we have a continuous function $\omega:\mathbb{R}_+\rightarrow\mathbb{R}_+ $ such that $$ p'(t)=\omega(p(t))\\ p(0)=0$$
has only the trivial solution $p=0$ on $J=[0,b]$. Now the author claims, that there are functions $p_n$ such that $$p_n'(t)=\omega(p_n(t))+\frac{1}{n}\\ p(0)=\frac{1}{n}$$
for all $t\in J$ and $\lim\limits_{n\rightarrow\infty}p_n=0$.
I understand the limit, but i really dont know why we have such functions $p_n$ on the whole interval $J$. Thanks for your help!
Edit: The statment should only be true for $n>N$.
The authors did so because they wanted to used Peano existence theorem. Note that the Peano's theorem requires the domain to be an open subset.
The such existence is not true for all $n$. The solution for any induced ODE may explode before reaching time $b$. For example, consider $\omega (x)=x^2$.
For $n$ sufficiently large, the mentioned existence is valid. The proof idea is to replace $\omega$ by another bounded continuous function $\tilde{\omega}$ such that these two functions are identical in a neighborhood of $0$. Then thanks to the boundedness of $\tilde{\omega}$, the coresponding solutions $(\tilde{p}_n)$ will not explode. After that prove what you have claimed $\lim \tilde{p}_n=0$. From which you can show that when $n$ is sufficiently large, $\tilde{p}_n$ is also a solution of ODE $p'=\omega(p)+\frac{1}{n}$.