Could you explain a little more?
Facebook applications do NOT have the ability to run scheduled tasks. But facebook apps are mostly related to facebook or iframe related websites integrated into facebook.
Thus, you will need to run scheduled tasks on your server directly using something like CRON jobs .
Quick example:
Let's say your facebook application is located at: http://yourserver.com/facebookapp1/ You can schedule a task on your server that will call the page http://yourserver.com/facebookapp1/scheduledtaks/
So basically, he will call this page every week or day, and this page will do the work necessary for your service ...
Do you have a web server? Can you add cron jobs? What is your server technology? If you give us more information, we can give you a more specific answer ...
To answer your edit:
Perhaps you need to create a script that you will call every day, for example. Let me call it check_and_send. Now your script will need to check whether it is necessary to send something (it depends on your logic), for example, check the DB to find out if you need to publish something today or not. When he wants to post something, he can do it using facebook graph api . You will need to request permission to publish_stream for the user.
You will also need a preliminary step where you ask the user to authenticate with facebook and ask him to give your application permission to "publish_stream", but ALSO permission "offline_access" so that you can publish to the user's wall on anyone (and not only when he is connected ) ... see facebook rights
After that, you will need to save the access_token (for each user) in your database or somewhere, and you will use it when you need to publish this particular user on the wall ...
Shortly speaking:
1) (once) authenticate the user, request permission, save it to facebook UID and access_token
2) (periodically) check your database if something is to be published, if so, publish it using the facebook api chart and saved access_token.