The question isn't Cap's determination... it's whether he knows enough about hand-to-hand combat to counter Batman's encycolpiedic knowledge, plus his huge arsenal, plus his prpensity for setting traps. It's pretty clear that Bats could take Cap here, but it would be a tough one.
Batman vs. The Hulk is impossible to predict because I've seen Hulk take nuclear weapons and keep coming... and then be brought down by a dozen tranq rounds.
Like I mentioned in an earlier post, Cap is considered to be the best trained fighter on Marvel's earth. Bats and Cap are equal as far as fighting and tactics are concerned, Cap is just physically and mentally tougher, but Batman is smarter and is sneakier. These two would wind up fighting forever.
As for the Hulk one, in this months Avengers, Jack o Spades detonated himself (the equivalent of a low level nuclear weapon) and the Hulk was knocked out turned back into Banner. So if Bats has a nuclear battarang with him...