special sequences in a product metric space
I came across the following problem and I hope somebody can help me: I have some complete metric space [itex](X,d)[/itex] (non-compact) and its product with the reals [itex](R\times X, D)[/itex] with the metric [itex]D[/itex] just being [tex]D((t,x),(s,y))=|s-t|+d(x,y)[/tex] for [itex]x,y\in X; s,t\in R[/itex]. Then I have some sequences [itex]x_n\subset X[/itex], which converges to [itex]x[/itex] and [itex]t_n\subset R[/itex], which goes to infinity. These two give rise to sequences [itex](t_n,x)[/itex] and [itex](t_n,x_n)[/itex] in [itex]R\times X[/itex] which do not converge either, but nevertheless [tex]D((t_n,x),(t_n,x_n))\to 0[/tex] holds as [itex]n\to\infty[/itex]. Moreover I have now some continuous map [itex]f\colon R\times X\to X[/itex] and my problem is under what conditions does [tex]d(f(t_n,x),f(t_n,x_n))\to 0[/tex] hold as [itex]n\to\infty[/itex]?
I know that it does not hold in general as there are simple counterexamples and it does hold if [itex]f[/itex] 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 [itex]f[/itex] fulfills the following type of 'group condition': [tex]f(t+s,x)=f(t,f(s,x))[/tex] for all [itex]s,t\in R; x\in X[/itex]. 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.
PS: I think that [itex]X[/itex] is even a manifold and [itex]f[/itex] is smooth, but I don't think that makes much of a difference...