Strong Convergence Theorem for Two Commutative Asymptotically Nonexpansive Mappings in Hilbert Space

Abstract

𝐶 is a bounded closed convex subset of a Hilbert space 𝐻, 𝑇 and 𝑆∶𝐶→𝐶 are two asymptotically nonexpansive mappings such that 𝑆𝑇=𝑇𝑆. We establish a strong convergence theorem for 𝑆 and 𝑇 in Hilbert space by hybrid method. The results generalize and unify many corresponding results.

1. Introduction

Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻. Recall that a mapping 𝑇∶𝐶→𝐶 is said to be asymptotically nonexpansive mapping if‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑡_𝑛‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶, (1.1)where 𝑡_𝑛→1(𝑛→∞). We may assume that 𝑡_𝑛≥1 for all 𝑛=1,2,3,…. Denote by 𝐹(𝑇) the set of fixed points of 𝑇. Throughout this paper 𝑇 and 𝑆∶𝐶→𝐶 are two commutative asymptotically nonexpansive mappings with asymptotical coefficients {𝑡_𝑛} and {𝑠_𝑛}, respectively. Suppose that 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅ ([1,  Goebel and Kirk's theorem] makes it possible). It is well known that 𝐹(𝑇) and 𝐹(𝑆) are convex and closed [1, 2], so is 𝐹. 𝑃_𝐾 denotes the metric projection from 𝐻 onto a closed convex subset 𝐾 of 𝐻 and 𝜔_𝑤(𝑥_𝑛) denotes the weak 𝑤-limit set of {𝑥_𝑛}. It is well known that a Hilbert space 𝐻 satisfies Opial's condition [3], that is, if a sequence {𝑥_𝑛} converges weakly to an element 𝑦∈𝐻 and 𝑦≠𝑧, thenliminf_𝑛→∞‖𝑥_𝑛−𝑦‖<liminf_𝑛→∞‖𝑥_𝑛−𝑧‖. (1.2)

Up to now, fixed points iteration processes for nonexpansive and asymptotically nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities [4–6]. There are many strong convergence theorems for nonexpansive and asymptotically nonexpansive mappings in Hilbert space [7, 8].

Especially, Shimizu and Takahashi [7] studied the following iteration process of nonexpansive mappings for arbitrary 𝑥₀∈𝐶:𝑥_𝑛+1=𝛼_𝑛𝑥₀+(1−𝛼_𝑛)2(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛, (1.3)where {𝛼_𝑛}⊆[0,1],lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=0𝛼_𝑛=∞. And then they proved that {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀). This result was extended to two commutative asymptotically nonexpansive mappings by Shioji and Takahashi [9].

Recently, some attempts to the modified Mann iteration method are made so that strong convergence is guaranteed. And for hybrid method proposed by Haugazeau [10], Kim and Xu [8] introduced the following iteration processes for asymptotically nonexpansive mapping 𝑇:𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛,𝐶_𝑛={𝑣∈𝐶∶‖𝑦_𝑛−𝑣‖²≤‖𝑥_𝑛−𝑣‖²+𝜃_𝑛𝑄},_𝑛={𝑣∈𝐶∶⟨𝑥_𝑛−𝑣,𝑥₀−𝑥_𝑛𝑥⟩≥0},_𝑛+1=𝑃_{𝐶𝑛⋂𝑄𝑛}(𝑥₀), (1.4)where 𝜃_𝑛=(1−𝛼_𝑛)(𝑡²_𝑛−1)(diam𝐶)²→0 as 𝑛→∞. Then proved that {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀). This result was generalized to two asymptotically nonexpansive mappings by Plubtieng and Ungchittrakool [11].

