△-Convergence Problems for Asymptotically Nonexpansive Mappings in CAT(0) Spaces

Abstract

New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping 𝑇 of a closed convex subset 𝐶 of a CAT(0) space 𝑋. Consider the iteration process {𝑥_𝑛}, where 𝑥₀∈𝐶 is arbitrary and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑇^𝑛𝑦_𝑛 or 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛 for 𝑛≥1, where {𝛼_𝑛},{𝛽_𝑛}⊂(0,1). It is shown that under certain appropriate conditions on 𝛼_𝑛,𝛽_𝑛,{𝑥_𝑛}  △-converges to a fixed point of 𝑇.

1. Introduction and Preliminaries

Let 𝐶 be a nonempty subset of a metric space (𝑋,𝑑). A mapping 𝑇∶𝐶→𝐶 is a contraction if there exists 𝑘∈[0,1) such that for all 𝑥,𝑦∈𝐶, we have 𝑑(𝑇𝑥,𝑇𝑦)<𝑘𝑑(𝑥,𝑦). It is said to be nonexpansive if for all 𝑥,𝑦∈𝐶, we have 𝑑(𝑇𝑥,𝑇𝑦)≤𝑑(𝑥,𝑦). 𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}∈[1,∞) with 𝑘_𝑛→1 such that 𝑑(𝑇^𝑛𝑥,𝑇^𝑛𝑦)≤𝑘_𝑛𝑑(𝑥,𝑦) for all integers 𝑛≥1 and all 𝑥,𝑦∈𝐶. Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is asymptotically nonexpansive with sequence 𝑘_𝑛=1, for all 𝑛≥1. There are, however, asymptotically nonexpansive mappings which are not nonexpansive (see, e.g., [1]). As a generalization of the class of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972 and has been studied by several authors (see, e.g., [3–5]). Goebel and Kirk proved that if 𝐶 is a nonempty closed convex and bounded subset of a uniformly convex Banach space (more general than a Hilbert space, i.e., CAT(0) space), then every asymptotically nonexpansive self-mapping of 𝐶 has a fixed point. The weak and strong convergence problems to fixed points of nonexpansive and asymptotically nonexpansive mappings have been studied by many authors.

We will denote by 𝐹(𝑇) the set of fixed points of 𝑇. In 1967, Halpern [6] introduced an explicit iterative scheme for a nonexpansive mapping 𝑇 on a subset 𝐶 of a Hilbert space by taking any point 𝑢,𝑥₁∈𝐶 and defined the iterative sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,for𝑛≥1, (1) where 𝛼_𝑛∈[0,1]. He pointed out that under certain appropriate conditions on 𝛼_𝑛,{𝑥_𝑛} converges strongly to a fixed point of 𝑇. In 1994, Tan and Xu [7] introduced the following iterative scheme for asymptotically nonexpansive mapping on uniformly convex Banach space: 𝑥₀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑦_𝑛𝑦,𝑛≥0,_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥0, (2) where {𝛼_𝑛},{𝛾_𝑛}⊆(0,1). They proved that under certain appropriate conditions on 𝛼_𝑛,𝛾_𝑛,{𝑥_𝑛} converges weakly to a fixed point of 𝑇.

In 2012, we [8] studied the viscosity approximation methods for nonexpansive mappings on CAT(0) space. For a contraction 𝑓 on 𝐶, consider the iteration process {𝑥_𝑛}, where 𝑥₀∈𝐶 is arbitrary and 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛, (3) for 𝑛≥1, where {𝛼_𝑛}⊂(0,1). We proved that under certain appropriate conditions on 𝛼_𝑛,{𝑥_𝑛} converges strongly to a fixed point of 𝑇 which solves some variational inequality.

The purpose of this paper is to study the iterative scheme defined as follows: consider an asymptotically nonexpansive self-mapping 𝑇 of a closed convex subset 𝐶 of a CAT(0) space 𝑋 with coefficient 𝑘_𝑛. consider the iteration process {𝑥_𝑛}, where 𝑥₀∈𝐶 is arbitrary and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛, (4) or 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛, (5) for 𝑛≥1, where {𝛼_𝑛},{𝛽_𝑛}⊂(0,1). We show that {𝑥_𝑛}  △-converges to a fixed point of 𝑇 under certain appropriate conditions on 𝛼_𝑛,𝛽_𝑛, and 𝑘_𝑛.

