Some Convergence Theorems for Contractive Type Mappings in CAT(0) Spaces

Abstract

We establish theorems of strong convergence, for the Ishikawa-type (or two step; cf. Ishikawa, 1974) iteration scheme, to a fixed point of a uniformly L-Lipschitzian asymptotically demicontractive mapping and a uniformly L-Lipschitzian hemicontractive mapping in CAT(0) space. Moreover, we will propose some open problems.

1. Introduction

Let (𝑋,𝑑) be a metric space. One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [1] introduced a convex structure in a metric space (𝑋,𝑑). A mapping 𝑊∶𝑋×𝑋×[0,1]→𝑋 is a convex structure in 𝑋 if 𝑑𝑢,𝑊𝑥,𝑦,𝜆≤𝜆𝑑(𝑢,𝑥)+(1−𝜆)𝑑𝑢,𝑦 (1) for all 𝑥,𝑦∈𝑋 and 𝜆∈[0,1]. A metric space together with a convex structure 𝑊 is known as a convex metric space. A nonempty subset 𝐾 of a convex metric space is said to be convex if 𝑊𝑥,𝑦,𝜆∈𝐾 (2) for all 𝑥,𝑦∈𝐾 and 𝜆∈[0,1]. In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [1]).

Example 1 (see [2]). Let 𝑋={(𝑥₁,𝑥₂)∈ℝ²∶𝑥₁>0,𝑥₂>0}, for all 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂)∈𝑋, and 𝜆∈[0,1]. We define a mapping 𝑊∶𝑋×𝑋×[0,1]→𝑋 by 𝑊𝑥,𝑦,𝜆=𝜆𝑥₁+(1−𝜆)𝑦₁,𝜆𝑥₁𝑥₂+(1−𝜆)𝑦₁𝑦₂𝜆𝑥₁+(1−𝜆)𝑦₁, (3) and define a metric 𝑑∶𝑋×𝑋→[0,∞) by 𝑑𝑥,𝑦=𝑥₁−𝑦₁+𝑥₁𝑥₂−𝑦₁𝑦₂. (4) Then we can show that (𝑋,𝑑,𝑊) is a convex metric space, but it is not a normed linear space.

A metric space 𝑋 is a 𝐶𝐴𝑇(0) space (the term is due to Gromov [3] and it is an acronym for E. Cartan, A. D. Aleksandrov, and V. A. Toponogov) if it is geodesically connected and if every geodesic triangle in 𝑋 is at least as “thin” as its comparison triangle in the Euclidean plane (see, e.g., [4], page 159). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a 𝐶𝐴𝑇(0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [4] or Burago et al. [5].

Let (𝑋,𝑑) be a metric space. A geodesic path joining 𝑥∈𝑋 to 𝑦∈𝑋 (or, more briefly, a geodesic from 𝑥 to 𝑦) is a mapping 𝑐 from a closed interval [0,𝑙]⊂ℝ to 𝑋 such that 𝑐(0)=𝑥, 𝑐(𝑙)=𝑦, and 𝑑(𝑐(𝑡),𝑐(𝑡^′))=|𝑡−𝑡^′|, for all 𝑡,𝑡^′∈[0,𝑙]. In particular, 𝑐 is an isometry and 𝑑(𝑥,𝑦)=𝑙. The image 𝛼 of 𝑐 is called a geodesic (or metric) segment joining 𝑥 and 𝑦. When it is unique, this geodesic is denoted by [𝑥,𝑦]. The space (𝑋,𝑑) is said to be a geodesic space if every two points of 𝑋 are joined by a geodesic, and 𝑋 is said to be uniquely geodesic if there is exactly one geodesic joining 𝑥 and 𝑦 for each 𝑥,𝑦∈𝑋. A subset 𝑌⊆𝑋 is said to be convex if 𝑌 includes every geodesic segment joining any two of its points.

