Hi there, man2than.
I see you're getting notifications for old text messages whenever you restart your iPhone 7 on iOS 11.0.3. They're continuous for a while, but then it stops. I'm happy to help.
Force an app to close on your iPhone, iPad, or iPod touch -- First, force close the Messages application if it's open, then -- Restart your iPhone, iPad, or iPod touch -- Once it's returned to the Home screen, do you get those notifications still?
How to back up your iPhone, iPad, and iPod touch -- If restarting your phone didn't resolve the issue, please be sure to have a fresh backup to keep your personal information safe. In order to back up your Health data, the backup needs to be encrypted. iCloud does this automatically, but iTunes doesn't. If you'd like to use iTunes to back up your phone, refer to this article: About encrypted backups in iTunes
If you can't update or restore your iPhone, iPad, or iPod touch -- Even though the title of the article isn't specific to your issue, the section titled "Put your device in recovery mode and set it up again" will walk you through all the steps to reinstall the iOS software on your device. Test the issue once your phone has returned to the Home screen.
Let me know how that goes.
Take care!