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 1. Let 𝑋 be a CAT(0) space. Then, one has the following:(i)(see [9, Lemma 2.4])  for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has 𝑑(1−𝑡)𝑥⊕𝑡𝑦,𝑧≤(1−𝑡)𝑑(𝑥,𝑧)+𝑡𝑑𝑦,𝑧, (6)(ii) (see [10]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠∈[0,1] one has 𝑑(1−𝑡)𝑥⊕𝑡𝑦,(1−𝑠)𝑥⊕𝑠𝑦≤|𝑡−𝑠|𝑑𝑥,𝑦, (7)(iii) (see [5, Lemma 3]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has 𝑑(1−𝑡)𝑧⊕𝑡𝑥,(1−𝑡)𝑧⊕𝑡𝑦≤𝑡𝑑𝑥,𝑦, (8)(iv) (see [9]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has 𝑑²(1−𝑡)𝑥⊕𝑡𝑦,𝑧≤𝑡𝑑²(𝑥,𝑧)+(1−𝑡)𝑑²𝑦,𝑧−𝑡(1−𝑡)𝑑²𝑥,𝑦. (9)

Let 𝑋 be a complete CAT(0) space and let {𝑥_𝑛} be a bounded sequence in a complete 𝑋 and for 𝑥∈𝑋 set 𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛. (10)

The asymptotic radius 𝑟({𝑥_𝑛}) of {𝑥_𝑛} is given by 𝑟𝑥_𝑛=inf𝑟𝑥,𝑥_𝑛∶𝑥∈𝑋, (11) and the asymptotic center 𝐴({𝑥_𝑛}) of {𝑥_𝑛} is the set 𝐴𝑥_𝑛=𝑥∈𝑋∶𝑟𝑥,𝑥_𝑛=𝑟𝑥_𝑛. (12)

It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, 𝐴({𝑥_𝑛}) consists of exactly one point.

A sequence {𝑥_𝑛} in 𝑋 is said to △-converge to 𝑥∈𝑋 if 𝑥 is the unique asymptotic center of {𝑢_𝑛} for every subsequence {𝑢_𝑛} of {𝑥_𝑛}. In this case, we write △-lim_𝑛𝑥_𝑛=𝑥 and call 𝑥 the △-limit of {𝑥_𝑛}.

Lemma 2. Assume that 𝑋 is a CAT(0) space. Then, one has the following:(i)(see [12]) every bounded sequence in 𝑋 has a △-convergent subsequence;(ii)(see [13]) if 𝐾 is a closed convex subset of 𝑋 and 𝑇∶𝐾→𝑋 is an asymptotically nonexpansive mapping, then the conditions {𝑥_𝑛}  △-converge to 𝑥 and 𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))→0, imply 𝑥∈𝐾 and 𝑥∈𝐹(𝑇).

Lemma 3 (see [14, 15]). Let {𝑎_𝑛},{𝑏_𝑛},and{𝑐_𝑛} be three nonnegative real sequences satisfying the following condition: 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛+𝑐_𝑛,∀𝑛≥𝑛₀, (13) where 𝑛₀ is some nonnegative integer, ∑^∞_𝑛=1𝑏_𝑛<∞, ∑^∞_𝑛=1𝑐_𝑛<∞. Then the limit lim_𝑛→∞𝑎_𝑛 exists.

2. △-Convergence of the Iteration Sequences

In this section, we will study the △-convergence of the iteration sequence for asymptotically nonexpansive mappings in CAT(0) spaces.

Suppose that 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛. Firstly, we consider the iteration process: 𝑥₀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇^𝑛𝑦_𝑛𝑦,𝑛≥0,_𝑛=𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥0, (14) where {𝛼_𝑛},{𝛽_𝑛}⊆(0,1) and 𝑘_𝑛 satisfy the following.(i)There exist positive integers 𝑛₀, 𝑛₁, and 𝛿>0, 0<𝑏<min{1,1/𝐿}, where 𝐿=sup_𝑛𝑘_𝑛, such that 0<𝛿<𝛼_𝑛<1−𝛿,𝑛≥𝑛₀,0<1−𝛽_𝑛<𝑏,𝑛≥𝑛₁, (15)(ii)Consider ∑^∞_𝑛=1(𝑘²_𝑛−1)<∞.

We will prove that {𝑥_𝑛}  △-converges to a fixed point of 𝑇.

Lemma 4. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛, and ∑^∞_𝑛=1(𝑘²_𝑛−1)<∞. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑇^𝑛𝑦_𝑛, 𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛, 𝑛≥0. Then the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists for all 𝑝∈𝐹(𝑇).

