Using `date` command to get previous, current and next month

the main problem occur when you don't have date --date option available and you don't have permission to install it, then try below -

Previous month
#cal -3|awk 'NR==1{print toupper(substr($1,1,3))"-"$2}'
DEC-2016 
Current month
#cal -3|awk 'NR==1{print toupper(substr($3,1,3))"-"$4}'
JAN-2017
Next month
#cal -3|awk 'NR==1{print toupper(substr($5,1,3))"-"$6}'
FEB-2017

The problem is that date takes your request quite literally and tries to use a date of 31st September (being 31st October minus one month) and then because that doesn't exist it moves to the next day which does. The date documentation (from info date) has the following advice:

The fuzz in units can cause problems with relative items. For example, `2003-07-31 -1 month' might evaluate to 2003-07-01, because 2003-06-31 is an invalid date. To determine the previous month more reliably, you can ask for the month before the 15th of the current month. For example:

 $ date -R
 Thu, 31 Jul 2003 13:02:39 -0700
 $ date --date='-1 month' +'Last month was %B?'
 Last month was July?
 $ date --date="$(date +%Y-%m-15) -1 month" +'Last month was %B!'
 Last month was June!

If you happen to be using date in a MacOS environment, try this:

ST1:~ ejf$ date
Mon Feb 20 21:55:48 CST 2017
ST1:~ ejf$ date -v-1m +%m
01
ST1:~ ejf$ date -v+1m +%m
03

Also, I'd rather calculate the previous and next month on the first day of each month, this way you won't have issues with months ending the 30/31 or 28/29 (Feb/Feb leap year)

the following will do:

date -d "$(date +%Y-%m-1) -1 month" +%-m
date -d "$(date +%Y-%m-1) 0 month" +%-m
date -d "$(date +%Y-%m-1) 1 month" +%-m

or as you need:

LAST_MONTH=`date -d "$(date +%Y-%m-1) -1 month" +%-m`
NEXT_MONTH=`date -d "$(date +%Y-%m-1) 1 month" +%-m`
THIS_MONTH=`date -d "$(date +%Y-%m-1) 0 month" +%-m`

you asked for output like 9,10,11, so I used the %-m

%m (without -) will produce output like 09,... (leading zero)

this also works for more/less than 12 months:

date -d "$(date +%Y-%m-1) -13 month" +%-m

just try

date -d "$(date +%Y-%m-1) -13 month"

to see full result