Polynomial-like iterative equation on Riesz spaces (Q6612324)

The paper deals with the following polynomial-like iterative equation:\N\[\N\lambda_1 f+\lambda_2 f^2+\cdots+\lambda_m f^m=F,\N\]\Nwhere \(F\) is given, \(f:X \to X\) is the unknown function, \(X\) a linear space, and \(f^k\) denotes the \(k\)-iterate of \(f\). The main part of the paper consists in the investigation of the above equation in Riesz spaces, i.e., an ordered vector space which is also a lattice. After a section devoted to the properties of order-preserving self-maps, in Section 3 results are given on the existence and uniqueness of order-preserving solutions on nonempty convex complete sublattices of a Riesz space \(X\). In Section 4 semi-continuous solutions on \(\mathbb{R}\) are investigated. The last section of the paper contains various examples and remarks.