You cannot do it without something calling your action via url. And that's what is cron job made for.
You put a cron job (crontab), and make it execute your action via curl every monday.
Note: you need Linux for this, either on your server or elsewhere as long as external system can access your URL.
P.S. and there is another solution, basically to have your PHP CLI program daemonized, so it will run in an endless loop and do the cron tasks for you.