Value distribution of meromorphic solutions of certain difference Painlevé III equations

In this paper, we investigate the difference Painleve III equations w ( z + 1 ) w ( z − 1 ) ( w ( z ) − 1 ) 2 = w 2 ( z ) − λ w ( z ) + μ $w(z+1)w(z-1)(w(z)-1)^{2}=w^{2}(z)-\lambda w(z)+\mu$ ( λ μ ≠ 0 $\lambda\mu\neq 0$ ) and w ( z + 1 ) w ( z − 1 ) ( w ( z ) − 1 ) 2 = w 2 ( z ) $w(z+1)w(z-1)(w(z)-1)^{2}=w^{2}(z)$ , and obtain some results about the properties of the finite order transcendental meromorphic solutions. In particular, we get the precise estimations of exponents of convergence of poles of difference Δ w ( z ) = w ( z + 1 ) − w ( z ) $\Delta w(z)=w(z+1)-w(z)$ and divided difference Δ w ( z ) w ( z ) $\frac{\Delta w(z)}{w(z)}$ , and of fixed points of w ( z + η ) $w(z+\eta)$ ( η ∈ C ∖ { 0 } $\eta\in C\setminus\{0\}$ ).