Oh yes, we do have problems. Most of the cats haven't been a problem as far as fighting, as they come for food & in most cases they are young cats not yet ruled by hormones. In most cases they come from one of two neighbors, neither of whom believe in spaying/neutering or FEEDING their cats. A few are older toms that stop in here during their travels--some of those come a few times and then we never see them again, others keep coming back. With the young cats from the neighbors, if they are no problem, not picking fights with our cats, we have taken them in to be neutered/spayed. Most of them simply move in here permanently; the odd one does go back "home" and just comes here regularly for food. If the cat is a fighter, we euthanize him. The same two options apply to any strays/drop offs that show up here--the nice ones get fixed, the mean ones get euthanized. In the case of the wild toms, if they are fighters, beating up our cats (and there have only been a couple of those) they get live trapped & sent to animal control. If they are not inclined to beat up our cats, they get left alone for however long they stay here. If they stay indefinitely & end up getting sick or injured, then we have to trap them & have them euthanized.
We have a cat door into the house and it's not uncommon for the strays to find their way into the house--we've had a couple wild strays that will sneak into the house to sleep on cold winter nights. It's a little startling to walk into the bedroom & turn on the light & have a cat come streaking out from under a dresser & race down the hall to the cat door!!
I have no idea why certain people in the neighborhood keep getting cats when they have no interest in feeding those cats.