Special sequences in a product metric space

[holy_toaster]

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...

 

[zhentil]

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.

 

[holy_toaster]

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?