I don't have time to try this, but this code should work.

#!/usr/bin/perl -w

  use strict;
  open (FILE, "+<users.txt") or die "File not found\n"; #Open the file
+ in read/write mode
  flock FILE,2; # It is good practice to lock a file you will write to
+.
  my $name = '';
  my $yn = '';

  print "Enter your name--> ";
  chomp ($name = <STDIN>);

  while (<FILE>) {

     chomp;
     if (/^$name/) {
        print "Your name was already found\n";
        exit; }
  }
 
  print "Name not found. Would you like to add it? (y/n)--> ";
  chomp ($yn = <STDIN>);
  if ($yn eq 'y') {
        print FILE "$name\n"; #Since we went through the file we can n
+ow print to the end of the file
        close FILE; #Closing a file will unlock it
  }else {
        print "FINE!\n"; ;
  }
   
  print "end.\n";
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