A geodesic triangle  Δ(𝑥₁,𝑥₂,𝑥₃) is a geodesic metric space (𝑋,𝑑) that consists of three points 𝑥₁,𝑥₂,𝑥₃∈𝑋 (the vertices of Δ) and is a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(𝑥₁,𝑥₂,𝑥₃) in (𝑋,𝑑) is a triangle Δ(𝑥₁,𝑥₂,𝑥₃)=Δ(𝑥₁,𝑥₂,𝑥₃) in ℝ² such that 𝑑_ℝ²(𝑥_𝑖,𝑥_𝑗)=𝑑(𝑥_𝑖,𝑥_𝑗) for 𝑖,𝑗∈{1,2,3}. Such a triangle always exists (see, [4]).

A geodesic metric space is said to be a 𝐶𝐴𝑇(0)  space if all geodesic triangles of appropriate size satisfy the following 𝐶𝐴𝑇(0) comparison axiom.

Let Δ be a geodesic triangle in 𝑋 and let Δ⊂ℝ² be a comparison triangle for Δ. Then Δ is said to satisfy the 𝐶𝐴𝑇(0)  inequality if for all 𝑥,𝑦∈Δ and all comparison points 𝑥,𝑦∈Δ, 𝑑𝑥,𝑦≤𝑑𝑥,𝑦. (5) Complete 𝐶𝐴𝑇(0) spaces are often called Hadamard spaces (see, [6]). If 𝑥,𝑦₁,𝑦₂ are points of a 𝐶𝐴𝑇(0) space and if 𝑦₀ is the midpoint of the segment [𝑦₁,𝑦₂], which we will denote by 𝑦₁⊕𝑦₂/2, then the 𝐶𝐴𝑇(0) inequality implies 𝑑²𝑦𝑥,¹⊕𝑦₂2≤12𝑑²𝑥,𝑦₁+12𝑑²𝑥,𝑦₂−14𝑑²𝑦₁,𝑦₂. (6) This inequality is the (CN) inequality of Bruhat and Tits [7]. In fact, a geodesic space is a 𝐶𝐴𝑇(0) space if and only if it satisfies the (CN) inequality (cf. [4], page 163). The previous inequality has been extended by Khamsi and Kirk [8] as 𝑑²𝑧,𝛼𝑥⊕(1−𝛼)𝑦≤𝛼𝑑²(𝑧,𝑥)+(1−𝛼)𝑑²𝑧,𝑦−𝛼(1−𝛼)𝑑²𝑥,𝑦, CN^∗ for any 𝛼∈[0,1] and 𝑥,𝑦,𝑧∈𝑋. The inequality CN^∗ also appeared in [9].

Let us recall that a geodesic metric space is a 𝐶𝐴𝑇(0) space if and only if it satisfies the (CN) inequality (see, [4], page 163). Moreover, if 𝑋 is a 𝐶𝐴𝑇(0) metric space and 𝑥,𝑦∈𝑋, then for any 𝛼∈[0,1], there exists a unique point 𝛼𝑥⊕(1−𝛼)𝑦∈[𝑥,𝑦] such that 𝑑𝑧,𝛼𝑥⊕(1−𝛼)𝑦≤𝛼𝑑(𝑧,𝑥)+(1−𝛼)𝑑𝑧,𝑦 (7) for any 𝑧∈𝑋 and [𝑥,𝑦]={𝛼𝑥⊕(1−𝛼)𝑦∶𝛼∈[0,1]}. In view of the previous inequality, 𝐶𝐴𝑇(0) space has Takahashi's convex structure 𝑊(𝑥,𝑦,𝛼)=𝛼𝑥⊕(1−𝛼)𝑦. It is easy to see that for any 𝑥,𝑦∈𝑋 and 𝜆∈[0,1], 𝑑(),𝑑()=()𝑑.𝑥,1−𝜆𝑥⊕𝜆𝑦=𝜆𝑑𝑥,𝑦𝑦,1−𝜆𝑥⊕𝜆𝑦1−𝜆𝑥,𝑦 (8) As a consequence, ()()1⋅𝑥⊕0⋅𝑦=𝑥,1−𝜆𝑥⊕𝜆𝑥=𝜆𝑥⊕1−𝜆𝑥=𝑥. (9) Moreover, a subset 𝐾 of 𝐶𝐴𝑇(0) space 𝑋 is convex if for any 𝑥,𝑦∈𝐾, we have [𝑥,𝑦]⊂𝐾.

