“Give up” is a very common phrasal verb in English.
When you say you “give up doing something”, it means that you quit or stop trying to do it.
When you say you give up on something, you stop trying to do or achieve it.
She advised me to give up smoking.
It’s hard to give up gambling.
We don’t intend to give up.
I think it’s time for me to give up on this relationship.