Common fixed points and iteration of commuting nonexpansive mappings
Shiro Ishikawa · Pacific Journal of Mathematics · 1979
The following result is shown. Let Tt(i = 1, 2, , v) be commuting nonexpansive self-mappings on a compact convex subset D of a Banach space and let x be any point in Zλ Then the sequence IΓ π Γ&, l ί T Γs, π \\s2 it IL v-i-iL ^- 2 = i L L w i = 1 L % o = 1 converges to a common fixed point of {Γ}ϊ=i, where S< = (1 — αjJ + (XiTi, 0 < at < 1, I is the identity mapping. In [2], DeMarr proved that if Tt(i eJ,J is an index set) are commuting nonexpansive self-mappings on a compact convex subset D of a Banach space (i.e., | | Tx — Ty\\ \\ ̂ ||a? — y\\ \\ for all α?,? / in D, and T ^ = TjT, for all i, i e J) , then TlίeJ) have a common fixed point in Ώ. The problem we shall consider in this paper is that of constructing a sequence of points {x n}n=ι in D that converges to the common fixed point of Ti (i e J, J is a finite index set). If a Banach space is strictly convex (i.e., \\\\ax + (1 — a)y\\ \\ < max {| | a? 11, \\\\y\\\\) for a? Φ y, 0 < α: < 1), the problem was solved in [5]. Throughout this paper, we denote an identity mapping by I and the set of fixed points of T by F[T]. And we define UtS Tt = T n+1(Ut=i Ti) for any positive integer n and Π U Tt = 2V We have the following main theorem. THEOREM. Let Tt{i = 1, 2, v) be commuting nonexpansive mappings from a compact convex subset D of a Banach space into itself, and let x be any point in D. Then Cΐί=ιF[Ti] is nonempty and the sequence {x^J converges to a point in Πϊ^-PΊTiL where x n ^ is defined for each positive integer nt by Γ π Γ-s, ϊ ϊ 1 Γsu •. Γs3 π \\s2 π sj\\... J\\]x where S< = (1- α4) / + α^T*, 0 < α * < l(i = 1, 2, , v). Before proving the theorem, we first prove the following lemmas on which the proof of theorem is based. LEMMA 1. Let T and P be nonexpansive mappings from a 493