For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and

$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$

Let $y_n$ be the unique solution of

$$\begin{cases} y_N(0)=0 \\ y_n'=f_n(y_n) \text{ on [0,1]}. \end{cases}$$

Question: Is there a function $y\in\mathcal{C}^1([0,1],\mathbb{R})$ such that

$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|y_n(x)-y(x)\big|= 0$$

which is solution of the system (which has itself an infinity of solutions)

$$\begin{cases} y(0)=0 \\ y'=\sqrt{y} \text{ on [0,1]} \end{cases}$$

and satisfies the condition: $y(x)>0$ for $x\in\,]0,1]$ (which, I hope, permits to characterize $y$).