Definition 2. Let 𝐶 be a nonempty subset of a metric space (𝑋,𝑑). Let 𝐹(𝑇) denote the fixed point set of 𝑇. Let 𝐹(𝑇)≠∅.(1) A mapping 𝑇∶𝐶→𝐶 is said to be 𝑘-strict asymptotically pseudocontractive with sequence {𝑎_𝑛} if lim_𝑛→∞𝑎_𝑛=1 for some constant 𝑘, 0≤𝑘<1 and 𝑑²𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑎²_𝑛𝑑²𝑥,𝑦+𝑘𝑑𝑥,𝑇^𝑛𝑥−𝑑𝑦,𝑇^𝑛𝑦², (10) for all 𝑥,𝑦∈𝐶,  𝑛∈ℕ. If 𝑘=0, then 𝑇 is said to be asymptotically nonexpansive with sequence {𝑎_𝑛}, that is, 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑎_𝑛𝑑𝑥,𝑦,∀𝑥,𝑦∈𝐶. (11)(2) A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically demicontractive with sequence {𝑎_𝑛} if lim_𝑛→∞𝑎_𝑛=1 for some constant 𝑘, 0≤𝑘<1, and 𝑑²𝑇^𝑛𝑥,𝑝≤𝑎²_𝑛𝑑²𝑥,𝑝+𝑘⋅𝑑²𝑥,𝑇^𝑛𝑥,∀𝑝∈𝐹(𝑇), (12) for all 𝑥∈𝐶,  𝑛∈ℕ. If 𝑘=0, then 𝑇 is said to be asymptotically quasi-nonexpansive with sequence {𝑎_𝑛}, that is, 𝑑𝑇^𝑛𝑥,𝑝≤𝑎_𝑛𝑑𝑥,𝑝,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇). (13)(3) A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically pseudocontractive with sequence {𝑎_𝑛} if lim_𝑛→∞𝑎_𝑛=1 and 𝑑²𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑎_𝑛𝑑²𝑥,𝑦+𝑑𝑥,𝑇^𝑛𝑥−𝑑𝑦,𝑇^𝑛𝑦², (14) for all 𝑥,𝑦∈𝐶,  𝑛∈ℕ.(4) A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically hemicontractive with sequence {𝑎_𝑛} if lim_𝑛→∞𝑎_𝑛=1 and 𝑑²𝑇^𝑛𝑥,𝑝≤𝑎_𝑛𝑑²𝑥,𝑝+𝑑²𝑥,𝑇^𝑛𝑥,∀𝑝∈𝐹(𝑇), (15) for all 𝑥∈𝐶,  𝑛∈ℕ.(5) A mapping 𝑇∶𝐶→𝐶 is said to be uniformly  𝐿-Lipschitzian if for some constant 𝐿>0, 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝐿⋅𝑑𝑥,𝑦,∀𝑥,𝑦∈𝐶, (16) for all 𝑛∈ℕ.

Liu [10] has proved the convergence of Mann and Ishikawa iterative sequence for uniformly 𝐿-Lipschitzian asymptotically demicontractive and hemicontractive mappings in Hilbert space (cf. [11]). The existence of (common) fixed points of one mapping (or two mappings or family of mappings) is not known in many situations. So the approximation of fixed points of one or more nonexpansive, asymptotically nonexpansive, or asymptotically quasi-nonexpansive mappings by various iterations have been extensively studied in Banach spaces, convex metric spaces, 𝐶𝐴𝑇(0) spaces, and so on (see, [2, 6, 8, 9, 12–27]).

In this paper, we establish theorems of strong convergence for the Ishikawa-type (or two step, cf. [28]) iteration scheme to a fixed point of a uniformly 𝐿-Lipschitzian asymptotically demicontractive mapping and a uniformly 𝐿-Lipschitzian asymptotically hemicontractive mapping in 𝐶𝐴𝑇(0) space. Moreover, we will propose some open problems.

2. Preliminaries

We introduce the following iteration process.

Let 𝐶 be a nonempty convex subset of a 𝐶𝐴𝑇(0) space (𝑋,𝑑) and let 𝑇∶𝐶→𝐶 be a given mapping. Let 𝑥₁∈𝐶 be a given point.

