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Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a uniformly bounded sequence (i.e., there exists $C>0$: $|f_n| < C$ for every $n$) such that $$ f_n \in C^2_x \times C^1_t, $$ and, as $n \to \infty$, \begin{equation} \tag{1} f_n(x,t) \longrightarrow f(x,t), \quad \text{a.e.} -(x,t) \in \mathbb{R} \times [0,\infty). \end{equation} Let $x_n: [0,\infty) \to \mathbb{R} $ be a sequence of absolutely continuous functions that satisfies \begin{equation} \tag{2} \frac{d}{dt} x_n(t) = f_n(x_n(t),t), \quad \text{a.e.} - t, \; x_n(0)=0, \end{equation} and \begin{equation} \tag{3} x_n(\cdot) \to x(\cdot), \quad \text{uniformly in any bounded interval in } [0,\infty), \end{equation} for some function $x: [0,\infty) \to \mathbb{R} $.

Question.

  1. Is the following statement true: As $n \to \infty$, \begin{equation} \tag{4} f_n(x_n(t),t) \longrightarrow f(x(t),t), \quad \text{a.e.} - t \in [0,\infty)? \end{equation}
  2. Is (4) true for any $x_n$ not necessarily satisfies (2)?
  3. If (4) is wrong, what are additional requirements needed on $f_n$ to make it true?

Remark.

  • The limit function $f$ does not have to be $C^1$, even $C^0$.
  • The conclusion (4) is true if (1) is true uniformly, without the assumption (2).

Updated. The second question is answered negatively thanks to the counterexample by Iosif Pinelis.

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The answer is no. E.g., for all natural $n$ and all $(u,t)\in\mathbb R\times[0,\infty)$, let $f_n(u,t)=e^{-nu^2}$, $f(u,t)=0$, and $x_n(t)=x(t)=0$. Then $f_n\to f$ a.e. on $\mathbb R\times[0,\infty)$ and $x_n\to x$ uniformly on $[0,\infty)$, whereas for all $t\in[0,\infty)$ $$f_n(x_n(t),t)=1\to1\ne0=f(x(t),t). $$

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  • $\begingroup$ Thanks for your negative answer to my first question. So I shouldn't expect too much if I allow such a freedom on $x_n$. What if now $x_n$ is closely related to $f_n$, namely, for every $n$, $\frac{d}{dt} x_n(t) = f_n(x_n(t),t)$, a.e-$t$ with $x_n(0) = x(0)$? Would it be enough to draw a positive conclusion? $\endgroup$ – Manolis D Aug 29 '19 at 13:18
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    $\begingroup$ @ManolisD : I think asking multiple questions in one post is not encouraged on MO. Some people may feel able and committed enough to answer one of the questions but not all of them. Also, posting several questions in several posts should give the questions greater exposure. $\endgroup$ – Iosif Pinelis Aug 29 '19 at 17:21

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