the Safari browser does not open?

Have you tried opening it from the Applications folder? The Dock alias may be corrupt.

Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

I suggest you use the program created by Etresoft, a frequent contributor.  It will provide a snapshot of your system which we can analyze to possibly determine the cause of your problem. Please use copy and paste as screen shots can be hard to read. On the screen with Options, please open Options and check the bottom 2 boxes before running. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown.

https://itunes.apple.com/us/app/etrecheck/id1423715984?mt=12

Use the Add Text button to include the Etrecheck report in your reply.