Unicity of meromorphic functions concerning differences and small functions

In this paper, we study the unicity of meromorphic functions concerning differences and small functions and mainly prove two results: 1. Let f be a transcendental entire function of finite order with a Borel exceptional entire small function a(z), and let eta be a constant such that Delta(2)(eta) f not equivalent to 0. If Delta(2 )(eta)f and Delta(eta )f share Delta(eta)a CM, then a(z) is a constant a and f (z) = a + Be-Az, where A, B are two nonzero constants; 2. Let f be a transcendental meromorphic function with rho(2) (f) < 1, let a(1), a(2) be two distinct small functions of f, let L(z, f) be a linear difference polynomial, and let a(1) not equivalent to L(z, a(2)). If 8(a(2), f) > 0, and f and L(z, f) share a(1) and infinity CM, then L(z, f)-a(1)/f-a(1) = c, for some constant c not equal 0. The results improve some results following C. X. Chen f-a(1) and R. R. Zhang [Uniqueness theorems related difference operators of entire functions, Chinese Ann. Math. Ser. A 42 (2021), no. 1, 11-22] and R. R. Zhang, C. X. Chen, and Z. B. Huang [Uniqueness on linear difference polynomials of meromorphic functions, AIMS Math. 6 (2021), no. 4, 3874-3888].