I have two separate Memorex model MVD2042 A DVD players. They both quit just like yours did. Being an engineer, I took them both apart to troubleshoot, and found that they had failed for precisely the same reason: optoisolator failure in the power supply.
For safety reasons, the power supply design totally isolates the primary side including the AC line cord from the secondary side including the chassis, the signal cables, and everything else you might touch. Power from the AC line cord goes through a regulator on the primary side, then through the power transformer to the secondary side and through a set of rectifiers and filters to provide all of the various DC voltages needed by the MVD 2042.
The windings of the transformer are insulated and isolated from one another, so no connection exists through the transformer between primary and secondary.
However, measurements of the secondary voltages must be sent back to the primary to control the regulator. This cannot be done using a wire, because then the isolation would be lost, so an optoisolator is used to send the voltage measurements back to the primary.
The optoisolator is a device into which you can put information on the secondary side, and the information will come out on the primary side. The connection is made using a beam of light, so the primary and secondary remain isolated. When the optoisolator fails, it tells the regulator to turn down the voltage all the way, and the DVD player quits.
You can download a PDF data sheet for the optoisolator here: http://www.cosmo-ic.com/object/products/K1010.pdf. I have been unable to find a place to buy just a few of these devices, and my two DVD players remain unrepaired. I have heard that the HCPL-817-060E optoisolator can be substituted; it is made by Avago, and sold by Digi-Key.