If you go to SO-H can you bring up the invoice and see the lines on it for all the items?  Are there shipment transactions listed in IN-E?  If you go to IN-A and click Shipments, are they there?

Then the invoices are in fact posted and the On-Hand in IN-A should be less than it was before the invoice posted.  The "Available" will not be dfiferent but the On-Hand will be.  
 
Next time you are getting ready to post an invoice, go to IN-A for the part number on the invoice before and after posting and check the On-Hand quantity.  Posting the invoice should both reduce the On-Hand and reduce the On Sales Order or On Back Order.