Some of them said, "By the power of Beelzebul, the prince of demons, he drives out demons." Others, to test him, asked him for a sign from heaven. But he knew their thoughts and said to them, "Every kingdom divided against itself will be laid waste and house will fall against house…
But if it is by the finger of God that (I) drive out demons, then the kingdom of God has come upon you… (Lk 11:15-26)
Jesus was busy establishing the kingdom of God. One of the activities of establishing the kingdom of God was by chasing the demons. This activity of Jesus was misunderstood by many. Some thought that he was doing this by the power of Beelzebul. In effect they were accusing that Jesus was possessed and with the help of the head of the demons he was working.
There were others who believed but they wanted some assurance. So they asked for signs from heaven. They accepted but they wanted some confirmation. These people were also not belonging to the kingdom. Jesus refuted them as well. He used parables to tell them that even in ordinary events their arguments would not hold. No one will work against himself or his own kingdom.
Jesus argues from the point of view of God. If demons are cast out then it is the work of God. This is the prime argument to say that he was working for the kingdom of God. God was working for him and in him. He was sent by God and his work was to establish the kingdom of God. By exorcising the demons the work of the kingdom is already established among them.