Well you can go under the house and make sure the joist are sound, if they are you could add a few studs underneath the soft spots this will help. The only real solution is to have the floor torn up in that spot and replace the sub flooring. This will mean replacing the linoleum or tile. I'd try the joist first.
Without knowing the type of home you are in, I suspect it isn't on a slab. It may be that the sub floor is MDF, and would take quite awhile, given any finish floor over it, to weaken from splashes and spills. With all due respect they should be wiped up. You also don't state the length of time the situation has existed, which could make a major difference in the cause. If you have access from under the kitchen, certainly begin your search there, as one other answer suggests, especially where any plumbing might also be in close proximety to the SOFT spot. To create any significant damage would take a considerable amount of water SOAKING over an extended period of time unless the finish floor isn't sealed, or if the substrate is MDF. Without knowing the age of the house, or duration of the softening, or the substrate for the finished floor, or the material of the finished floor, you may have easy options or a larger repair/replacement problem. Steven Wolf
it depends on what your flooring is made out of.. more info please..