Q: how to check if FQDN host-name is with only small letters we want to check the hostname on machines
the rule should be like this:
each hostname must be only with small letters
each hostname is FQDN
we try this
hostname=master01.nassa.com
[[ $hostname  =~ ^[a-z.a-z.a-z]+$ ]] && hostname is ok

but in spite the hostname is valid , it isn't print "hostname is ok"
what is the right syntax , where we are wrong?

A: Your regex is somehow wrong and the part after && misses echo. 
Instead of writting complex regexs to validate the hostname you can consider to do the opposite:
hostname=master01.naSsa.com
[[ $hostname  =~ [A-Z] ]] && echo "hostname is not ok"
hostname is not ok

Or 
hostname=master01.nassa.com
[[ $hostname  =~ [A-Z] ]] || echo "hostname is ok"
hostname is ok

Moreover to avoid complex situation you better define locale to C or posix using LC_ALL=C
See : Why does [A-Z] match lowercase letters in bash?

A: For a /bin/sh solution:
hostname=master01.nassa.com

case $hostname in
    *[A-Z]*)
        echo 'Not ok' ;;
    *)
        echo 'Ok'
esac

The "Not ok" branch is taken if the value of $hostname contains any uppercase characters (this may depend on your locale settings), while the "Ok" branch is taken if the "Not ok" branch isn't taken.
This is similar to the bash-specific (and also ksh and zsh)
if [[ "$hostname" == *[A-Z]* ]]; then
    echo 'Not ok'
else
    echo 'Ok'
fi

Your test with the regular expression ^[a-z.a-z.a-z]+$ is identical to a test with ^[a-z.]+$ (since a [...] character class matches a single character and repeated a-z inside it won't make a difference).   This will actually do what you say it does, which is to allow only hostnames with lowercase letters (and dots).  The issue is that your hostname also contains digits.  To allow for these too, you would have to use ^[a-z0-9.]+$.  In this case though, it may be easier to test for the smaller set of characters that are not allowed, which is what I do above.

A: hostname=master01.nassa.com
[[ "$hostname"  =~ ^[a-z]+\.[a-z]+\.[a-z]+$ ]] && echo hostname is ok

is your example with some bugs fixed but for more accurite validation look at https://stackoverflow.com/questions/11809631/fully-qualified-domain-name-validation
