This paper concerns difference equations $y(x+1)=G(x,y)$ where $G$ takes values in ${\mathbf{C}}^n$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $\mathbf{C}$ and holomorphic in a neighborhood of 0 in ${\mathbf{C}}^n$. It is shown that under certain conditions on the linear part of $G$, formal power series solutions in $x^{-1/p},p\in \mathbf{N},$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

In this paper a proof is given of a theorem of J. Écalle that formal power series solutions of nonlinear meromorphic differential equations are multisummable.