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When the phone is unlocked the screen does not turn on. The touchscreen is dark but I can answer calls and turn off the phone. The screen goes dark for a while and then returns to normal. The time the screen is dark is getting bigger.
I appreciate any help.
That seems to be a hardware issue, you could repair the phone with Xperia Companion and see if that solves the issue, if it doesn't then it's time to contact your Local support to get it inspected.