Introduction
Jaya is one of the two gatekeepers of Vishnu's heaven, Vaikuntha, together with Vijaya. In Puranic tradition, the two were cursed to be born on earth three times as enemies of Vishnu.
This story is not part of Valmiki's Ramayana.
The curse
According to the Bhagavata Purana, Jaya and Vijaya stopped the sages Sanaka and his brothers from entering Vaikuntha, and were cursed to be born on earth. They chose to be born as enemies of Vishnu, so that they would return sooner. In the second of their three births, they were born as Ravana and Kumbhakarna, and Rama ended their lives.