Should the wife's children from her first husband inherit from the husband's properties that he wrote in the wife's name, given that he instructed her that these properties should be for his children after his death, and that she did not write this will?

Merely writing down properties does not indicate ownership, and a will in favor of an heir is not valid, due to the saying of the Prophet, peace be upon him: "Indeed, Allah has given every rightful person their due, so there is no will for an heir."

If the father dies before the mother, his wealth is a right for all his heirs, and the mother is among them. Her share is then निकालो and distributed among her heirs.

If a dispute arises, it is best to refer the matter to the Sharia court.

One must fear Allah and not take what one does not deserve.