What is the ruling on continuing to try to wake up one's family members for prayer, especially Fajr, if they do not wake up until after the time for the obligatory prayer has passed, and this is after a year of unsuccessful attempts?

Scholars have differed on the ruling of waking a sleeping person for prayer. Some have said it is recommended, provided the sleep was not a transgression. Others have said it is obligatory, especially if the time for prayer is limited. You should strive to wake your family members for prayer, and if they do not wake up, then you have fulfilled your duty.