For LBAs it's rather easy to prove the decidability of the halting problem, as there can only be a finite number of different configurations when using limited space.
But what about PDAs with $\epsilon$-transitions? Is the halting problem decidable for deterministic PDA's with $\epsilon$-transitions? Given such a DPDA $P$ and an input $x$, can we decide whether $P$ will halt on input $x$?
The stack may be infinitely large, so it seems like it might be much harder here to see if the PDA is in an infinite loop or not.
I don't think it's as easy as answered here for non-$\epsilon$ PDAs:
[For] DFAs or PDAs, the halting problem is decidable: the machine always halts because it halts when it reaches the end of its input, the input is finite and the machine consumes one character of input at every step.
The answer to my earlier question (Is a PDA's stack size linear bounded in input size?) also seems to point in a direction of a higher difficulty of proving this.