On Convergence of Quasi ρ-preserving Locally Related Quasi-nonexpansive Mappings. (17th February 2019)

• Record Type: Journal Article
• Authors: Al-Rawashdeh, Ahmed; Mehmood, Nayyar
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 3(2019)
• Page Start: 326
• Page End: 340
• Publication Date: 2019-02-17
• Subjects: Picard -- Mann -- Ishikawa iterations -- quasi ρ-preserving -- locally related quasi-nonexpansive mappings -- convergence criteria; Primary 46B20 -- 45D05 -- Secondary 47E10 -- 34A12; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1554585
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The concept of quasi-nonexpansive mappings was initiated in Diaz and Metcalf (Bull. Amer. Math. Soc. 73 (1967), 516–519), further it was generalized by Petryshyn and Williamson (Journal of Mathematical Analysis and Applications 43, no. 2 (1973): 459-497). Further investigating the above work, in this article a Banach space with a relation ρ has been considered, the notion of ρ -preserving and quasi ρ -preserving locally related quasi-nonexpansive mappings has been introduced. It has been shown that every ρ -preserving mapping is a quasi ρ -preserving and every nonexpansive mapping is locally related quasi-nonexpansive but converse may not hold. The necessary and sufficient conditions for the convergence of Picard, Mann and Ishikawa iterations of ρ -preserving, quasi ρ -preserving locally related quasi-nonexpansive mappings are presented.