What is the ruling on the actions of a husband who frequently swears by divorce and then claims it did not occur, treats his wife poorly, abandons her in bed, and neglects his children? Is the wife entitled to seek a divorce?

If the scholars have issued a fatwa to your husband that he has pronounced two divorces, then he must comply with that. As for the harm inflicted upon you by his treatment, you have the right to refer the matter to the Sharia courts to remove it, such as his refusal to have intercourse or his turning away from you in bed, as that is harm which permits seeking divorce. The maintenance of the children is obligatory upon him until they reach adulthood and are able to earn a living, and he must undertake their care and upbringing. The cure for anxiety is through piety towards Allah, and it is not permissible to wish for death due to harm, but rather to say: "O Allah, grant me life as long as life is better for me, and cause me to die if death is better for me." You have the right to sue your husband to remove the harm from you, in accordance with the rule: "There shall be no harm nor reciprocating harm."