Give: O.E. giefan (W. Saxon) "to give, bestow; allot, grant; commit, devote, entrust," class V strong verb (past tense geaf, pp. giefen), from P.Gmc. *gebanan (cf. O.Fris. jeva, M.Du. gheven, Du. geven, O.H.G. geban, Ger. geben, Goth. giban), from PIE *ghabh- "to take, hold, have, give" (see habit). It became yiven in M.E., but changed
Give up is found in Icelandic book Landnáma (written ~1200) where old people were 'given up' in bad times. The meaning is similar to what we have today, but in Landnáma it describes a decision that was made about old people of a certain district being pushed of cliffs during fanime.