On the basis of (1.3) and (1.4), we propose a new iteration processes for two commutative asymptotically nonexpansive mappings 𝑆 and 𝑇:𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)2(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛,𝐶_𝑛={𝑣∈𝐶∶‖𝑦_𝑛−𝑣‖²≤‖𝑥_𝑛−𝑣‖²+𝜃_𝑛𝑄},_𝑛={𝑣∈𝐶∶⟨𝑥_𝑛−𝑣,𝑥₀−𝑥_𝑛𝑥⟩≥0},_𝑛+1=𝑃_{𝐶𝑛⋂𝑄𝑛}(𝑥₀), (1.5)where 𝜃_𝑛=(1−𝛼_𝑛)(𝑔²_𝑛−1)(diam𝐶)²,𝑔_𝑛=(2/(𝑛+1)(𝑛+2))∑^𝑛_𝑘=0∑_{𝑖+𝑗=𝑘}𝑠_𝑖𝑡_𝑗, for every 𝑛=1,2,…. The purpose of this paper is to prove {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀).

2. Auxiliary Lemmas

This section collects some lemmas which will be used to prove the main results in the next section.

Lemma 2.1 (see [7]). Letting 𝐿_𝑛=(𝑛+1)(𝑛+2)/2, there holds the identity in a Hilbert space 𝐻:‖𝑦_𝑛−𝑣‖²=1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑥_𝑖,𝑗−𝑣‖²−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑥_𝑖,𝑗−𝑦_𝑛‖² (2.1)for {𝑥_𝑖,𝑗}^∞_𝑖,𝑗=0⊆𝐻, 𝑦_𝑛=(1/𝐿_𝑛)∑^𝑛_𝑘=0∑_{𝑖+𝑗=𝑘}𝑥_𝑖,𝑗∈𝐻 and 𝑣∈𝐻.

Lemma 2.2. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻, 𝑆 and 𝑇 two commutative asymptotically nonexpansive mappings of 𝐶 into itself with asymptotical coefficients {𝑠_𝑛} and {𝑡_𝑛}, respectively. For any 𝑥∈𝐶, put 𝐹_𝑛(𝑥)=(2/(𝑛+1)(𝑛+2))∑^𝑛_𝑘=0∑_{𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥. Thenlim_𝑙→∞limsup_𝑛→∞sup_𝑥∈𝐶‖𝐹_𝑛(𝑥)−𝑆^𝑙𝐹_𝑛(𝑥)‖=0,lim_𝑙→∞limsup_𝑛→∞sup_𝑥∈𝐶‖𝐹_𝑛(𝑥)−𝑇^𝑙𝐹_𝑛(𝑥)‖=0. (2.2)

Proof. Put 𝑥_𝑖,𝑗=𝑆^𝑖𝑇^𝑗𝑥, 𝑣=𝑆^𝑙𝐹_𝑛(𝑥) and 𝐿_𝑛=(𝑛+1)(𝑛+2)/2. It follows from Lemma 2.1 that‖𝐹_𝑛(𝑥)−𝑆^𝑙𝐹_𝑛(𝑥)‖²=1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²=1𝐿_𝑛^𝑙−1_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≤𝑙−1}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≥𝑙}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²≤1𝐿_𝑛^𝑙−1_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≤𝑙−1}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≥𝑙}𝑠²_𝑙‖𝑆^𝑖−𝑙𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²=1𝐿_𝑛^𝑙−1_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≤𝑙−1}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛−𝑙_{𝑘=0𝑖+𝑗=𝑘}𝑠²_𝑙‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖²≤1𝐿_𝑛^𝑙−1_{𝑘=0𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛_{𝑘=𝑙𝑖+𝑗=𝑘,𝑖≤𝑙−1}‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖²+1𝐿_𝑛^𝑛−𝑙_{𝑘=0𝑖+𝑗=𝑘}(𝑠²_𝑙−1)‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖². (2.3)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)Choose 𝑝∈𝐹, then there exists a constant 𝑀>0 such that‖𝑆^𝑖𝑇^𝑗𝑥−𝑝‖≤𝑠_𝑖𝑡_𝑗𝑀‖𝑥−𝑝‖≤2,‖𝐹_𝑛1(𝑥)−𝑝‖≤𝐿_𝑛^𝑛_𝑘∑₌₀∑_{𝑖+𝑗=𝑘}‖𝑆^𝑖𝑇^𝑗𝑀𝑥−𝑝‖≤2,‖𝑆^𝑙𝐹_𝑛(𝑥)−𝑝‖≤𝑠_𝑙‖𝐹_𝑛𝑀(𝑥)−𝑝‖≤2, (2.4) for all nonnegative integer 𝑖,𝑗,𝑙, and 𝑛. Hence, ‖𝑆^𝑖𝑇^𝑗𝑥−𝑆^𝑙𝐹_𝑛(𝑥)‖≤𝑀,‖𝑆^𝑖𝑇^𝑗𝑥−𝐹_𝑛(𝑥)‖≤𝑀 for all nonnegative integer 𝑖,𝑗,𝑙, and 𝑛. Sosup_𝑥∈𝐶‖𝐹_𝑛(𝑥)−𝑆^𝑙𝐹_𝑛(𝑥)‖²≤(𝑙+1)𝑙𝑀(𝑛+2)(𝑛+1)²+2(𝑛+1−𝑙)𝑙𝑀(𝑛+2)(𝑛+1)²+(𝑠²_𝑙−1)(𝑛+2−𝑙)(𝑛+1−𝑙)𝑀(𝑛+2)(𝑛+1)²→0(𝑛→∞,𝑙→∞). (2.5)Similarly, we can provelim_𝑙→∞limsup_𝑛→∞sup_𝑥∈𝐶‖𝐹_𝑛(𝑥)−𝑇^𝑙𝐹_𝑛(𝑥)‖=0. (2.6)

