...I agree that must have been poorly installed, my kitchen vinyl is around 25 years old and still has some wear left in it yet. You would do best to replace it with some heavy duty stuff, try the old battle ship vinyl (it lasts decades!).
For now, however, you can just paste the edges back down with regular vinyl adhesive. Like the stuff used to hold down the old style vinyl tiles that weren't already glued on the back. It might take a trip around a few very old hardware stores to find some, but it should work.
A word of warning through, if you intend to ever use the floor surface below again it may be a bad idea to put thick glue all over it. Wood can be sanded, but if it's a real nice tile or stip oak, don't put glue anywhere near it! Still, have you looked under there yet, it might be quite a nice old floor waiting to be uncovered again? It could save a lot of time and money if there is.