Two consoles charging, but not turning on

I was rooting through my collection of consoles in the attic this weekend looking for my PSP Go, and I discovered that I have not one, but two of them (I buy far too much junk). Unfortunately, the consoles are not working. I bought them both off eBay a few years ago. I can’t remember if one of these units ever worked, so it’s not unlikely that both of them might have been dead when I received them. This means that I don’t know their history, so can’t tell whether they’ve ever been modded.

Both units exhibit similar, but not identical behaviour:

- Both units appear to charge when plugged in (the orange light comes on)

- On Unit A, the green light comes on when I switch it on, and stays on until I power down the unit (by activating the power slider for a few seconds)

- On Unit B, the green light comes on when I switch it on, but it switches itself off after a few moments

- Neither unit displays anything on the screen

- Neither unit makes any sound at all

I fully disassembled and reassembled Unit A last night, checking for any damage and making sure all the cables are firmly seated. Unfortunately, I couldn’t find anything and the exercise made no difference.

Can anybody offer any advice?