Email accounts cant view for more than 3 seconds??? Weird
Couple of things to try:
First, a simple reset. You won't lose any data or settings.
Press the sleep/wake button & home button at the same time, keep pressing until you see the apple logo(ignore the slide to power off), then release.
See it that fixes the issue. If not proceed:
Delete the accounts in question from your phone. Turn the phone off, then back on & recreate the accounts. See if that fixes the issue. If not, proceed:
Restore your phone as a "new" device, not from a backup. Recreate your accounts & see if that fixes the issue. If it does, re-sync your content. You will lose all saved data in apps, messages, etc if you restore as new.