Genuine Ford starter solenoid with Motorcraft markings and part number.

NOTE: New solenoids must have a good power source, and a good earth, to prevent them from sticking. Please charge your battery, make sure your cables are in good condition without internal oxidation (high resistance) and improve the earth point between the bracket and the inner apron. Clean off some paint, or use a nut, bolt and panelwasher on the inner apron, with the paint cleaned off.

In basic terms, what happens is when the power is low, the contacts move slower, and the current draw starts arcing across the terminals before they touch, welding it contacted. When it has good power, the solenoid moves faster and it doesn't have a chance to arc across. 

As this is an installation issue, this isn't covered by warranty.