IOS – Has anyone had troubles with an iPhone 4’s battery life failing after upgrading from iOS 6 to iOS 7

After I upgraded my iPhone 4 from iOS 6 to 7, I have experienced my phone running out of power before the end of the day on a night charge (I know it's easy to run a full charge down during the day if you use an iPhone heavily enough, but my usage has been light enough to usually have less than 1GB data transfer per month–I don't use my iPhone much by general standards, but it is extremely useful to me when I do use it).

Also, it takes longer than usual to charge enough to use while connected, and I have had at least one time when I started it charging, powered it up, and worn down the battery enough that it shut down while powered down.

Is this an iOS 7 issue? Or an iPhone 4 issue? Or an iOS 7 + iPhone 4 issue? Or an old battery issue?

I am not aware of any significant changes besides the OS migration; I treat technology, including iPhones, carefully, and I am not aware of having dropped it or otherwise caused damage.

I went on the Apple site and gave the serial number, and the site said my iPhone 4 is no longer under warranty, but I could purchase a support incident for $20. Is that $20 likely to cover my phone with earlier battery life, or does the $20 buy, "We'll look at it and give you an estimate for replacement parts or other repairs"?

I'd be interested in knowing what the diagnosis and prognosis are likely to be. I'm happy to continue with this phone, but the battery issues are a marked problem for now.

Thanks,