How do you deak with a situation of continued check, but never checkmate?  For example, a situation I got myself in the other day.  I was white, had rook, bishop and pawn vs a queen and pawn.  The diagram below was constructed form memory and is not exact, but it illustrates the scenario I'm talking about.  Black has the white king in check.  Pretty much no matter how I get out of the check, the queen can get to a spot and put white in check again.  However, without bringing the king in, black can never get mate.  It becomes an endless game of cat mouse between the black queen and white king.  If black does not keep the king in check, white can make an offensive move.

 Does one consider offering a draw?  Is this a type of situation where an oppurtunity will open up for white eventually(after 10-15+ moves)?

If it is a perpetual check, then it is counted as a draw.  Typically, the draw is demonstrated by repeating the position three times.  However, in OTB play, stating that you have a perpetual check is often considered a draw offer.

In the position you show, I'm not positive it is a "perpetual check."  It may be possible to get your King on the a-file in such a way as to block any reasonable potential checks. 

Black's certainly winning...and I think a good player would be able to weasel out a win in such a position. I suggest (at least in this position) offering a draw and hoping he accepts. Because I really can't see how white could win...considering the kind of power black has over the potential continuous check until he gets you in a position he likes.

All black needs to do is set you up in such a way that black can either nag a free piece OR force-push his pawn to the other end. It seems like it's just a matter of time, personally.

> an endless game of cat mouse between the black queen and white king.

If 50 moves pass without a capture or a pawn move, either side can claim a draw.