Is it possible for a Facebook application to send a message to its users ? Sending the Message that we get next to notification(inbox) .On behalf of the Application. ?
The facebook api does not allow sending private messages (inbox) at all, not for an application or a user by an application.
For user (with a user access token) applications can use the Send Dialog or the Chat API.
As for sending user notifications from the application, check the Social Channels and Requests documentation.