Remark 2.3. Lemma 2.2 extends [7, Lemma 1].

Lemma 2.4. Let 𝑆 and 𝑇 be two commutative asymptotically nonexpansive mappings defined on a bounded closed convex subset 𝐶 of a Hilbert space 𝐻 with asymptotical coefficients {𝑠_𝑛} and {𝑡_𝑛}, respectively. Let 𝐿_𝑛=((𝑛+1)(𝑛+2)/2). If {𝑥_𝑛} is a sequence in 𝐶 such that {𝑥_𝑛} converges weakly to some 𝑥∈𝐶 and {𝑥_𝑛−(1/𝐿_𝑛)∑^𝑛_𝑘=0∑_{𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛} converges strongly to 0, then 𝑥∈𝐹(𝑆)∩𝐹(𝑇).

Proof. We claim that {𝑆^𝑙𝑥} converges strongly to 𝑥 as 𝑙→∞. If not, there exist a positive number 𝜀₀ and a subsequence {𝑙_𝑚} of {𝑙} such that ‖𝑆^𝑙𝑚𝑥−𝑥‖≥𝜀₀ for all 𝑚. However, we have‖𝑥_𝑛−𝑆^𝑙𝑚𝑥‖≤‖𝑥_𝑛−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛1‖+‖𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑆^𝑙𝑚(1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛)‖+‖𝑆^𝑙𝑚(1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛)−𝑆^𝑙𝑚𝑥‖≤‖𝑥_𝑛−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛1‖+‖𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑆^𝑙𝑚(1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛)‖+𝑠_𝑙𝑚‖1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑥‖≤‖𝑥_𝑛−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛1‖+‖𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑆^𝑙𝑚(1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛)‖+𝑠_𝑙𝑚‖1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑥_𝑛‖+𝑠_𝑙𝑚‖𝑥_𝑛−𝑥‖. (2.7)By Opial's condition, for any 𝑦∈𝐶 with 𝑦≠𝑥, we haveliminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖. (2.8)Let 𝑟=liminf_𝑛→∞‖𝑥_𝑛−𝑥‖ and choose a positive number 𝜌 such that𝜌<𝑟²+𝜀²₀4−𝑟. (2.9)Then, there exists a subsequence {𝑥_𝑛𝑝} of {𝑥_𝑛} such that lim_𝑝→∞‖𝑥_𝑛𝑝−𝑥‖=𝑟 