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An equation in nonlinear combination of iterates

Chaitanya Gopalakrishna, Weinian Zhang

Abstract

In this paper we deal with an equation in nonlinear combination of iterates. Although it can be reduced by the logarithm conjugacy to a form for application of Schauder's or Banach's fixed point theorems, a difficulty called Zero Problem is encountered for continuous solutions because the domain does not contain $0$. So we consider solutions with weaker regularity, using the Knaster-Tarski fixed point theorem for complete lattices to give order-preserving solutions. Then we give semi-continuous solutions and integrable solutions.

An equation in nonlinear combination of iterates

Abstract

In this paper we deal with an equation in nonlinear combination of iterates. Although it can be reduced by the logarithm conjugacy to a form for application of Schauder's or Banach's fixed point theorems, a difficulty called Zero Problem is encountered for continuous solutions because the domain does not contain . So we consider solutions with weaker regularity, using the Knaster-Tarski fixed point theorem for complete lattices to give order-preserving solutions. Then we give semi-continuous solutions and integrable solutions.
Paper Structure (5 sections, 11 theorems, 42 equations)

This paper contains 5 sections, 11 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Continuous solutions
  3. Order-preserving solutions
  4. Semi-continuous solutions
  5. Examples and remarks

Key Result

Proposition 1

Let $G \in \mathcal{X}\subseteq \mathcal{C}(J,J)$. Then a map $g$ is a solution (resp. unique solution) of prod in $\mathcal{X}'\subseteq \mathcal{C}(J,J)$, where $\Psi_k \in \mathcal{Y}_k\subseteq \mathcal{C}(J,\mathbb{R}_+)$ and $\psi_k \in \mathcal{Z}_k\subseteq \mathcal{C}(J,J)$ for all $1\le

Theorems & Definitions (14)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Proposition 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 4 more