Does divorce take effect if a man falsely swears an oath of divorce to his wife to avoid problems and the loss of the children?

Whoever swears an oath by divorce while being untruthful, their divorce has taken effect, as mentioned by Al-Kasani in Bada'i' al-Sana'i', by Al-Mudawwanah, and by Al-Ansari in Asna al-Matalib.

Accordingly, your wife has been divorced with one pronouncement. If this pronouncement is the first or second, you may take her back. If it is the third, she is not permissible for you until she marries another husband in a sincere marriage, not a tahlil (analytical) marriage. If the other husband then divorces her, it becomes permissible for you to marry her.