Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space

Jing Quan,Shih-Sen Chang,Min Liu

doi:10.1186/1687-1812-2012-16

Jing Quan, Shih-Sen Chang + Show 1 more

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https://doi.org/10.1186/1687-1812-2012-16

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Abstract

The purpose of this article is to study the strong and weak convergence of implicit iterative sequence to a common fixed point for pseudocontractive semigroups in Banach spaces. The results presented in this article extend and improve the corresponding results of many authors.

Highlights

Introduction and preliminariesThroughout this article we assume that E is a real Banach space with norm ||·||, E* is the dual space of E; 〈·, ·〉 is the duality pairing between E and E*; C is a nonempty closed convex subset of E; N denotes the natural number set; R+ is the set of nonnegative real numbers; The mapping J : E → 2E∗ defined byJ(x) = f ∗ ∈ E∗ : x, f ∗ = x 2; f ∗ = x, x ∈ E (1)is called the normalized duality mapping
In 1974, Deimling [1] proved the following existence theorem of fixed point for a continuous and strong pseudocontraction in a nonempty closed convex subset of Banach spaces
The problems of convergence of an implicit iterative algorithm to a common fixed point for a family of nonexpansive mappings or pseudocontractive mappings have been considered by several authors, see [2,3,4,5]

Summary

Introduction

In 1974, Deimling [1] proved the following existence theorem of fixed point for a continuous and strong pseudocontraction in a nonempty closed convex subset of Banach spaces. Let E be a Banach space, C be a nonempty closed convex subset of E and T: C ® C be a continuous and strong pseudocontraction. The problems of convergence of an implicit iterative algorithm to a common fixed point for a family of nonexpansive mappings or pseudocontractive mappings have been considered by several authors, see [2,3,4,5].

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