Re: "Incorrect last name in the app store and game center
I recently changed my last name and changed it to my apple id."
You may want to try re-starting your iPhone: (deletes temporary files, re-establishes connections, may update recent setting changes, and often restores order if something is not working as expected.
You may need to allow 24 hours for the name-change to come into effect everywhere it is in use, including FaceTime and Messages etc.
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Re: "when logging into the application through the game center, the old last name is displayed."
You may want to try signing out, and signing back in again (check any payment methods match the name)
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More info. / support / contact options etc: Apple Account Support Team: Web-Page:
(NB: There is a "Get support" link at the bottom)
Apple Account - Official Apple Support