Entire functions sharing a small entire function‎ ‎with their difference operators

In this paper‎, ‎we mainly investigate the uniqueness of the entire function sharing a‎ ‎small entire function with its high difference operators‎. ‎ < W > e obtain‎ ‎ < a > result‎, ‎which can give a negative answer to an uniqueness question‎ ‎relate to the Br$\ddot{u}$ck conjecture dealt by Liu and Yang‎. ‎Meanwhile‎, ‎we also establish a difference analogue of the‎ ‎Br$\rm\ddot{u}$ck conjecture for entire functions of order less than 2‎, ‎which improves some results obtained by Liu and Yang.