Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONVEX SOLUTIONS OF THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON OPEN SET

  • Gong, Xiaobing (Key Laboratory of Numerical Simulation of Sichuan Province, College of Mathematics and Information Science Neijiang Normal University)
  • Received : 2012.05.25
  • Published : 2014.05.31
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Abstract

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.

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