Apple TV 3rd Generation stopped working and cannot be properly restored using iTunes.

I have two Apple TV 3rd Generation. Both stopped working and cannot be properly restored using iTunes. The restore appears to work but when either is plugged into the TV they just intermittently flash the Apple TV screen. They try to give a stable TV picture input but can't. I have tried restore 3 times. The final message is always the same - restore complete, disconnect from the Mac and connect the Apple TV to TV set.