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On difference Riccati equation and continued fractions

Alexey V. Ivanov

Abstract

We study a difference Riccati equation $Φ(x) + ρ(x)/Φ(x-ω) = v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.

On difference Riccati equation and continued fractions

Abstract

We study a difference Riccati equation with periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.

Paper Structure

This paper contains 5 sections, 13 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Continued fractions
  3. Decomposition of cocycles
  4. Hyperbolicity and convergence
  5. Critical set method

Key Result

Theorem 1

Assume that a given functional continued fraction $\{(a_{j}, b_{j}), f_{j}\}$ is generated by a pair $(b, \omega)$ with function $b\in C(\mathbb{T}^{1},\mathbb{R})$ and $\omega\in (0,1)\cap \mathbb{R}\setminus\mathbb{Q}$. If there exist positive constants $C_{0}$ and $\lambda_{0}>1$ such that the co then the continued fraction converges and $f_{*} = \lim\limits_{n\to \infty}f_{n}\in C(\mathbb{T}^{

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Corollary 1
  • Lemma 4
  • Lemma 5
  • ...and 7 more