I had the same problem on my L535 so here is what i know:
Lumia line REALLY have a problem with dead displays!
This problem is not limited only to L535, as I found during my research when i faced this problem. Some guys face it on L820, L920 and some other Lumias. So my guess is there is a problem on an assembling line from a Nokia/Microsoft plant.
On L535 this problem is faced more often!
From all devices on which this problem is present, L535 has it the most frequently. BUT, what would you expect from a device which was clearly bad-crafted? I mean, after all the problems L535 comes with, i'm not surprised it has this problem more often than any other Lumia device.
Cause AND Solution
The cause of this problem is....-drums-....bad/broken motherboard. When i faced the problem, 1-2 days after i bought the device, i sent it for replacement, and they told me, before granting the replacement, that the motherboard was the problem. SO, the fault is MS/Nokia's not users.
Solution? Send it back for repair or replacement. Its not your fault and there is almost nothing you can do. Almost? Yea. You could try a soft reset (volume down+lock) or a reflash from Lumia Recovery Tool, but there were just a few ppl who solved the problem this way, and my guess is that their problem with dead screen was not caused by a broken motherboard.
PS: I now have a L535 again, working decent(i wont lie, is not perfect). It is still hardware 3.0.0 not that 5.0.1 variant and i have the latest firmware, which i recommend you to get even from Lumia Recovery Tool if the OTA is not ready for your region. Sure, you'll loose some of your data even if you back up things, but the improvement with the latest firmware is BIG so its worth it.