Proof. Taking 𝑝∈𝐹(𝑇), we have 𝑑𝑥_𝑛+1,𝑝=𝑑𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑝≤𝛼_𝑛𝑑𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑𝑇^𝑛𝑦_𝑛,𝑝≤𝛼_𝑛𝑑𝑥_𝑛,𝑝+1−𝛼_𝑛𝑘_𝑛𝑑𝑦_𝑛,𝑝≤𝛼_𝑛𝑑𝑥_𝑛+,𝑝1−𝛼_𝑛𝑘_𝑛𝛽_𝑛𝑑𝑥_𝑛+,𝑝1−𝛽_𝑛𝑑𝑇^𝑛𝑥_𝑛,𝑝≤𝛼_𝑛𝑑𝑥_𝑛+,𝑝1−𝛼_𝑛𝑘_𝑛𝛽_𝑛𝑑𝑥_𝑛+,𝑝1−𝛽_𝑛𝑘_𝑛𝑑𝑥_𝑛=,𝑝1+1−𝛼_𝑛𝑘_𝑛×𝑘−1_𝑛1−𝛽_𝑛+1𝑑𝑥_𝑛≤𝑘,𝑝1+²_𝑛−1𝑑𝑥_𝑛,𝑝. (16) By Lemma 3, we can get that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists.

Remark 5. The above lemma implies that {𝑥_𝑛} is bounded and so is the sequence {𝑇𝑥_𝑛}. Moreover, let 𝐿=sup_𝑛𝑘_𝑛, then we have 𝑑𝑇^𝑛𝑥_𝑛,𝑝≤𝑘_𝑛𝑑𝑥_𝑛,𝑝≤𝐿𝑑𝑥_𝑛,𝑑𝑦,𝑝_𝑛,𝑝≤𝛽_𝑛𝑑𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑𝑇^𝑛𝑥_𝑛𝑥,𝑝≤𝐿𝑑_𝑛𝑑𝑇,𝑝_𝑛𝑦_𝑛,𝑝≤𝑘_𝑛𝑑𝑦_𝑛,𝑝≤𝐿²𝑑𝑥_𝑛,𝑝. (17) It follows that the sequences {𝑇^𝑛𝑥_𝑛}, {𝑦_𝑛}, {𝑇^𝑛𝑦_𝑛} are bounded.

Proposition 6. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑇^𝑛𝑦_𝑛, 𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛, 𝑛≥0. Then under the hypotheses (i) and (ii), one can get that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑦_𝑛)=0.

Proof. By the assumption, 𝐹(𝑇) is nonempty. Take 𝑝∈𝐹(𝑇), by Lemma 1(iv), we have 𝑑²𝑥_𝑛+1,𝑝=𝑑²𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑝≤𝛼_𝑛𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑²𝑇^𝑛𝑦_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛≤𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑²𝑇^𝑛𝑦_𝑛,𝑝−𝑑²𝑦_𝑛+,𝑝1−𝛼_𝑛𝑑²𝑦_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛,𝑑²𝑦_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝=𝑑²𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝≤𝛽_𝑛𝑑²𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑝−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−𝑑²𝑥_𝑛,𝑝≤𝛽_𝑛𝑑²𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝, (18) which implies that 𝑑²𝑦_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝≤1−𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑝−𝑑²𝑥_𝑛≤,𝑝1−𝛽_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝. (19)
Therefore, we have 𝑑²𝑥_𝑛+1,𝑝≤𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛𝑘²_𝑛−1𝑑²𝑦_𝑛+,𝑝1−𝛼_𝑛1−𝛽_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛. (20)
Since {𝑥_𝑛} and {𝑦_𝑛} are bounded and 0<𝛿<𝛼_𝑛<1−𝛿 for all 𝑛≥𝑛₀. we have 𝛿²𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛≤𝑑²𝑥_𝑛,𝑝−𝑑²𝑥_𝑛+1+,𝑝1−𝛼_𝑛𝑘²_𝑛−1𝑑²𝑦_𝑛+,𝑝1−𝛼_𝑛1−𝛽_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝. (21)
By the conditions (i) and (ii), we have ^∞_𝑛=1𝛿²𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛<∞, (22) which implies that lim_𝑛→∞𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛=0. (23)

Theorem 7. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑇^𝑛𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛,𝑛≥0. Then under the hypotheses (i) and (ii), one can get that {𝑥_𝑛}  △-converges to a fix point of 𝑇.

