You need to send the app-to-user request using the application access_token
. For example:
$param = array(
'message' => 'Check out the latest update',
'data' => 'some_data_string',
'access_token' => 'app_id|app_secret',
);
$tmp = $facebook->api("/[user_id]/apprequests", "POST", $param);
EDIT:
See the facebook documentation for generating an app access_token. The documentation states:
There is another method to make calls to the Graph API that doesn't
require using a generated app token. You can just pass your app id and
app secret as the access_token parameter when you make a call:
https://graph.facebook.com/endpoint?key=value&access_token=app_id|app_secret