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Abstract In this paper, we investigate the difference Painlevé III equations w(z+1)w(z−1)(w(z)−1)2=w2(z)−λw(z)+μ $w(z+1)w(z-1)(w(z)-1)^{2}=w^{2}(z)-\lambda w(z)+\mu$ ( λμ≠0 $\lambda\mu\neq 0$) and w(z+1)w(z−1)(w(z)−1)2=w2(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 Δw(z)=w(z+1)−w(z) $\Delta w(z)=w(z+1)-w(z)$ and divided difference Δw(z)w(z) $\frac{\Delta w(z)}{w(z)}$, and of fixed points of w(z+η) $w(z+\eta)$ ( η∈C∖{0} $\eta\in C\setminus\{0\}$).