Proof. We first show that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)=0. Indeed 𝑑𝑥_𝑛,𝑦_𝑛=𝑑𝑥_𝑛,𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝑑𝑇^𝑛𝑦_𝑛,𝑇^𝑛𝑥_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝐿𝑑𝑦_𝑛,𝑥_𝑛; (24) it follows that 1−𝐿1−𝛽_𝑛𝑑𝑥_𝑛,𝑦_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛. (25) By the conditions (i) and (ii) and Proposition 6, we get lim_𝑛→∞𝑑(𝑥_𝑛,𝑦_𝑛)=0.
And then, 𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝑑𝑇^𝑛𝑦_𝑛,𝑇^𝑛𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝐿𝑑𝑦_𝑛,𝑥_𝑛. (26) By Proposition 6, we get that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)=0.
We claim that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Indeed we have 𝑑𝑦_𝑛,𝑇^𝑛𝑥_𝑛=𝑑𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛𝑑𝑥→0._𝑛+1,𝑥_𝑛=𝑑𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑥_𝑛≤1−𝛼_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑦_𝑛𝑑𝑥→0._𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1+𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1+𝐿𝑑𝑥_𝑛−1,𝑥_𝑛𝑑𝑥→0._𝑛,𝑇^𝑛−1𝑥_𝑛≤𝑑𝛼_𝑛−1𝑥_𝑛−1⊕1−𝛼_𝑛−1𝑇^𝑛−1𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝛼_𝑛−1𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛+1−𝛼_𝑛−1𝑑𝑇^𝑛−1𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝛼_𝑛−1𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛+1−𝛼_𝑛−1𝐿𝑑𝑦_𝑛−1,𝑥_𝑛≤𝛼_𝑛−1𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛+1−𝛼_𝑛−1𝐿𝑑𝑦_𝑛−1,𝑥_𝑛−1+𝑑𝑥_𝑛−1,𝑥_𝑛→0. (27) Thus, 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝑑𝑇^𝑛𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝐿𝑑𝑇^𝑛−1𝑥_𝑛,𝑥_𝑛→0. (28)
Since {𝑥_𝑛} is bounded, we may assume that {𝑥_𝑛}  △-converges to a point 𝑥. By Lemma 2, we have 𝑥∈𝐹(𝑇).

Next we will consider another iteration process: 𝑥₀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑦_𝑛𝑦,𝑛≥0,_𝑛=𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥0, (29) where {𝛼_𝑛},{𝛽_𝑛}⊆(0,1), and 𝑘_𝑛 satisfy the following(H1) There exist positive integers 𝑛₀ and 𝛿>0, such that 0<𝛿<𝛼_𝑛<1−𝛿,𝑛≥𝑛₀;1−𝛽_𝑛→0; (30)(H2)∑^∞_𝑛=1(𝑘_𝑛−1)<∞.

We will prove that {𝑥_𝑛} also △-converges to a fixed point of 𝑇.

Lemma 8. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛, and ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛,𝑛≥0. Then the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists for all 𝑝∈𝐹(𝑇).

Proof. Taking 𝑝∈𝐹(𝑇), we have 𝑑𝑥_𝑛+1,𝑝=𝑑𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑦_𝑛,𝑝≤𝛼_𝑛𝑘_𝑛𝑑𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑𝑦_𝑛,𝑝≤𝛼_𝑛𝑘_𝑛𝑑𝑥_𝑛+,𝑝1−𝛼_𝑛𝛽_𝑛𝑑𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑𝑇^𝑛𝑥_𝑛,𝑝≤𝛼_𝑛𝑘_𝑛𝑑𝑥_𝑛+,𝑝1−𝛼_𝑛𝛽_𝑛𝑑𝑥_𝑛,𝑝+1−𝛽_𝑛𝑘_𝑛𝑑𝑥_𝑛=𝑘,𝑝1+_𝑛−11−1−𝛼_𝑛𝛽_𝑛𝑑𝑥_𝑛,𝑝. (31) By Lemma 3, we can get that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists.

Next, we will prove lim_𝑛→∞𝑑(𝑇^𝑛𝑥_𝑛,𝑦_𝑛)=0.

Proposition 9. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛,𝑛≥0. Then under the hypotheses (H1) and (H2), one can get that lim_𝑛→∞𝑑(𝑇^𝑛𝑥_𝑛,𝑦_𝑛)=0.

