NONLINEAR PERRON-FROBENIUS PROBLEM FOR ORDER-PRESERVING MAPPINGS .1.

We consider the eigenvalue problem of an order-preserving mapping defined on a positive cone of an ordered Banach space. Among other things, we prove the existence and, in some cases, the uniqueness of the positive eigenvalue. We also discuss other properties of eigenvalues and eigenvectors. The notion of indecomposability for nonlinear mappings that we introduce in an infinite dimensional setting will play a key role in our argument. We apply the results in this paper to boundary value problems for a class of partial differential equations in part II.