Rates of convergence for asymptotically weakly contractive mappings in normed spaces

Thomas Powell, Franziskus Wiesnet

Research output: Contribution to journalArticlepeer-review

Abstract

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalized convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically ψ-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalize and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the proof mining program.

Original languageEnglish
JournalNumerical Functional Analysis and Optimization
Volume42
Issue number15
Early online date25 Nov 2021
DOIs
Publication statusPublished - 31 Dec 2021

Keywords

  • Krasnoselskii-Mann iteration
  • Weakly contractive mappings
  • proof mining
  • rates of convergence
  • uniformly smooth Banach spaces

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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