Multisummability for some classes of difference equations

This paper concerns difference equations y(x + 1) = G(x, y) where G takes values in C-n and G is meromorphic in x in a neighborhood of infinity in C and holomorphic in a neighborhood of 0 in C-n. It is shown that under certain conditions on the linear part of G, formal power series solutions in x(-1/p),p is an element of 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.