= Solution
For $s>t$, the coefficient bound gives $A(s)-A(t)\geq C_1(s-t)$. Thus for $0\leq s\leq t$ the positive kernel satisfies
$$
K(t,s):=e^{-(A(t)-A(s))}\leq e^{-C_1(t-s)},\qquad
\int_0^t K(t,s)\,ds\leq\frac{1-e^{-C_1t}}{C_1}\leq\frac1{C_1}.
$$
The starting <Picard iterate> is continuous and has <supremum norm> $M_0=|b|$. If the $n$th <Picard iterate> is bounded and continuous, its integral defines another <continuous function>, and
$$
|y_{n+1}(t)|\leq |b|e^{-A(t)}+M_n^2\int_0^tK(t,s)\,ds
\leq |b|+\frac{M_n^2}{C_1}.
$$
Taking the <supremum norm> and applying <mathematical induction> proves boundedness of every iterate and
$$
\boxed{M_{n+1}\leq |b|+C_1^{-1}M_n^2.}
$$
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