I searched Google and found a few references to A. W. Sullivan. One hit referenced "New Ireland" by A. W. Sullivan in the Hawkesbury Chronicle and Farmers Advocate (Windsor, NSW) Sat 11 Sep 1886 - Page 4 - Advertising - and another hit reference to "Summer Day" by A. W. Sullivan. Unfortunately - absolutely no paintings displayed...
I find an A W Sullivan listed as an annual member of the board of trustees for the Chicago Art Institute 1895-96. That would be Albert W. Sullivan. His brother was the famous architect, Louis Sullivan. This is the only relation I can find to that name and art.
Is there a signature? I did find some of his handwriting here...
Program for the May 27th, 1876 athletic sports competition organized by the Chicago Football Club in which Louis H. Sullivan and his brother, Albert W. Sullivan participated. With annotations in Albert W. Sullivan's hand. Source: Art Institute of Chicago