_0^tK(t,s)\bigl(y_n(s)+y_{n-1}(s)\bigr)\bigl(y_n(s)-y_{n-1}(s)\bigr)\,ds.
$$
The preceding bound and the kernel estimate therefore give
$$
\boxed{\|y_{n+1}-y_n\|_\infty\leq\frac{4B}{C_1}\|y_n-y_{n-1}\|_\infty
=\gamma\|y_n-y_{n-1}\|_\infty.}
$$
Moreover $D_1:=\|y_1-y_0\|_\infty\leq B^2/C_1$. Summing the resulting <geometric series>, for $m>n$,
$$
\|y_m-y_n\|_\infty\leq D_1\sum_{j=n}^{m-1}\gamma^j
\leq\frac{D_1\gamma^n}{1-\gamma}.
$$
Thus there is a <uniform limit> $y$ on $[0,\infty)$, continuous by the <uniform limit theorem>, with $\|y\|_\infty\leq2B$. The integral passes to this limit because
$$
\sup_{t\geq0}\left|\int_0^tK(t,s)\bigl(y_n(s)^2-y(s)^2\bigr)\,ds\right|
\leq\frac{4B}{C_1}\|y_n-y\|_\infty\longrightarrow0.
$$
Therefore $y$ satisfies the <Volterra integral equation>, and hence the <initial value problem>. This proves global existence by <Global Picard iteration with a decaying kernel>. The same estimate gives the useful error bound
$$
\boxed{\|y-y_n\|_\infty\leq\frac{B^2\gamma^n}{C_1(1-\gamma)}.}
$$
If $b=0$, every iterate is zero and the convergence assertion is immediate.
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