Applications of optimal error bounds for some generalized two-step iterative processes in Banach spaces

Abstract

In a recent paper~\cite{paper2}, we proposed the concept of optimal error bounds for an iterative process, which allows us to obtain the convergence result of the iterative sequence to the common fixed point of the nonexpansive mappings in Banach spaces. Moreover, we also achieve the comparison results between different iterative processes via optimal error bounds. In this paper, we continue to determine optimal error bounds for more general iterative processes which were studied by many authors, such as in~\cite{DungHieu} and references therein. From there, the convergence results are obtained and the convergence rates of these iterative processes are determined under some sufficient conditions on sequences of parameters.

Theorems & Definitions (16)

• Definition 2.1: $(\kappa,\alpha)$-nonexpansive mapping
• Definition 2.2: Abstract iterative process
• Definition 2.3: Optimal error bounds
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Corollary 3.5
• Theorem 3.6
• Theorem 3.7