K+R+B vs K+R

Is there a mate for this situation?

According to the book, Fundamental Chess Endings, by Muller & Lamprecht, it's generally a draw. There are some positions, however, in which a mate can be administered, but they require more than 50 moves, so they can be claimed to be a draw too under the 50-move rule

You'd have to be an expert. If I was the defending player I'd swap off the rooks first opportunity I get and thereby draw the game.

Theoretically, this is usually a draw. However, unless you know the drawing technique it can be very difficult to defend in a practical game. I've had this endgame twice in serious competition play and managed to win both fairly easily.