Am I permitted to use the internet service that I need for my studies, while one of my relatives is using it for disobedience to Allah, knowing that I have tried repeatedly to stop him without success and I cannot do without the service? And do I bear the sin of my relative's use of it, given that I was the one who introduced the service?

If you are using the service for permissible matters, then you are not obliged to disconnect it, especially if you cannot do without it. You must prevent others from using it for forbidden things to the best of your ability, or expend whatever effort you can in that regard. You should advise your relative, enjoin what is good upon him, and forbid what is evil from him. After that, you bear no sin for him, as Allah Almighty says: "So remind, for you are only a reminder. You are not over them a controller" (Qur’an 88:21-22), and His saying: "O you who have believed, upon you is [the responsibility for] yourselves. No harm will come to you from those who have gone astray when you have been guided" (Qur’an 5:105). So, whoever fulfills the duties within his capacity has been guided.