Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions.pdf

Nonlinear Analysis 68 (2008) 412–419 /locate/na Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions Shin-ya Matsushitaa,?, Wataru Takahashib aDepartment of Information Engineering, Matsue National College of Technology, Nishi-ikuma, Matsue, Shimane, 690-8518, Japan bDepartment of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo, 152-8552, Japan Received 15 November 2005; accepted 1 November 2006 Abstract In this paper, we study boundary conditions for nonexpansive nonself-mappings in a Banach space. Using this, we prove two strong convergence theorems for nonexpansive nonself-mappings in a Banach space without boundary conditions. c? 2006 Elsevier Ltd. All rights reserved. MSC: primary 47H10; secondary 47H09 Keywords: Nonexpansive nonself-mapping; Fixed point; Boundary condition; Sunny nonexpansive retraction; Banach limit; Approximation of fixed points 1. Introduction LetC be a closed convex subset of a Banach space E and let T be a nonexpansive mapping fromC into E . Recently, the problem of approximating fixed points of nonexpansive nonself-mappings has been studied by many researchers. In [9], Marino and Trombetta introduced two contraction mappings in a Hilbert space and studied the behaviour of the fixed points of such two mappings. Further, for if T satisfies the weakly inward condition and F(T ) 6= ?, Xu and Yin [24] proved that two sequences converge strongly to fixed points of T . Using Banach limits and sunny nonexpansive retractions, Takahashi and Kim [22] extended Xu and Yin’s results to a reflexive Banach space E with a uniformly Ga?teaux differentiable norm. For relevant papers on these results, see Singh and Watson [17], Matsushita and Kuroiwa [10–12], Alber, Reich and Yao [1], Shahzad [15], Chidume, Zegeye and Shahzad [3] and Song and Chen [18] and so on. The purpose of this paper is to study boundary conditions for nonexpansive nonself-mappings in a Banach space.