and ‖𝑥_𝑛𝑝−𝑥‖<𝑟+(𝜌/4) for all 𝑝. By definition of {𝑠_𝑙𝑚}, there exists a positive integer 𝑚₀ such that𝑠_𝑙𝑚‖𝑥_𝑛𝑝𝜌−𝑥‖<𝑟+4, (2.10)for all 𝑚>𝑚₀. Sincelim_𝑛→∞‖𝑥_𝑛−1𝐿_𝑛^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛‖=0 (2.11)and {𝑠_𝑙𝑚} is bounded, there exists a positive integer 𝑝₀ such that‖𝑥_𝑛𝑝−1𝐿_𝑛𝑝^𝑛𝑝_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛𝑝𝜌‖<4,𝑠_𝑙𝑚‖1𝐿_𝑛𝑝^𝑛𝑝_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛𝑝−𝑥_𝑛𝑝𝜌‖<4 (2.12)for all 𝑚 and 𝑝>𝑝₀. By {𝑥_𝑛𝑝}⊂𝐶 is bounded and Lemma 2.2, there exist 𝑚₁>𝑚₀ and 𝑝₁>0 such that‖1𝐿_𝑛𝑝^𝑛𝑝_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛𝑝−𝑆^𝑙𝑚1(1𝐿_𝑛𝑝^𝑛𝑝_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛𝑝𝜌)‖<4 (2.13)for all 𝑝>𝑝₁. By (2.7), (2.10), (2.12), and (2.13), we have‖𝑥_𝑛𝑝−𝑆^𝑙𝑚1𝜌𝑥‖<4+𝜌4+𝜌4𝜌+𝑟+4=𝑟+𝜌 (2.14)for all 𝑝>max{𝑝₀,𝑝₁}. However,‖𝑥_𝑛𝑝−𝑆^𝑙𝑚1𝑥+𝑥2‖²=12‖𝑥_𝑛𝑝−𝑆^𝑙𝑚1𝑥‖²+12‖𝑥_𝑛𝑝−𝑥‖²−14‖𝑆^𝑙𝑚1𝑥−𝑥‖²<(𝑟+𝜌)²2+(𝑟+𝜌/4)²2−𝜀²₀4<(𝑟+𝜌)²−𝜀²₀4<𝑟² (2.15)for all 𝑝>max{𝑝₀,𝑝₁}. This contradicts (2.8). So {𝑆^𝑙𝑥} converges strongly to 𝑥 and then 𝑥∈𝐹(𝑆). Similarly, we can get 𝑥∈𝐹(𝑇). Hence, 𝑥 is a common fixed point of 𝑆 and 𝑇.

Lemma 2.5 (see [12]). Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻. The set 𝐷∶={𝑣∈𝐶∶‖𝑦−𝑣‖²≤‖𝑥−𝑣‖²+⟨𝑧,𝑣⟩+𝑏} is convex and closed for given 𝑥,𝑦,𝑧∈𝐶 and 𝑏∈ℝ.

3. Main Results

In this section, we prove our main theorem.

Theorem 3.1. Let 𝐶 be a bounded closed convex subset of a Hilbert 𝐻, 𝑇 and 𝑆∶𝐶→𝐶 be two commutative asymptotically nonexpansive mappings with asymptotical coefficients {𝑡_𝑛} and {𝑠_𝑛}, respectively. Suppose that 0≤𝛼_𝑛≤𝑎 for all 𝑛, where 0<𝑎<1. If 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅, then the sequence generated by (1.5) converges strongly to 𝑃_𝐹(𝑥₀).

