What is the ruling on one who vowed not to do a specific thing and emphatically swore an oath while prostrating, invoking the names of Allah Almighty, and vowed to fast perpetually if he broke it, then fell into the sin and made the same emphatic vow again, then likewise succumbed to weakness?

Sincere repentance is necessary for all sins and transgressions, along with adherence to Allah's law. As for the vow, it is a vow of obstinacy (nadhir lajaj), and the fatwa concerning it permits a choice between fulfilling the vowed act or performing an expiation for breaking an oath (kaffarat yamin). If you do not wish to fast perpetually, you may pay the expiation for breaking an oath. An expiation is due for each time you broke the vow. The expiation is to feed ten needy persons, or clothe them, or free a slave. If you are unable to do that, then fast three days for each expiation, and it is more cautious for them to be consecutive.