Thank you for your reply. That is correct, I think that Henry B. and William I. Harrison were half-brothers to George, Mary J. and James W. Lumpkins based on the 1850 census of Weakley County, TN. Right now it is just a theory, not enough documentation yet to be sure.
In the 1860 census, Ulysses Harrison was 4 years old and living in the Lumpkins household in Weakley County, TN. During the Civil War, the three Lumpkins siblings moved to Conway County, AR where Henry and William Harrison had moved in the late 1850's.
In 1870 Conway County, AR census, Ulysses Harrison was living in household of William I. Harrison. Mary J. Lumpkins, who by this time had married John Hamilton, was living next door to Henry B. Harrison and William I. Harrison in this same census. In the 1900 census of Ellis County, TX Ulysses and his family were listed on same page as William I. Harrison and his family. I do not know how Ulysses was related to William at this time.
I found a biography of James W. Lumpkins in an Ellis County, TX history from the mid-1890's. It stated that his mother Malinda Fiser was first married to a Harrison, before she married James's father George H. Lumpkins. I have not yet found a record for either of Mailnda's marriages.