Have you ever had your Facebook logged out by itself?
It's probably something wrong with internet browser's cookies.
Set the cookies to active and let cookies has high enough of capacity of byte.
Your cookies also brobably removes itself, try to look at the internet browser configuration. If you are using Mozilla Firefox, you can click "Tools" on menubar and find something about cookies right there. Internet Explorer, Opera Browser, Netscape Navigator, and some other internet browsers has similar method.
