Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a sequence of bounded variation functions such that there exists $C>0$: $|f_n| < C$ for every $n$ and, as $n \to \infty$, for $t>0$, \begin{equation} \tag{1} f_n(\cdot,t) \longrightarrow f(\cdot,t), \quad \text{in } L^1(\mathbb{R}). \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 $x_n(\cdot) \to x(\cdot)$ uniformly in any bounded interval in $[0,\infty)$, for some $x: [0,\infty) \to \mathbb{R} $.

Is the following claim true?

Claim. For a.e.-$t$, there exists a sequence $y_n$ such that, $y_n \ge x_n (t)$, $y_n \to x(t)$ and $$f_n(y_n,t) \to f(x(t)+,t),$$ where $f(x(t)+,t)$ stands for the right limit in the nomal sense.

Remark. This is an attempt to solve a slightly stronger version of this problem.


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