Algorithm 3. The sequences {𝑥_𝑛} and {𝑦_𝑛} defined by the iterative process 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (17) is called an Ishikawa-type iterative sequence (cf. [28]).

If 𝛽_𝑛≡0, then Algorithm 3 reduces to the following

Algorithm 4. The sequence {𝑥_𝑛} defined by the iterative process 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (18) is called a Mann-type iterative sequence (cf. [29]).

Lemma 5 (see [10]). Let sequences {𝑎_𝑛}, {𝑏_𝑛} satisfy that 𝑎_𝑛+1≤𝑎_𝑛+𝑏_𝑛, (19)𝑎_𝑛≥0, for all 𝑛∈ℕ, ∑^∞_𝑛=1𝑏_𝑛 is convergent, and {𝑎_𝑛} has a subsequence {𝑎_𝑛𝑘} converging to 0. Then, we must have lim_𝑛→∞𝑎_𝑛=0. (20)

3. Convergence Theorems

Lemma 6. Let (𝑋,𝑑) be a 𝐶𝐴𝑇(0) space and let 𝐶 be a nonempty convex subset of 𝑋. Let 𝑇∶𝐶→𝐶 be an uniformly 𝐿-Lipschitzian mapping and let {𝛼_𝑛}, {𝛽_𝑛} be sequence in [0,1]. Define the iteration scheme {𝑥_𝑛} as Algorithm 3. Then 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝐿1+2𝐿+𝐿²𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1, (21) for all 𝑛≥1.

Proof. Let 𝐶_𝑛=𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛). We have 𝑑𝑥_𝑛−1,𝑦_𝑛−1=𝑑𝑥_𝑛−1,1−𝛽_𝑛−1𝑥_𝑛−1⊕𝛽_𝑛−1𝑇^𝑛−1𝑥_𝑛−1≤𝛽_𝑛−1⋅𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1=𝛽_𝑛−1𝐶_𝑛−1. (22) From (22), we get 𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑦_𝑛−1≤𝑑𝑥_𝑛−1,𝑇^𝑛−1𝑥_𝑛−1+𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑇^𝑛−1𝑦_𝑛−1≤𝐶_𝑛−1+𝐿⋅𝑑𝑥_𝑛−1,𝑦_𝑛−1≤𝐶_𝑛−1+𝛽_𝑛−1⋅𝐿⋅𝐶_𝑛−1. (23) From (22) and (23), we get 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝑑𝑇^𝑛𝑥_𝑛,𝑇𝑥_𝑛≤𝐶_𝑛+𝐿⋅𝑑𝑇^𝑛−1𝑥_𝑛,𝑥_𝑛≤𝐶_𝑛+𝐿𝑑𝑇^𝑛−1𝑥_𝑛,𝑇^𝑛−1𝑥_𝑛−1+𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑥_𝑛≤𝐶_𝑛+𝐿²⋅𝑑𝑥_𝑛,𝑥_𝑛−1+𝐿⋅𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑥_𝑛≤𝐶_𝑛+𝐿²⋅𝑑1−𝛼_𝑛−1𝑥_𝑛−1⊕𝛼_𝑛−1𝑇^𝑛−1𝑦_𝑛−1,𝑥_𝑛−1+𝐿⋅𝑑𝑇^𝑛−1𝑥_𝑛−1,1−𝛼_𝑛−1𝑥_𝑛−1⊕𝛼_𝑛−1𝑇^𝑛−1𝑦_𝑛−1≤𝐶_𝑛+𝐿²⋅𝛼_𝑛−1⋅𝑑𝑇^𝑛−1𝑦_𝑛−1,𝑥_𝑛−1+𝐿1−𝛼_𝑛−1𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑥_𝑛−1+𝛼_𝑛−1⋅𝑑𝑇^𝑛−1𝑥_𝑛−1,𝑇^𝑛−1𝑦_𝑛−1≤𝐶_𝑛+𝐿²⋅𝛼_𝑛−1𝐶_𝑛−1+𝛽_𝑛−1⋅𝐿⋅𝐶_𝑛−1+𝐿1−𝛼_𝑛−1𝐶_𝑛−1+𝐿²⋅𝛼_𝑛−1⋅𝛽_𝑛−1⋅𝐶_𝑛−1≤𝐶_𝑛+𝐿1+2𝐿+𝐿²𝐶_𝑛−1,𝑛≥1. (24) This completes the proof of Lemma 6.

