relent
To finally become less strict or stop refusing something.
To relent means to finally give in or become less strict after refusing or resisting for a while. When parents relent, they might say yes to something they initially said no to, usually because their child made a good argument or showed real responsibility. When a teacher relents on a deadline, she extends it after students explain they need more time.
The word suggests a softening, like ice melting. A storm that has been pounding your town might relent by late afternoon, letting the sun break through. A coach who has been pushing the team hard through practice might relent and give them a water break.
Relenting isn't the same as being weak or inconsistent. Sometimes relenting shows wisdom: recognizing when circumstances have changed or when holding firm serves no good purpose. A king in an old story might relent and pardon a prisoner after hearing new evidence. A big brother might relent and let his little sister join his game after she promises to follow the rules.
When someone refuses to relent, they stay firm no matter what. The word relentless describes someone who never gives up or never softens, like a relentless detective who won't stop investigating until she solves the case.