iPhone 5s, shuts down suddenly and won’t start

Hello guys
I have in iPhone 5S which suddenly went off.
I started it with home+lock and it worked for 10′ and then again went off.
I took the phone to a service , they checked the battery(it was ok) , the problem persisted.
They changed the battery, the problem didn’t disappeared.(phone was shutting down by it self)
Now the phone is not even starting on, and they say the battery does not get the proper amount of power…..(they charged it with a charger that had a sort of display)
From time to time the phone starts, works really well for some minutes and then (even with full battery) the phone shuts down again.
The phone stars (when it does) only with lock+home…
The problem appeared when I updated the phone to iOS 10( the 2nd day apple released the iOS 10)

Ps: a strange thing is that when the phone "goes off", theoretically is not shut down as if you try to call the phone number that’s in the phone, you don’t text the message the phone is out of the network/area, but in the headset you hear that it rings…..

Does anyone else have this problem?
Thank you.

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