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When a door is opened on my 1988 Vanden Plas that door lamp, the dome lamps, and the driver headrest lamp illuminate. The passenger headrest lamp doesn't . However it does come on by the roller switch. With help from Don B of this forum I diagnosed the problem as a failed diode on the lamp. Power remains at all times to the lamp so it can be turned on and off manually by the roler switch. The diode is responsible for communicating with the CPU (central processing unit) to automatically turn on the lamp only when a door is open and turn it off when the door is closed.
I've read where some people replaced the diode, but it was so tiny (pictured at the red arrow) and needs to be soldered (which I would have to buy equipment to do) that I decided to order a new lamp.