Try a reset by pressing the home and sleep buttons until you see the Apple logo, ignoring the slider. Takes about 5-15 secs of button holding and you won't lose any data or settings. You may need to hold the buttons longer.
If still a problem restore with your backup. You probably will need to start here.
If still a problem restore as new, i.e. without your backup. See how it runs with nothing synced to it.
If still a problem, it's likely a hardware issue.