Abstract
Publisher Summary This chapter introduces fixpoints and describes the exposed fixpoints in order-structures. Fixpoints exist only under additional assumptions on the maps that are involved. The results are applied to the lattice of superlinear vector-valued functionals on some abelian semigroup. There, the fixpoint theorem yields strong combinations of Hahn-Banach and Krein-Milman type theorems. The chapter presents an iteration theorem for almost commuting families of decreasing maps, generalizing the iteration theorem which was proved for the case of a single map. This special case had turned out to be a rather efficient tool in many areas of analysis. The chapter also discusses some applications that include boundaries for compact sets and iterations.
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