but it is still not making the expected result

what is the expected result? what is the observed result?

a couple observations:

• use warnings; in addition to use strict;
• LWP::Simple can make the code a little more, well, simple ;) (at a glance, i don't see anything that requires LWP::UserAgent directly (oh, now i see the user agent string change...)

  use LWP:::Simple;
  my $res = get( $target );
  if( is_success($res) ){
   ...
  }else{
   ...
  }
  [download]

• where is check_url() called from? Seems like it should return a boolean based on the result of the check.
• where is $response_limit defined?
• can avoid some repetativeness, since the printf format is always the same:

  sub check_url {  # subroutine who check given URL
      my $target = shift;
      my $response_limit = shift;
      my $time = time;      # Start timer
      my($result, $rc, $error);
      my $res = get( $target );
      if( is_success($res) ){
            $time = sprintf "Response %d seconds", (time - $time);
            if ($response_limit && ($time >= $response_limit)) {  # note
  + bugfix of direction of inequality
               $error = "Slow response from $target\: $time seconds";
               ($rc, $result) = (-1, "SLOW");
            } else {
               ($rc, $result) = (1, "ACCESSED");
            }
      } else {                 # Error .... Site is DOWN and script send
  + e-mail to you..
            ($rc, $result, $time) = (0, "DOWN", "N/A");
            $error = "$target is DOWN." . $res;
      }
      push @errors, $error if defined $error;
      my $out_format = "| %-50.50s %-10s %-20s |\n";
      printf OUT $out_format, $target, $result, $time;
      printf     $out_format, $target, $result, $time;
      return $rc;
  }
  [download]