Explicit iterations for Lipschitzian semigroups with the Meir-Keeler type contraction in Banach spaces

Siwaporn Saewan,Poom Kumam

doi:10.1186/1029-242x-2012-279

Abstract

In this paper, we introduce and prove strong convergence theorems for a new viscosity iteration scheme for approximating common fixed points of a Lipschitzian semigroup on a compact and convex subset of a smooth Banach space. Our results extend and improve recent results. MSC:47H09, 47H10.

Highlights

Let C be a nonempty, closed and convex subset of a Banach space E
Recall that a self mapping f : E → E is an α-contraction on C if there exists a constant α ∈ (, ) such that for any x, y ∈ E, we have f (x) – f (y) ≤ α x – y
A function ψ : R+ → R+ is said to be an L-function if ψ( ) =, ψ(t) > for any t >, and for every t > and s >, there exists u > s such that ψ(t) ≤ s for all t ∈ [s, u]

Summary

Introduction

Let C be a nonempty, closed and convex subset of a Banach space E. For a representation of S as Lipschitzian mappings on a compact and convex subset C of a smooth Banach space E with respect to a left regular sequence {μn} of means defined on an appropriate invariant subspace of l∞(S); for some related results, we refer the readers to [ , ].

Results

Conclusion