Re: the name of the calling person does not appear.
Welcome to iMore! If the number on their contact card matches exactly to the number they're calling from, you should see the contact's name instead of the phone number. When you go to Phone app > Recents, do their names show or only their phone numbers? Try resetting your iPhone by holding down the power button and home button until you see the Apple boot logo, then going back to Phone app > Recents and seeing if their names start filtering in.