New text message notification will not go away, but there are no new text messages

Try stopping and restarting the Messages app: double tap the home button to bring up the recently used apps list at the bottom, locate the Messages app icon (swiping if necessary), press and hold it until it jiggles, tap the red minus sign. Open and close messages again and see if that clears it. If not, stop it again followed by resetting your phone: hold on/off and home buttons simultaneoulsy for 10-15 seconds until you see the Apple logo (ignore the off slider that appears first), release and wait for the phone to restart.

So what if I have tried what you have said and it still doesn't work? Any suggestions?

You're welcome.

Not sure if it will work, but you could try going to Settings>Notifications>Messages and turn Badge App Icon to Off, then reset your phone (hold the on/off and home buttons at the same time until you see the Apple logo, then release). After the phone restarts go back and turn Badge App Icon back to On and see of the erroneous badge app notification will stay off this time.

Nope that doesn't work either. I've been trying every way I can think of to get rid of the **** notification but nothing works.

Depending on how much it bothers you, the "nuclear option" would be to delete your existing text message conversations (but there's no guarantee this will work either). Not what else to suggest.