what is causing the lino to come up? Is this an exterior door that is collecting water? What is under the lino? Is it another floor, concrete, or wood? if its particle or compressed dust, this could be the problem mixed with water.
is this square lino or sheet?
If its compressed saw dust subfloor, then glue will not hold to this if its been mixed with water. The outer smooth skin of this material will hold glue, but when its gone or damaged, the underneath stuff will not work
You may have a buildup of debris also, Get a scraper and clean off the under neath tile and the flooring area. Apply some "special projects" liquid nails in that area, this should work, but will be something else to remove if needed.