From Longman Dictionary of Contemporary Englishgive up on somebody/something phrasal verbHOPEto stop hoping that someone or something will change or improve He’d been in a coma for six months, and doctors had almost given up on him. At that point, I hadn’t completely given up on the marriage. → give→ See Verb table
Examples from the Corpus
give up on • For that reason Labour planners would not like to give up on a May election until they have to.• She was loyal and intelligent and she never gave up on anything.• Sarah Brady said she hasn't given up on federal legislation.• Even then, Sarah had not given up on her marriage.• Meanwhile John Lawer insists he won't be giving up on his epic adventure or his one-man campaign against three-wheel discrimination.• Kovacs had given up on the pencils and slipped them back into the box.• Around the turn of the century, the Crown gave up on the Quaker colony but then later restored it to Penn. 