Proof. Note that 𝐶_𝑛 is convex and closed for all 𝑛≥0 by Lemma 2.5. On the other hand, 𝑄_𝑛 is convex and closed. So is 𝐶_𝑛∩𝑄_𝑛.
By definition of {𝑡_𝑛} and {𝑠_𝑛}, there exists 𝑀>0 such that ‖𝑠_𝑖𝑡_𝑗−1‖≤𝑀 for all 𝑖,𝑗≥0. On the other hand, for arbitrary 𝜀>0, there exists 𝑁>0 such that ‖𝑠_𝑖𝑡_𝑗−1‖<𝜀 for all 𝑖,𝑗>𝑁. Hence‖𝑔_𝑛2−1‖=‖(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}(𝑠_𝑖𝑡_𝑗≤2−1)‖(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}‖𝑠_𝑖𝑡_𝑗≤2−1‖(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘,𝑖≤𝑁}‖𝑠_𝑖𝑡_𝑗2−1‖+(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘,𝑗≤𝑁}‖𝑠_𝑖𝑡_𝑗+2−1‖(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘,𝑖≥𝑁+1,𝑗≥𝑁+1}‖𝑠_𝑖𝑡_𝑗<−1‖2(𝑁+1)𝑀+(𝑛+2)2(𝑁+1)𝑀(𝑛+2)+𝜀. (3.1)Thus lim_𝑛→∞𝑔_𝑛=1. Obviously, lim_𝑛→∞𝜃_𝑛=0.
Next, we prove that 𝐹⊂𝐶_𝑛∩𝑄_𝑛. Indeed, first of all‖𝑦_𝑛−𝑝‖²≤𝛼_𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑛2)‖(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑝‖²≤𝛼_𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑛)𝑔²_𝑛‖𝑥_𝑛−𝑝‖²=‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑛)(𝑔²_𝑛‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛−𝑝‖²)≤‖𝑥_𝑛−𝑝‖²+𝜃_𝑛 (3.2)for all 𝑝∈𝐹. So 𝐹⊂𝐶_𝑛. It suffices to show that 𝐹⊂𝑄_𝑛 for all 𝑛≥0. We prove this by induction. For 𝑛=0, we have 𝐹⊂𝐶=𝑄₀. Assume that 𝐹⊂𝑄_𝑛. Since 𝑥_𝑛+1 is the projection of 𝑥₀ onto 𝐶_𝑛∩𝑄_𝑛, we have⟨𝑥_𝑛+1−𝑧,𝑥₀−𝑥_𝑛+1⟩≥0∀𝑧∈𝐶_𝑛𝑄_𝑛 (3.3)As 𝐹⊂𝐶_𝑛∩𝑄_𝑛, (3.3) holds for all 𝑧∈𝐹, in particular. This together with the definition of 𝑄_𝑛+1 implies that 𝐹⊂𝑄_𝑛+1. Hence, 𝐹⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛≥0.
We will show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. By the definition of 𝑄_𝑛, we have that 𝑥_𝑛=𝑃_𝑄𝑛(𝑥₀). It follows from 𝑥_𝑛+1∈𝐶_𝑛∩𝑄_𝑛⊂𝑄_𝑛 that ‖𝑥_𝑛−𝑥₀‖≤‖𝑥_𝑛+1−𝑥₀‖. This shows that the sequence {‖𝑥_𝑛−𝑥₀‖} is increasing. Since 𝐶 is bounded, we obtain that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. Notice again that from 𝑥_𝑛=𝑃_𝑄𝑛(𝑥₀) and 𝑥_𝑛+1∈𝑄_𝑛, we have ⟨𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀⟩≥0. Hence‖𝑥_𝑛+1−𝑥_𝑛‖²=‖(𝑥_𝑛+1−𝑥₀)−(𝑥_𝑛−𝑥₀)‖²=‖𝑥_𝑛+1−𝑥₀‖²+‖𝑥_𝑛−𝑥₀‖²−2⟨𝑥_𝑛+1−𝑥₀,𝑥_𝑛−𝑥₀⟩=‖𝑥_𝑛+1−𝑥₀‖²−‖𝑥_𝑛−𝑥₀‖²−2⟨𝑥_𝑛+1−𝑥₀−(𝑥_𝑛−𝑥₀),𝑥_𝑛−𝑥₀⟩=‖𝑥_𝑛+1−𝑥₀‖²−‖𝑥_𝑛−𝑥₀‖²−2⟨𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀⟩≤‖𝑥_𝑛+1−𝑥₀‖²−‖𝑥_𝑛−𝑥₀‖²→0(𝑛→∞). (3.4)
