Is the mother obligated to write over a portion of the house to her son who made improvements to it without her knowledge, given that she does not have the necessary funds for this, and that the rest of her children consider this an injustice to them?

As long as the son spent on furnishing his mother's apartment without her permission, he has no right to demand a share in it. Some scholars are of the opinion that he is not entitled to claim anything of what he spent, even if he intended to be reimbursed. Others are of the opinion that he is entitled to be reimbursed for what he spent if he intended to be reimbursed and did not spend as a donation. Therefore, the mother is not obligated to write a part of the house to her son in exchange for what he spent on it, and it is not permissible for her to favor him with that, even as a donation from her; because equality is obligatory in giving gifts to children.