There are two possibilities. Either B is telling the truth, or B is lying.
If B's statement is true, then the date is even. Also, C and D are lying, since their statements are inconsistent with B's. Also, because exactly two of the friends must be telling the truth and two lying on any given date, A must be telling the truth. And given what he says about lying the previous day, A is the one who tells the truth on even days and lies on odd days.
If B's statement is false, then C and D have to be true, because C and D must have the same truth value and only two of the four can be telling the truth. This means the date is odd (as C implies), and also that A is lying. But if A is lying about lying yesterday, this means he told the truth yesterday. Therefore, he has to be the one who tells the truth on even days and lies on odd days.
So to sum up, there are only two possibilities -- B or not B -- and either way, Abe ends up being the one who tells the truth on even days but lies on odd days.
This is more of a puzzle for the newspaper than a real logic problem, but the two go hand in hand. If you want to learn real formal logic, message me here on Wyzant.