Theorem 7. Let (𝑋,𝑑) be a complete 𝐶𝐴𝑇(0) space, let 𝐶 be a nonempty bounded closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be a completely continuous and uniformly 𝐿-Lipschitzian and asymptotically demicontractive with sequence {𝑎_𝑛}, 𝑎_𝑛∈[1,∞), ∑^∞_𝑛=1(𝑎²_𝑛−1)<∞, 𝜀≤𝛼_𝑛≤1−𝑘−𝜀, for all 𝑛∈ℕ and some 𝜀>0. Given 𝑥₀∈𝐶, define the iteration scheme {𝑥_𝑛} by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (25) Then {𝑥_𝑛} converges strongly to some fixed point of 𝑇.

Proof. Since 𝑇 is a completely continuous mapping in a bounded closed convex subset 𝐶 of complete metric space, from Schauder's theorem, 𝐹(𝑇) is nonempty. It follows from CN^∗ inequality that 𝑑²𝑥_𝑛+1,𝑝=𝑑²1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑥_𝑛≤,𝑝1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑇^𝑛𝑥_𝑛−,𝑝1−𝛼_𝑛𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛, (26) for all 𝑝∈𝐹(𝑇). Since 𝑇 is a asymptotically demicontractive, we get 𝑑²𝑥_𝑛+1,𝑝≤1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑎²_𝑛𝑑²𝑥_𝑛,𝑝+𝑘⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛=𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑎²_𝑛−1𝑑²𝑥_𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛−𝑘𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛,∀𝑝∈𝐹(𝑇). (27) Since 0<𝜀≤𝛼_𝑛≤1−𝑘−𝜀, we have 1−𝑘−𝛼_𝑛≥𝜀. Thus, 𝛼_𝑛1−𝑘−𝛼_𝑛≥𝜀². (28) From (27), we have 𝑑²𝑥_𝑛+1,𝑝≤𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑎²_𝑛−1𝑑²𝑥_𝑛,𝑝−𝜀²⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛, (29) for all 𝑝∈𝐹(𝑇). Since 𝐶 is bounded and 𝑇 is self-mapping in 𝐶, there exist some 𝑀>0 so that 𝑑²(𝑥_𝑛,𝑝)≤𝑀, for all 𝑛∈ℕ. Since 0≤𝛼_𝑛≤1, it follows from (29) that 𝑑²𝑥_𝑛+1,𝑝≤𝑑²𝑥_𝑛,𝑝+𝑎²_𝑛−1𝑀−𝜀²⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛,∀𝑝∈𝐹(𝑇). (30) Therefore, 𝜀²⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝑑²𝑥_𝑛,𝑝−𝑑²𝑥_𝑛+1,𝑝+𝑎²_𝑛−1𝑀. (31) So ^𝑚_𝑛=1𝜀²⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝑑²𝑥₁,𝑝−𝑑²𝑥_𝑚+1,𝑝+^𝑚_𝑛=1𝑎²_𝑛−1𝑀≤2𝑀+^∞_𝑛=1𝑎²_𝑛−1𝑀, (32) for all 𝑚∈ℕ. Since ∑^∞_𝑛=1(𝑎²_𝑛−1)<∞, we get ^∞_𝑛=1𝜀²⋅𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛<∞. (33) Therefore, lim_𝑛→∞𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛=0,lim_𝑛→∞𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛=0. (34) Since 𝑇 is a uniformly 𝐿-Lipschitzian, it follows from Lemma 6 that lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=0. (35) Since {𝑥_𝑛} is a bounded sequence and 𝑇 is completely continuous, there exist a convergent subsequence {𝑇𝑥_𝑛𝑘} of {𝑇𝑥_𝑛}. Therefore, from (35), {𝑥_𝑛} has a convergent subsequence {𝑥_𝑛𝑘}. Let lim_𝑘→∞𝑥_𝑛𝑘=𝑞. It follows from the continuity of 𝑇 and (35), we have 𝑞=𝑇𝑞. Therefore, {𝑥_𝑛} has a subsequence which converges to the fixed point 𝑞 of 𝑇. Let 𝑝=𝑞 in the inequality (30). Since ∑^∞_𝑛=1(𝑎²_𝑛−1)<∞ and ∑^∞_𝑛=1𝜀²⋅𝑑²(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)<∞, from (30) and Lemma 5, we have lim_𝑛→∞𝑑²𝑥_𝑛,𝑞=0. (36) Therefore, lim_𝑛→∞𝑥_𝑛=𝑞. (37) This completes the proof of Theorem 7.

