Q: IOS 9.3.5 internet connection

Take a look at this Apple Support article:

https://support.apple.com/en-us/HT204051

Reset your network settings.

Settings - General - Reset - Reset Network Settings

Reset your device. This won't delete your data stored on the phone.

• Press and hold the Sleep/Wake button
• Press and hold on the Home button
• Keep holding both buttons until the display turns off and back on with Apple logo on it.

Alternatively, you may go to Settings - General - Reset - Reset All Settings

The same thing is happening for us.  On Friday one of our iPhone 6 quite working.  Constantly searching for service or saying no service.  Went to spring and spent hours trying to figure it out.  phone was activated, network settings updated, new sim card, even changed the number.  Nothing made it work.  Then they talked with Apple and reset the phone, connected to a computer, still nothing.  Only option they left us with was to upgrade and get a new phone. 

Then today, my daughters iPhone 6 started doing the same exact thing.  We have a case number with Apple directly, they worked with us for hours today and still the phone has no service and does not work even on wifi!  The rep helping us has to take the case to the next level and will be contacting us back to send it to them for repair or replacement because there are no options left.

This is not a simple issue.  One phone doing it, ok... we bit the bullet and upgraded buying a new one.  But now two doing the same thing right after installing the update..... this is a major apple problem and needs to be fixed NOW

It worked!!!  thanks so much.  excellent directions