What is the ruling of Islamic law regarding a mother's 'urfi marriage to a man without witnesses, and to what extent are the children sinful who used abusive language toward their mother after discovering this marriage?

The questioner and his brother must repent to Allah for their ill-treatment of their mother. Repentance involves abandoning the sin, regretting it, and resolving never to return to it, along with seeking their mother's satisfaction. The mother's right is immense, and it is not permissible to curse or insult her, no matter how badly she may have behaved.

The marriage contract that took place without a guardian (wali) and witnesses is invalid. If the man does not divorce her, she may adopt the view of those who believe there is no need for divorce or annulment in فاسد (corrupt/invalid) contracts, as Imam Al-Shafi'i said: "There is no need for annulment or divorce, because it is a marriage that was not properly concluded." If the mother needs to get married, it is not permissible to prevent her; rather, she should be assisted.