How can the younger brother be appeased after the death of the father who willed his apartment to the questioning brother, knowing that the father had helped the other brothers purchase their apartments?

First: If your father's act of writing the apartment in your name was intended for you to take possession after his death, it is considered a bequest to an heir (وصية لوارث). This bequest is not effective unless the other heirs consent. If they do not consent, the apartment reverts to the estate and is divided according to Islamic law. For an heir's consent to be valid, they must be an adult of sound judgment. However, if your father intended it as a gift during his lifetime, but you did not take possession of it until after his death, the gift becomes void upon his death before its قبض (taking possession).

Second: Regarding the money your father paid to buy and furnish an apartment for your two brothers: If it was a loan (قرض), then it is a debt upon them and must be returned to the estate. If it was a donation (تبرع) or a gift (هبة), then it must be seen whether he acted equitably (عدل) in it. If he acted equitably by giving the remaining children a gift that achieved fairness, then it is permissible. If he did not act equitably, then upon his death, the gift becomes effective, and they are not obliged to return it. However, some scholars hold the view that an inequitable gift (هبة جائرة) must be returned to the estate even after the death of the giver.

Third: In case of disagreement among the heirs, they should refer to the Sharia court or to a scholar qualified for judicial matters.