Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Yonghong Yao,Jung Im Kang,Yeong-Cheng Liou,Yeol Je Cho

doi:10.1186/1687-1812-2013-31

Abstract

Abstract In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others. MSC:47H05, 47H10, 47H17.

Highlights

Let H be a real Hilbert space with the inner product ·, · and the norm ·
We denote by Fix(T(s)) the set of fixed points of T(s) and by Fix(S) the set of all common fixed points of S, i.e., Fix(S) = s≥ Fix(T(s))
Approximation of fixed points of nonexpansive mappings by a sequence of finite means has been considered by many authors; see, for instance, [ – ]

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the norm ·. Recall that a family S := {T(s)}s≥ of mappings of C into itself is called a nonexpansive semigroup if it satisfies the following conditions: (S ) T( )x = x for all x ∈ C; (S ) T(s + t) = T(s)T(t) for all s, t ≥ ; (S ) T(s)x – T(s)y ≤ x – y for all x, y ∈ C and s ≥ ; (S ) for each x ∈ H, s → T(s)x is continuous. It is known that Fix(S) is closed and convex [ , Lemma ].

Results

Conclusion