Is it permissible for the family of a man who died in a non-Muslim country to claim three children a woman alleges she bore him, knowing that there is no evidence except for the testimony of a man who saw the woman with him in his house, without knowing if the relationship was legitimate?

The marriage of a woman or the lineage of her children to a deceased man cannot be established by her testimony and the testimony of the aforementioned man, due to the absence of conditions for proving marriage or lineage.

However, if the family of the deceased man believes the woman is truthful and wishes to attribute the children to him, this is permissible according to the Shafi'is and Hanbalis, under certain conditions, including: that the person to whom the lineage is attributed is deceased, that it is possible for the children to be his lineage, that the one acknowledging the attribution is an heir who possesses the deceased's estate, and that none of the heirs who possess the estate deny the lineage.

As for the Hanafis and Malikis, the acknowledgment of lineage by anyone other than the father has no effect in establishing lineage. The acknowledged person is considered related only to the acknowledging party, and the one who acknowledges inherits from them, provided that the deceased has no other heir.