That's obviously not supposed to happen. There's a few things I'd recommend you check, but I don't have a specific answer as this isn't a common problem:
* You look to be using JomSocial. If so, please make sure you are using at least v3.2.0.3. If you're using an earlier version than that, you're likely affect by
this bug in JomSocial
* In the Global Configuration area of Joomla, please set Caching to Disabled to test. This isn't a permanent fix, but will help narrow things down.
* In the case where you look logged out, can you do a refresh in your browser and see if you are then shown as logged in properly? That could be an indication of browser caching for the HTML of your site, which shouldn't be used on sites that allow authentication.
Finally, you mention:
After i will logon with my username or my facebook account
Are you saying this happens even when you use your standard Joomla username and password and not only when logging in through a social network? If so, I'd recommend trying to disable the JFBCSystem, JFBCUser and JFBCAuthentication plugins. That will disable JFBConnect entirely. If the problem persists, it means that JFBConnect is not the cause of the problem. We'll gladly still help provide suggestions on how to fix the issue in this case, but there's only so much we'd be able to do.
I hope that helps get you started,
Alex