I changed my Apple ID but during update or download app it ask previous id password I current Apple ID automatic replace with other phone Apple ID ,also this other phone id relaxes iCloud id. After that I made new Apple id but it always ask password of old Apple ID . But old Apple ID shows not exist. Also during delete of iCloud ID it ask old ID password which does not exist. I made new Apple ID successfully but for updates it ask old ID password . How to solve this problem?
This is because each App on your phone is linked to an Apple ID. You can change your AppleID on your phone, but the app will still remembering the AppleID from was installed initially. I found similar behavior if you use a phone backup from other phone different than your own. 

The only solution you have in this case is to unninstall each installed with the old AppleID and reinstall it again with your new AppleID. This will solve the problem. 

I hope it helps.