I am trying to prove the followings:
1) Let $r$ be a continuous,nonnegative function on an interval $J=[a,b]\subset\mathbb{R}$ and $\delta$ and $k$ be nonnegative constants such that $$r(t)\le \delta+\int_a^tkr(s)ds$$ Then show that $$r(t)\le \delta\exp[k(t-a)]$$
2) Let $r,k,f\in C^1$ and nonnegative such that $$r(t)\le f(t)+\int_a^tk(s)r(s)ds,\,\,\,\,\text{for }a\le t\le b$$ Then show that $$r(t)\le f(t)+\int_a^tf(s)k(s)\exp\Big[\int_s^tk(u)du\Big]ds,\,\,\,\,\,a\le t\le b$$
I did manage to prove the first one (If anyone wants to see I can post it), but couldn't do the second one. Any help would be highly appreciated. Thanks in advance.
Let $$ E(t)=\exp\left(-\int_a^tk(s)ds\right) $$ Then $$ \frac{d}{dt}\left(E(t)\int_a^tk(s)r(s)ds\right) =-k(t)E(t)\int_a^tk(s)r(s)ds+E(t)k(t)r(t) =E(t)k(t)\left(r(t)-\int_a^tk(s)r(s)ds\right) \le E(t)k(t)f(t) $$ Thus after integration and comparing with the original inequality, the desired inequality follows.