Bulletin of the
Korean Mathematical Society BKMS

Because of difficulty of using Schauder's fixed point theorem to the polynomial-like iterative equation, a lots of work are contributed to the existence of solutions for the polynomial-like iterative equation on compact set. In this paper, by applying the Schauder-Tychonoff fixed point theorem we discuss monotone solutions and convex solutions of the polynomial-like iterative equation on an open set (possibly unbounded) in $\mathbb{R}^{N}$. More concretely, by considering a partial order in $\mathbb{R}^{N}$ defined by an order cone, we prove the existence of increasing and decreasing solutions of the polynomial-like iterative equation on an open set and further obtain the conditions under which the solutions are convex in the order.

Keywords: iterative equation, open set, order, increasing operator and decreasing operator

MSC numbers: 39B12, 58F08