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Suppose $F:\mathbb R^2 \to \mathbb R$ is continuous. Assume $f_n$ is a uniformly bounded sequence of real-valued functions on $[0,1]$ such that for each $n, f_n'(x) = F(x,f_n(x)),x\in [0,1].$

Is there a subsequence $f_{n_k}$ that converges uniformly on $[0,1]?$

This question seems to be coming down to prove $\{f'_n(x)\}$ is equicontinuous. If so, then it has a uniformly convergent subsequence and then so does $\{f_n\}$. We know each $f_n'(x)$ is uniformly continuous on $[0,1]$ by compactness. But in general, uniform continuity for each function doesn't imply equicontinuity, correct?

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  • $\begingroup$ That first paragraph needs a rewrite. $\endgroup$ – zhw. Dec 17 '15 at 0:49
  • $\begingroup$ Could you edit it or elaborate? $\endgroup$ – FTem Dec 17 '15 at 2:26
  • $\begingroup$ No it would be better if you did it. You can't see the problem? $\endgroup$ – zhw. Dec 17 '15 at 2:30
  • $\begingroup$ I fixed everything I could find... did I miss something? Also, any thoughts on the problem? I think I can show the this all true if $\{f'_n\}$ is uniformly bounded... $\endgroup$ – FTem Dec 17 '15 at 2:46
  • $\begingroup$ I edited the question to make it simpler. I am not sure why you think this comes down to showing $f_n'$ is equicontinuous. $\endgroup$ – zhw. Dec 17 '15 at 20:19
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The answer is yes. Proof: We are given that there exists a constant $M$ such that $|f_n|\le M$ on $[0,1]$ for all $n.$ Now $F$ is continuous on $[0,1]\times [-M,M],$ a compact set. Therefore $|F|$ is bounded by some constant $C$ on this set. It follows that

$$|f_n'(x)| = |F(x,f_n(x))| \le C$$

for all $n$ and all $x\in [0,1].$ By the mean value theorem, we then have $|f_n(y)-f_n(x)| \le C|y-x|$ for all $n$ and all $x,y \in [0,1].$ This shows $(f_n)$ is equicontinuous, and since $(f_n)$ is uniformly bounded, Arzela-Ascoli gives the desired uniformly convergent subsequence.

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