Solutions of certain Fermat-type partial differential-difference equations

The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: $${\left(\sum _{j=1}^m\frac{\mathrm{\partial }f(z_1,z_2,\dots ,z_m)}{\mathrm{\partial }{z}_j}\right)}^{m_1}+f^{m_2}(z_1+c_1,z_2+c_2,\dots ,z_m+c_m)=1,$$ where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1,c_2,\dots ,c_m)\in {\mathbb{C}}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from ${\mathbb{C}}^2$ to ${\mathbb{C}}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings.