Appointments will show up on your calendar even if they are not accepted, but updates will not be reflected if the original appointment has not been accepted. In the ipPhone native calendar they will appear slightly greyed with a blue dot preceding the Event name. Open the event to accept and then the blue dot will disappear and you will get updates for this event.
Turn off “Show Declined Events” setting in the iPhone native calendar. Open the iOS native Calendar and tap the word “Calendar” at the bottom center for the screen. Scroll to the bottom of the new screen that opens and at the very bottom there is an option to “Show Declined Events”. Turn that OFF, so that the button in white, not green.