From Wikipedia: "James B. Gillett (November 4, 1856 – June 11, 1937) was a lawman of the Old West, mostly well known due to his service as a Texas Ranger, and is a member of the Texas Rangers Hall of Fame."

Inscription Reads:

To Capt. Gillett with Sicere Good Wishes of J. Stockley Ligon October 30th, 1928.

I feel confident that the J.B. Gillett referred to in the inscription is the famous Old-West lawman. I purchased several other books in a lot with this one, and they all appear to be from Mr. Gillett's personal library. 

I am happy to send additional photos or answer questions. Please send me a message, and I will get back to you as soon as possible.