Corollary 8. Let (𝑋,𝑑) be a complete 𝐶𝐴𝑇(0) space, let 𝐶 be a nonempty bounded closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be a completely continuous and uniformly 𝐿-Lipschitzian and 𝑘-strict asymptotically pseudocontractive with sequence {𝑎_𝑛}, 𝑎_𝑛∈[1,∞), ∑^∞_𝑛=1(𝑎²_𝑛−1)<∞, and 𝜀≤𝛼_𝑛≤1−𝑘−𝜀, for all 𝑛∈ℕ and some 𝜀>0. Given 𝑥₀∈𝐶, define the iteration scheme {𝑥_𝑛} by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (38) Then {𝑥_𝑛} converges strongly to some fixed point of 𝑇.

Proof. By Definition 2, 𝑇 is 𝑘-strict asymptotically pseudocontractive; then 𝑇 must be asymptotically demicontractive. Therefore, Corollary 8 can be proved by using Theorem 7.

Lemma 9. Let (𝑋,𝑑) be a 𝐶𝐴𝑇(0) space and let 𝐶 be a nonempty convex subset of 𝑋. Let 𝑇∶𝐶→𝐶 be an uniformly 𝐿-Lipschitzian and asymptotically hemicontractive with sequence {𝑎_𝑛}⊂[1,∞), for all 𝑛∈ℕ, and 𝐹(𝑇) is nonempty. Define the iteration scheme {𝑥_𝑛} as follows: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (39) Then the following inequality holds: 𝑑²𝑥_𝑛+1,𝑝≤1+𝛼_𝑛𝑎_𝑛−1𝑎_𝑛𝛽_𝑛+1𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−𝛼_𝑛𝛽_𝑛−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛, (40) for all 𝑝∈𝐹(𝑇).

Proof. It follows from CN^∗ inequality that 𝑑²𝑥_𝑛+1,𝑝=𝑑²1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛≤,𝑝1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑇^𝑛𝑦_𝑛−,𝑝1−𝛼_𝑛𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛,𝑑²𝑦_𝑛,𝑝=𝑑²1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛≤,𝑝1−𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛−,𝑝1−𝛽_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛, (41)(42) for all 𝑝∈𝐹(𝑇). Since 𝑇 is asymptotically hemicontractive, we get 𝑑²𝑇^𝑛𝑦_𝑛,𝑝≤𝑎_𝑛𝑑²𝑦_𝑛,𝑝+𝑑²𝑦_𝑛,𝑇^𝑛𝑦_𝑛,𝑑²𝑇^𝑛𝑥_𝑛,𝑝≤𝑎_𝑛𝑑²𝑥_𝑛,𝑝+𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (43)(44) From (42) and (44), we have 𝑑²𝑦_𝑛,𝑝≤1−𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝑎_𝑛𝑑²𝑥_𝑛,𝑝+𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−1−𝛽_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛=1+𝑎_𝑛−1𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (45) From CN^∗ inequality, we have 𝑑²𝑦_𝑛,𝑇^𝑛𝑦_𝑛=𝑑²1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛≤1−𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛−1−𝛽_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (46) Substituting (45) and (46) into (43), we get 𝑑²𝑇^𝑛𝑦_𝑛,𝑝≤𝑎_𝑛1+𝑎_𝑛−1𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝑎_𝑛𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛+1−𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛−1−𝛽_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (47) From (41) and (47), we obtain 𝑑²𝑥_𝑛+1≤,𝑝1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑎_𝑛1+𝑎_𝑛−1𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑎_𝑛𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛+𝛼_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝛼_𝑛𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛−𝛼_𝑛1−𝛽_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−1−𝛼_𝑛𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛=1+𝛼_𝑛𝑎_𝑛−1+𝑎_𝑛𝑎_𝑛−1𝛽_𝑛𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−𝛼_𝑛𝛽_𝑛−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛+𝛼_𝑛𝛽_𝑛𝑑²𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛. (48) Since 𝑇 is uniformly 𝐿-Lipschitzian, we have 𝑑𝑇^𝑛𝑥_𝑛,𝑇^𝑛𝑦_𝑛≤𝐿⋅𝑑𝑥_𝑛,𝑦_𝑛=𝐿⋅𝑑𝑥_𝑛,1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛≤𝐿𝛽_𝑛⋅𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (49) Substituting (49) into (48), we obtain 𝑑²𝑥_𝑛+1,𝑝≤1+𝛼_𝑛𝑎_𝑛−11+𝑎_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛−𝛼_𝑛𝛽_𝑛−𝛼_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑦_𝑛,∀𝑝∈𝐹(𝑇). (50) This completes the proof of Lemma 9.

