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Quantitative asymptotic regularity and $T$-asymptotic regularity for the inexact generalized Halpern iteration

Nicoleta Dumitru, Laurentiu Leustean

Abstract

We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.

Quantitative asymptotic regularity and $T$-asymptotic regularity for the inexact generalized Halpern iteration

Abstract

We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.
Paper Structure (13 sections, 25 theorems, 68 equations)

This paper contains 13 sections, 25 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Quantitative notions and lemmas
  3. A generalized Halpern iteration
  4. Main technical lemma
  5. Quantitative hypotheses
  6. Main theorems
  7. Kanzow--Shehu iteration given by \ref{['KanShe-iteration']}
  8. The generalized Halpern iteration
  9. Sequential averaging method (SAM)
  10. Examples
  11. Example 1
  12. Example 2
  13. Example 3

Key Result

Lemma 2.1

FirLeu25a Let $(a_n)\subseteq [0,\infty)$ be such that $\sum\limits_{n=0}^\infty a_n$ converges with Cauchy modulus $\varphi$. Then

Theorems & Definitions (40)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 30 more