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. Because of diculty of using Schauder’s xed point theoremto the polynomial-like iterative equation, a lots of work are contributedto the existence of solutions for the polynomial-like iterative equation oncompact set. In this paper, by applying the Schauder-Tychono xedpoint theorem we discuss monotone solutions and convex solutions of thepolynomial-like iterative equation on an open set (possibly unbounded)in R N . More concretely, by considering a partial order in R N denedby an order cone, we prove the existence of increasing and decreasingsolutions of the polynomial-like iterative equation on an open set andfurther obtain the conditions under which the solutions are convex in theorder. 1. IntroductionAs indicated in the books [7, 19] and the surveys [3, 24], the polynomial-likeiterative equation(1.1) λ 1 f(x) +λ 2 f 2 (x) +···+λ n f n (x) = F(x), x ∈ S,where S is a subset of a linear space over R, F : S → S is a given function,λ i s (i = 1,...,n) are real constants, f : S → S is the unknown functionand f

CONVEX SOLUTIONS OF THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON OPEN SET