Lemma 10. Let (𝑋,𝑑) be a 𝐶𝐴𝑇(0) space and let 𝐶 be a nonempty bounded convex subset of 𝑋. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and asymptotically hemicontractive with sequence {𝑎_𝑛}⊂[1,∞), for all 𝑛∈ℕ and ∑^∞_𝑛=1(𝑎_𝑛−1)<∞. Let 𝐹(𝑇) be nonempty. Given 𝑥₁∈𝐶, define the iteration scheme {𝑥_𝑛} by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (51) If 𝜀≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝜀>0 and 𝑏∈(0,(1+𝐿²−1)/𝐿²), then lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=0. (52)

Proof. First, we will prove lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)=0. From Lemma 9 and 0≤𝛼_𝑛≤𝛽_𝑛, we have 𝑑²𝑥_𝑛+1,𝑝≤1+𝛼_𝑛𝑎_𝑛−11+𝑎_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (53) Thus 𝑑²𝑥_𝑛+1,𝑝−𝑑²𝑥_𝑛,𝑝≤𝛼_𝑛𝑎_𝑛−11+𝑎_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (54) Since ∑^∞_𝑛=1(𝑎_𝑛−1)<∞, we have lim_𝑛→∞(𝑎_𝑛−1)=0. Hence, {𝑎_𝑛} is bounded. By boundedness of 𝐶 and 0≤𝛼_𝑛≤𝛽_𝑛≤1, we obtain that {𝛼_𝑛(1+𝑎_𝑛𝛽_𝑛)𝑑²(𝑥_𝑛,𝑝)} is bounded. Therefore, there exists a constant 𝑀>0 such that 0≤𝛼_𝑛1+𝑎_𝑛𝛽_𝑛𝑑²𝑥_𝑛,𝑝≤𝑀. (55) From (54) and (55), we get 𝑑²𝑥_𝑛+1,𝑝−𝑑²𝑥_𝑛≤𝑎,𝑝_𝑛−1𝑀−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛. (56)
Let 𝐷=1−2𝑏−𝐿²𝑏²>0. Since lim_𝑛→∞𝑎_𝑛=1, there exists 𝑁∈ℕ such that 1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛≥1−𝑏−𝑎_𝑛𝑏−𝐿²𝑏²≥𝐷2>0, (57) for all 𝑛≥𝑁. Suppose that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)≠0, then there exist a 𝜀₀>0 and a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑑²𝑥_𝑛𝑖,𝑇^𝑛𝑖𝑥_𝑛𝑖≥𝜀₀. (58) Without loss of generality, we let 𝑛₁≥𝑁. From (56), we have 𝛼_𝑛𝛽_𝑛1−𝛽_𝑛−𝑎_𝑛𝛽_𝑛−𝐿²𝛽²_𝑛𝑑²𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤𝑎_𝑛−1𝑀+𝑑²𝑥_𝑛,𝑝−𝑑²𝑥_𝑛+1,𝑝, (59) so ^𝑖_𝑙=1𝛼_𝑛𝑙𝛽_𝑛𝑙1−𝛽_𝑛𝑙−𝑎_𝑛𝑙𝛽_𝑛𝑙−𝐿²𝛽²_𝑛𝑙𝑑²𝑥_𝑛𝑙,𝑇^𝑛𝑙𝑥_𝑛𝑙=^𝑛𝑖_𝑚=𝑛1𝛼_𝑚𝛽_𝑚1−𝛽_𝑚−𝑎_𝑚𝛽_𝑚−𝐿²𝛽²_𝑚𝑑²𝑥_𝑚,𝑇^𝑚𝑥_𝑚≤^𝑛𝑖_𝑚=𝑛1𝑎_𝑚−1𝑀+𝑑²𝑥_𝑛1,𝑝−𝑑²𝑥_𝑛𝑖+1,𝑝. (60) From (57)–(60) and 𝜀≤𝛼_𝑛≤𝛽_𝑛, we obtain 𝑖⋅𝜀²⋅𝐷2⋅𝜀₀≤^𝑛𝑖_𝑚=𝑛1𝑎_𝑚−1𝑀+𝑑²𝑥_𝑛1,𝑝−𝑑²𝑥_𝑛𝑖+1,𝑝. (61) Since ∑^∞_𝑛=1(𝑎_𝑛−1)<∞ and the boundedness of 𝐶, the right side of (61) is bounded. However, if we have 𝑖→∞, then the left side of (61) is unbounded. This is a contradiction. Therefore, lim_𝑛→∞𝑑𝑥_𝑛,𝑇^𝑛𝑥_𝑛=0. (62) Since 𝑇 is a uniformly 𝐿-Lipschitzian, from Lemma 6, we get lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=0. (63) This completes the proof of Lemma 10.

