Why are my lights not working on my ceiling fan

The lights on my ceiling fan in my living room went out yesterday, the light bulbs that go in it was different than I’m use to seeing and I had to order new ones on Amazon. I tried to take the bulbs out of the fixture in the kitchen to put in the one in the living room until the new bulbs arrived. When I did this the lights still didn’t work. So then I thought, hmm I wonder if the bulbs that were in it weren’t bad so I tried those (bulbs that were originally in the living room) in the fixture in the kitchen. I put them in and they blinked on for a second then went out. Now neither of them are working. I got the new bulbs in the mail today and still nothing. The fans on both work but neither of the lights will come on. What can my problem be?

When I put the old bulbs into the light fixture in the kitchen one bulb blinked on and then immediately went back out.here is what the bulbs look like, I’m not quite sure what type of fixture they are, they were already in the house when we bought it last year. How would I know if it’s the control circuit as you stated and what is the solution for that?