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Boot in safe mode:
/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
How is it?
Yes. Strange though,are you willing to perform a software repair?
Try force restarting the phone:
Press and hold the "Power Button" and "Volume Up" for 10 seconds until your Xperia vibrates 3 times. Now switch it on and observe the battery level.
You may need to perform a software repair using PC Companion.The process will erase all internal memory.You might need to charge your device for about 1-2 hours before so the device has at least 80% battery.
Make sure the device is not connected to computer or laptop and turned off (sometimes you have to press power button and volume up until the device vibrates 3 times or on some devices you have to press the small button near the SIM tray or on the back of the device).
Open PC companion->support zone->phone/tablet software update->repair my phone/tablet blue highlighted.Follow the steps given,do not connect your phone until prompted to press volume down while plugging USB cable. If PC companion does not start the repair process, you might need to run it in compatibility mode with Windows 7 (in some cases XP).
Hi,
The same here. Z3 Dual. Battery stuck at 50%.
On top, I am unable to start charging (original Sony charger + cable connected).
The same behaviour
- after repair via PCC
- booted in Safe mode.
- when switched off with Power + VolumeUp
- after orange button near sim pushed until 3 vibrations
I would take it to my local authorised service but I can't find any on the official Sony site... 
Maybe I will downgrade to 4.4.4. (it would be my last attempt to repair it myself)
Any other advices?
After I downgraded to 4.4.4 there was no change.
Upgraded back to 5.0.2. No change.
Now, after a couple of days the battery is drained. No possibility to get it charged. (the connected charger causes occasional blinks of red notification diode and picture of empty battery on the display)
I think it's hardware problem - searching for authorized repair service.