"How can check this? I'm using self-signed certificate which was included with F5 LTM VE."
- Nevermind. You're not using client certs and you're using the default server certs on the BIG-IP. Didn't catch that at first.
Okay, so it doesn't appear to be a validation/revocation issue. Do you not get certificate warnings with all of the browsers? Do you have any special cipher settings (or anything special) in your client SSL profile, or in your browsers?
Also, not sure this is related, as it talks about client cert, but take a look at this:
sol12313: The BIG-IP system responds with a full SSL handshake regardless of whether the client sends a session ID during SSL session renegotiation
http://support.f5.com/kb/en-us/solutions/public/12000/300/sol12313.html?sr=27615885
And ID 226971 from the 10.2.1 release notes
http://support.f5.com/kb/en-us/products/big-ip_ltm/releasenotes/product/relnote_10_2_1_ltm.html?sr=27616725