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Hyers-Ulam stability of the first order difference equation with average growth rate

Young Woo Nam

Abstract

The first order difference equation induced by the sequence of maps on $ \mathbb{C} $ has Hyers-Ulam stability where the limit of the geometric average of growth rate is convergent and not equal to one. %The average growth rate is a generalization of contracting or expanding constant of maps. We show no Hyers-Ulam stability where the average growth rate is (pre)periodic even though each periodic growth rate is strictly less than one. Examples of difference equation generated by time dependent maps which contains contracting maps and expanding maps are given.

Hyers-Ulam stability of the first order difference equation with average growth rate

Abstract

The first order difference equation induced by the sequence of maps on has Hyers-Ulam stability where the limit of the geometric average of growth rate is convergent and not equal to one. %The average growth rate is a generalization of contracting or expanding constant of maps. We show no Hyers-Ulam stability where the average growth rate is (pre)periodic even though each periodic growth rate is strictly less than one. Examples of difference equation generated by time dependent maps which contains contracting maps and expanding maps are given.
Paper Structure (5 sections, 9 theorems, 87 equations)

This paper contains 5 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction and preliminaries
  2. HU stability for exponential growth rate less than one
  3. HU stability for exponential growth rate greater than one
  4. No HU stability for periodic average growth rate
  5. Applications

Key Result

Lemma 1.1

Let $F \colon \mathbb N \times \mathbb C \rightarrow \mathbb C$ be a function which satisfies that for all $n \in \mathbb N$, $u, v \in \mathbb C$ and for $0<p<1$. For a given $\varepsilon>0$, suppose that the complex valued sequence $\{a_n\}_{n\in \mathbb N}$ satisfies the inequality for all $n \in \mathbb N$. Then the sequence $\{b_n\}_{n\in\mathbb N}$ defined as $b_{n+1} = F(n,b_n)$ satisfies

Theorems & Definitions (23)

  • Lemma 1.1: Corollary3 in jungnam
  • Lemma 2.1
  • proof
  • Lemma 2.2: Lemma 5.4 in nam2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 13 more