# Special sequences in a product metric space

Hi there,

I came across the following problem and I hope somebody can help me: I have some complete metric space $(X,d)$ (non-compact) and its product with the reals $(R\times X, D)$ with the metric $D$ just being $$D((t,x),(s,y))=|s-t|+d(x,y)$$ for $x,y\in X; s,t\in R$. Then I have some sequences $x_n\subset X$, which converges to $x$ and $t_n\subset R$, which goes to infinity. These two give rise to sequences $(t_n,x)$ and $(t_n,x_n)$ in $R\times X$ which do not converge either, but nevertheless $$D((t_n,x),(t_n,x_n))\to 0$$ holds as $n\to\infty$. Moreover I have now some continuous map $f\colon R\times X\to X$ and my problem is under what conditions does $$d(f(t_n,x),f(t_n,x_n))\to 0$$ hold as $n\to\infty$?

I know that it does not hold in general as there are simple counterexamples and it does hold if $f$ is (globally) Lipschitz. But for my setting globally Lipschitz is quite restrictive, so I am looking for milder assumptions. Specifically I would be interested if it does hold when $f$ fulfills the following type of 'group condition': $$f(t+s,x)=f(t,f(s,x))$$ for all $s,t\in R; x\in X$. I can not find a counterexample and cannot prove it either.

I know it's a quite specific problem, but I would be glad if somebody had an idea on that or could provide me with a source that helps because I am really stuck here with this.

Thanx.

PS: I think that $X$ is even a manifold and $f$ is smooth, but I don't think that makes much of a difference...

I think it might be difficult to formulate something so generally. Let X be the real numbers, and then we have a function from the plane to the line. Make it even simpler: let f(t,x)=t*h(x). When is infinity times zero zero?

It's an interesting question, and it's clear that one can do better than Lipschitz under certain conditions. But your group property doesn't do it: f(t,x)=xe^t is a counterexample.

But your group property doesn't do it: f(t,x)=xe^t is a counterexample.

I see. I now thought that in general uniform continuity could do it.

But in the case of my group property: Maybe it would be enough if I additionally demanded that $f(t_n,x_n)\to x$ as $n\to\infty$?