Rates of Convergence for Asymptotically Weakly Contractive Mappings in Normed Spaces. (18th November 2021)

• Record Type: Journal Article
• Authors: Powell, Thomas; Wiesnet, Franziskus
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 15(2021)
• Page Start: 1802
• Page End: 1838
• Publication Date: 2021-11-18
• Subjects: Weakly contractive mappings -- Krasnoselskii-Mann iteration -- uniformly smooth Banach spaces -- rates of convergence -- proof mining; 47H10, 47J25, 03F10; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2006696
• Languages: English
• ISSNs: 0163-0563

Abstract

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.