I have the same problem on my iPhone 5. My warranty is 5 days out of date. I can find no way to contact Apple without paying to do so.
My iPhone 4s failed in the same way, at about the same time in its life. For the 4s, I drove over an hour to the nearest Apple store, where the genius set up the software fix, she said I must have pushed the button too hard. Very unsatisfying for such an expensive product.
I was able to talk AT&T into giving me an early upgrade, also an expensive "fix", but I ended up with a shiny new iPhone 5 which is now, sadly, failing in the exact same way.
Should I have purchased Applecare? An additional expense of $178 to protect an already very expensive product against what is clearly a manufacturing defect? Apple is telling us that their product will work for one year. If we expect to have full use the product past the one year point we have to pay extra.
I want to trust Apple, I've bought into the ecosystem, purchased tons of apps, own a couple of iPads and another phone, (a 3gs, still working fine in the hands of an 8yr old!) but I can't justify buying/upgrading to another phone that is likely to have the same issue.