A Limit Set Trichotomy for Self-mappings of Normal Cones in Banach Spaces

(Here, we use the notation x > y and x 2 y to mean, respectively, x y E k” and x y E K”.) This result extends an earlier theorem of Smith [2] concerning “discrete dynamics of monotone, concave maps”; some interesting applications to differential equations can be found in [l, 21. Another extension of Smith’s theorem has been given by TakaE in [3]. In this paper we shall extend the above trichotomy in two directions. First, we shall allow general normal cones K with nonempty interior in a Banach space. Second, we shall allow a class of maps which is considerably more general than classes allowed in earlier results, even for K, the positive orthant in I?“. The key observation which we shall exploit centers about a metric p, called the part metric or Thompson’s metric (see [l, 4-61 and the references given there; and Section 2 below) which is defined on the interior I? of a cone. The proper class of maps to study seems to be those maps T: I? + k such that T”‘, the mth iterate of T, satisfies