"the app notifications" maybe is a good alternative
Instead of "notifications", I used "alerts" which should also relate to "ειδοποιήσεις".
On the app would be the more conversational term
the notifications of the app
The notifications of the app = Οι ειδοποιήσεις της εφαρμογής
"The application's notifications" is marked incorrect
That is a different translation...."Οι ειδοποιήσεις της εφαρμογής"