Now we claim that ‖(2/(𝑛+1)(𝑛+2))∑^𝑛_𝑘=0∑_{𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. By the definition of 𝑦_𝑛, we have‖2(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑥_𝑛‖=11−𝛼_𝑛‖𝑦_𝑛−𝑥_𝑛‖≤11−𝛼_𝑛(‖𝑦_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛≤1‖)1−𝑎(‖𝑦_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖). (3.5)Since 𝑥_𝑛+1∈𝐶_𝑛, ‖𝑦_𝑛−𝑥_𝑛+1‖²≤‖𝑥_𝑛−𝑥_𝑛+1‖²+𝜃_𝑛→0 as 𝑛→∞. So ‖𝑦_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞. This implies that‖2(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛−𝑥_𝑛‖→0(𝑛→∞). (3.6)
Since 𝐶 is bounded closed convex, 𝜔_𝑤(𝑥_𝑛)≠∅. It follows from (3.6) and Lemma 2.4 that 𝜔_𝑤(𝑥_𝑛)⊂𝐹. By the definition of 𝑄_𝑛, we have that ‖𝑥_𝑛−𝑥₀‖≤‖𝑃_𝐹(𝑥₀)−𝑥₀‖ for all 𝑛≥0. It follows from the weak lower semi-continuity of the norm that ‖𝑤−𝑥₀‖≤‖𝑃_𝐹(𝑥₀)−𝑥₀‖ for all 𝑤∈𝜔_𝑤(𝑥_𝑛). Since 𝜔_𝑤(𝑥_𝑛)⊂𝐹, we have 𝑤=𝑃_𝐹(𝑥₀) for all 𝑤∈𝜔_𝑤(𝑥_𝑛). Thus 𝜔_𝑤(𝑥_𝑛)={𝑃_𝐹(𝑥₀)}. Then, {𝑥_𝑛} converges to 𝑃_𝐹(𝑥₀) weakly. By the fact‖𝑥_𝑛−𝑃_𝐹(𝑥₀)‖²=‖𝑥_𝑛−𝑥₀+𝑥₀−𝑃_𝐹(𝑥₀)‖²=‖𝑥_𝑛−𝑥₀‖²+‖𝑥₀−𝑃_𝐹(𝑥₀)‖²+2⟨𝑥_𝑛−𝑥₀,𝑥₀−𝑃_𝐹(𝑥₀)⟩≤2(‖𝑃_𝐹(𝑥₀)−𝑥₀‖²+⟨𝑥_𝑛−𝑥₀,𝑥₀−𝑃_𝐹(𝑥₀)⟩)→0(𝑛→∞), (3.7)we have {𝑥_𝑛} converges to 𝑃_𝐹(𝑥₀) strongly. This completes the proof.

The following corollary follows from Theorem 3.1.

Corollary 3.2. Let 𝐶 be a bounded closed convex subset of a Hilbert 𝐻, 𝑇 and 𝑆∶𝐶→𝐶 be two commutative nonexpansive mappings. Suppose that 0≤𝛼_𝑛≤𝑎 for all 𝑛, where 0<𝑎<1. If 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅, then the sequence {𝑥_𝑛} generated by𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)2(𝑛+1)(𝑛+2)^𝑛_{𝑘=0𝑖+𝑗=𝑘}𝑆^𝑖𝑇^𝑗𝑥_𝑛,𝐶_𝑛={𝑣∈𝐶∶‖𝑦_𝑛−𝑣‖²≤‖𝑥_𝑛−𝑣‖²𝑄},_𝑛={𝑣∈𝐶∶⟨𝑥_𝑛−𝑣,𝑥₀−𝑥_𝑛𝑥⟩≥0},_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}(𝑥₀), (3.8)converges strongly to 𝑃_𝐹(𝑥₀).

Acknowledgments

This work is supported by National Natural Science Foundation of China (10771173).