Bar on screen unresponsive for the most part with iOS 7.1.1

About two weeks ago, I noticed a periodic issue on my iPhone where there is a half an inch long bar going horizontally across the screen a little over halfway down.I have tried resetting the phone seven times, and the issue either appears shortly after I reset the phone (i.e. when it's on any random screen), or becomes noticable once the phone prompts you to set your nation. Additionally, I have had the phone checked out by Apple (they suggested I perform the latest reset) and I have had different screens put on it by authorized phone repair persons to show that it wasn't a hardware issue. In the last week or so, the issue has occurred about 95% of the time, and prevents scrolling of articles, clicking on apps, and sending of messages in various apps. I thought it was originally a software issue at first, until I reset the phone independent of the computer and the same issue was happening, even before I restored what I had on there before. At this point, I am ready to jailbreak and downgrade to a slightly older iOS system just to see if it will be any different, as I am suspect that one of the recent updates might have something to do with it, but I am posting here first, just to see what others might think.