Periodic behavior of meromorphic functions sharing values with their difference operators in several complex variables

In the paper, we extend the results of Chen and Yi [4] and Fang et al. [6] concerning sharing three distinct values for meromorphic functions in $$\mathbb {C}^m$$ . We demonstrate that if f and $$\Delta _cf$$ share three distinct values $$a_1$$ , $$a_2$$ and $$\infty $$ CM, then either $$f(z) = e^{b_{1} z_{1} + \cdots + b_{m} z_{m}} h(z), \text { where h is c-periodic meromorphic function in } \mathbb {C}^m, (b_1, \dots , b_m) \in \mathbb {C}^m \setminus \{0\} \text { such that } b_{1} c_{1} + \cdots + b_{m} c_{m} = \ln 2 + 2k\pi i, k \in \mathbb {Z}$$ or $$f(z)\equiv f(z+2c)$$ for all $$z\in \mathbb {C}^m$$ , with the latter case arising when $$\limsup \limits _{r\rightarrow \infty }\frac{\log T(r,f)}{r}>0$$ . Also in the paper, we improve the recent result of Huang and Zhang [13] for entire function in $$\mathbb {C}$$ to meromorphic function into higher dimensions. Moreover, we show by plenty of examples that our results are best possible.