Does the marriage remain valid if the husband repeatedly swore to divorce without carrying out his oath, claiming it was an oath of anger, and is expiation (kaffarah) obligatory upon him, and what is the legitimate Islamic solution for this situation?

The ruling concerning whether a woman is still married to her husband or not depends on whether a divorce has occurred. If she has been divorced irrevocably (three times), then she is separated from her husband. Scholars have differed regarding the ruling when a man says, "Divorce be upon me if I do such-and-such," and then he does not do it. The majority of them hold that divorce occurs upon breaking the oath, while others (including Ibn Taymiyyah, Ibn Baz, and Ibn Uthaymeen) are of the opinion that if the divorce is in the form of an oath, then only expiation for an oath (kaffarat yamin) is required upon breaking it, and divorce does not occur. The husband's intention is considered: if he intended divorce, it occurs; but if he intended urging or preventing, then he is only obligated to pay expiation for an oath.

As for divorce pronounced in anger, its ruling varies depending on the intensity of the anger.

The husband must realize the seriousness of uttering divorce, as its frequent use has led to many instances of divorce, making his continued marriage to his wife a matter of dispute among scholars. The majority of them believe that he has divorced her irrevocably (three times). He must fear Allah and swear only by Him.