Surrender, as in The suspect gave himself up. [1100s]
Stop doing or performing something, as in They gave up the search, or She gave up smoking almost thirty years ago. [c. 1600]
Part with, relinquish, as in They gave up their New York apartment, or We gave up all hope of finding the lost tickets. [Mid-1500s]
Lose hope for, as in We had given you up as lost. [Late 1500s]
Admit defeat, as in I give upwhat's the right answer? [c. 1600]
give up on. Abandon, lose one's faith in, as in I gave up on writing a novel, or She gave up on religion years ago. [Colloquial; second half of 1900s] Also see give oneself up to.