Make sure you not running Firefox in (permanent) Private Browsing mode.

• https://support.mozilla.org/kb/Private+Browsing
• Tools > Options > Privacy > Firefox will: "Use custom settings for history"
• Deselect: [ ] "Always use private browsing mode"

If you not using Private Browsing Mode

• Tools > Options > Privacy > History>
• Firefox will: "Use custom settings for history":
• Select: [x] Accept Cookies from sites
  • Accept third party cookies: "From Visited"
• Keep until "They Expire"
• Deselect: [ ] "Clear history when Firefox closes"
• https://support.mozilla.org/kb/Clear+Recent+History

If you use cleanup software like CCleaner then check the settings for the Firefox application.

I have not changed anything in the settings of my Firefox browser! However I have 2 programs for Cleaning and There is nothing wrong in the settings?

check my Firefox browser settings ....

Details like websites remembering you (log you in automatically) are stored in a cookie.

• Create a cookie 'allow' exception to keep such cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Tools > Options > Privacy > Cookies: Exceptions

If clearing cookies doesn't work then it is possible that the cookies.sqlite file that stores the cookies is corrupted.

Rename (or delete) cookies.sqlite (cookies.sqlite.old) and delete other present cookies files like cookies.sqlite-journal in the Firefox profile folder in case the file cookies.sqlite got corrupted.

• http://kb.mozillazine.org/Cookies
• https://support.mozilla.org/kb/Deleting+cookies

The next time you log into a site, make sure the box ladled Remember Me is on.

Maybe also do a malware check with some malware scanning programs on the Windows computer.
Please scan with all programs because each program detects different malware.
All these programs have free versions.

Make sure that you update each program to get the latest version of their databases before doing a scan.

• Malwarebytes' Anti-Malware:
  http://www.malwarebytes.org/mbam.php
• AdwCleaner:
  http://www.bleepingcomputer.com/download/adwcleaner/
  http://www.softpedia.com/get/Antivirus/Removal-Tools/AdwCleaner.shtml
• SuperAntispyware:
  http://www.superantispyware.com/
• Microsoft Safety Scanner:
  http://www.microsoft.com/security/scanner/en-us/default.aspx
• Windows Defender: Home Page:
  http://www.microsoft.com/windows/products/winfamily/defender/default.mspx
• Spybot Search & Destroy:
  http://www.safer-networking.org/en/index.html
• Kasperky Free Security Scan:
  http://www.kaspersky.com/security-scan

You can also do a check for a rootkit infection with TDSSKiller.

• Anti-rootkit utility TDSSKiller:
  http://support.kaspersky.com/5350?el=88446

See also:

• "Spyware on Windows": http://kb.mozillazine.org/Popups_not_blocked

I followed what you suggested, but this doesn't fix the problem

Create a new profile as a test to check if your current profile is causing the problems. See "Creating a profile":

• https://support.mozilla.org/kb/profile-manager-create-and-remove-firefox-profiles
• http://kb.mozillazine.org/Standard_diagnostic_-_Firefox#Profile_issues

Profile Backup and Restore

• http://kb.mozillazine.org/Profile_backup
• https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles
• http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

for windows 7 ? https://support.mozilla.org/en-US/kb/profile-manager-create-and-remove-firefox-profiles

i have Create a new profile and that’s its okay now =) but how to fix the problem?

Profile is like create a new user, so everything creates newly. I hope it solves your issue

I know this and do not want to lose any of the open pages such as YouTube 'forum, etc! And a lot of pages, so I was thinking to do delete the Firefox browser completely and re-install it but now I have rollback!

So Is there a way to fix the problem without removing the Firefox browser?

Use firefox SYNC / Profile backup

Profile Backup and Restore

• http://kb.mozillazine.org/Profile_backup
• https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles
• http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

do i need to Create a new account for SYNC?

If you need.

• https://support.mozilla.org/en-US/kb/how-do-i-set-up-firefox-sync

Even YouTube keeps log my out of my account?

Using Firefox in Safe Mode keeps logs my out of websites/my Account?

In case you are using "Clear history when Firefox closes":

• do not clear the Cookies
• do not clear the Site Preferences

• Tools > Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
• https://support.mozilla.org/kb/Clear+Recent+History

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, and passwords.

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you would like to keep.

i don’t use "Clear history when Firefox closes" so is there something wrong ?

You seem to be losing cookies somehow if you get logged out and the website doesn't remember you.
If you aren't doing this in Firefox then other software (possibly security software) is likely to do this.
So you need to check the settings in that software to disable such a feature to cleanup the cookies.

Boot the computer in Windows Safe Mode with network support (press F8 on the boot screen) as a test to see if that helps.

i have both Malwarebytes Anti-Malware & SUPERAntiSpyware Free Edition Installed on my system, but I do not have any anti virus However I've cleaned up the Firefox browser + my system using programs (CCleaner. Etc.) So far, my problem is not solved?

and Boot the computer in Windows Safe Mode with network isn’t help