Proof. By the assumption, 𝐹(𝑇) is nonempty. Take 𝑝∈𝐹(𝑇), let 𝐿=sup_𝑛𝑘_𝑛, then we have 𝑑𝑇^𝑛𝑥_𝑛,𝑝≤𝑘_𝑛𝑑𝑥_𝑛,𝑝≤𝐿𝑑𝑥_𝑛,𝑑𝑦,𝑝_𝑛,𝑝≤𝛽_𝑛𝑑𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑𝑇^𝑛𝑥_𝑛𝑥,𝑝≤𝐿𝑑_𝑛𝑑𝑇,𝑝_𝑛𝑦_𝑛,𝑝≤𝑘_𝑛𝑑𝑦_𝑛,𝑝≤𝐿²𝑑𝑥_𝑛,𝑝. (32) It follows that the sequences {𝑥_𝑛},{𝑇^𝑛𝑥_𝑛},{𝑦_𝑛},{𝑇^𝑛𝑦_𝑛} are bounded.
By Lemma 1, we have 𝑑²𝑥_𝑛+1,𝑝=𝑑²𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑦_𝑛,𝑝≤𝛼_𝑛𝑘²_𝑛𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑²𝑦_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛≤𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑²𝑦_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛. (33)
Similar to the proof of Proposition 6, we can get 𝑑²𝑦_𝑛,𝑝−𝑑²𝑥_𝑛,𝑝≤1−𝛽_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝. (34)
Therefore, we have 𝑑²𝑥_𝑛+1,𝑝≤𝑑²𝑥_𝑛,𝑝+1−𝛼_𝑛1−𝛽_𝑛×𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛. (35) Since {𝑥_𝑛},{𝑦_𝑛} are bounded and 0<𝛿<𝛼_𝑛<1−𝛿 for all 𝑛≥𝑛₀. we have 𝛿²𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛≤𝑑²𝑥_𝑛,𝑝−𝑑²𝑥_𝑛+1+,𝑝1−𝛼_𝑛1−𝛽_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑘²_𝑛−1𝑑²𝑥_𝑛,𝑝. (36)
By the conditions (H1) and (H2), we have ∑^∞_𝑛=1(𝑘²_𝑛−1)<∞ and ^∞_𝑛=1𝛿²𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛<∞, (37) which implies that lim_𝑛→∞𝑑²𝑇^𝑛𝑥_𝑛,𝑦_𝑛=0. (38)

Theorem 10. Let 𝑋 be a CAT(0) space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with coefficient 𝑘_𝑛. If 𝐹(𝑇)≠∅, {𝛼_𝑛},{𝛽_𝑛}⊆(0,1). Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕(1−𝛼_𝑛)𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛⊕(1−𝛽_𝑛)𝑇^𝑛𝑥_𝑛,𝑛≥0. Then under the hypotheses (H1) and (H2), one can get that {𝑥_𝑛}  △-converges to a fix point of  𝑇.

Proof. We first show that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)=0. Indeed, by Lemma 1, and 𝛽_𝑛→1, we can get 𝑑𝑥_𝑛,𝑦_𝑛=𝑑𝑥_𝑛,𝛽_𝑛𝑥_𝑛⊕1−𝛽_𝑛𝑇^𝑛𝑥_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛→0. (39) And then, 𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝑑𝑥_𝑛,𝑦_𝑛+𝑑𝑦_𝑛,𝑇^𝑛𝑥_𝑛. (40) By Proposition 9, we obtain that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)=0.
We claim that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Indeed we have 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝛼_𝑛𝑇^𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑦_𝑛,𝑥_𝑛≤𝛼_𝑛𝑑𝑇^𝑛𝑥_𝑛,𝑥_𝑛+1−𝛼_𝑛𝑑𝑥_𝑛,𝑦_𝑛𝑑𝑥→0._𝑛,𝑇^𝑛−1𝑥_𝑛≤𝑑𝛼_𝑛−1𝑇^𝑛−1𝑥_𝑛−1⊕1−𝛼_𝑛−1𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝛼_𝑛−1𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛+1−𝛼_𝑛−1𝑑𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝛼_𝑛−1𝑘_𝑛−1𝑑𝑥_𝑛−1,𝑥_𝑛+1−𝛼_𝑛−1𝑑𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1+𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛≤𝛼_𝑛−1𝑘_𝑛−1𝑑𝑥_𝑛−1,𝑥_𝑛+1−𝛼_𝑛−1𝑑𝑦_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1+𝑘_𝑛−1𝑑𝑥_𝑛−1,𝑥_𝑛→0. (41) Thus, 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝑑𝑇^𝑛𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝐿𝑑𝑇^𝑛−1𝑥_𝑛,𝑥_𝑛→0. (42)
Since {𝑥_𝑛} is bounded, we may assume that {𝑥_𝑛}  △-converges to a point 𝑥. By Lemma 2, we have 𝑥∈𝐹(𝑇).

Acknowledgment

This research was supported by NSFC Grants nos. 11071279 and 11226125.