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If ƒ is an entire function of type a, bounded on the real axis, a well-known theorem of Bernstein gives sup^R \f'(x)\ < ffsup^ eR \f(x)\. Our objective is to construct a Banach space, based on this theorem, in which the existence problem for broad classes of partial differential equations can be solved with surprising ease. The idea of using Bernstein's theorem in the study of coupled systems is due to Nickel (3). 1. Notation. We denote real or complex ra-dimensional space by Rm or Cm respectively, and the norm of a vector z in one of these spaces by \z\. The norm of any function F: F—> Cp is ||F|| = sup^ GF|F(^)|. We choose o G Rm fixed, ot > 0, and use ƒ as a multi-index; thus: DJ = , a' = o(iaJi • • • >.

Existence theorems for strongly coupled systems of partial differential equations over Bernstein classes