Theorem 11. Let (𝑋,𝑑) be a complete 𝐶𝐴𝑇(0) space, let 𝐶 be a nonempty bounded closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be a completely continuous and uniformly 𝐿-Lipschitzian and asymptotically hemicontractive with sequence {𝑎_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑎_𝑛−1)<∞, for all 𝑛∈ℕ. Given 𝑥₁∈𝐶, define the iterative scheme {𝑥_𝑛} by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (64) If {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] with 𝜀≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝜀>0 and 𝑏∈(0,(1+𝐿²−1)/𝐿²), then {𝑥_𝑛} converges strongly to some fixed point of 𝑇.

Proof. Since 𝑇 is a completely continuous mapping in a bounded closed convex subset 𝐶 of complete metric space, from Schauder's theorem, 𝐹(𝑇) is nonempty. Since 𝑇 is completely continuous, there exist a convergent subset {𝑇𝑥_𝑛𝑖} of {𝑇𝑥_𝑛}. Let lim_𝑖→∞𝑇𝑥_𝑛𝑖=𝑝. (65) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, from Lemma 10, we have lim_𝑖→∞𝑥_𝑛𝑖=𝑝. (66) On the other hand, from the continuity of 𝑇, (66), and Lemma 10, we have 𝑑𝑝,𝑇𝑝=lim_𝑖→∞𝑑𝑥_𝑛𝑖,𝑇𝑥_𝑛𝑖=0. (67) This means that 𝑝 is a fixed point of 𝑇. From (55), (57), and 𝛼_𝑛≤𝛽_𝑛, we obtain Lemma 9 that 𝑑²𝑥_𝑛+1,𝑝≤𝑑²𝑥_𝑛,𝑝+𝑎_𝑛−1𝑀. (68) From (66), there exists a subsequence {𝑑²(𝑥_𝑛𝑖,𝑝)} of {𝑑²(𝑥_𝑛,𝑝)} which converges to 0. Therefore, from Lemma 5 and (68), lim_𝑛→∞𝑑²𝑥_𝑛,𝑝=0. (69) Hence, lim_𝑛→∞𝑥_𝑛=𝑝. (70) This completes the proof of Theorem 11.

Corollary 12. Let (𝑋,𝑑) be a complete 𝐶𝐴𝑇(0) space, let 𝐶 be a nonempty bounded closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be a completely continuous and uniformly 𝐿-Lipschitzian and asymptotically pseudocontractive with sequence {𝑎_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑎²_𝑛−1)<∞, for all 𝑛∈ℕ. Given 𝑥₁∈𝐶, define the iterative scheme {𝑥_𝑛} by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛⊕𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛⊕𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (71) If {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] with 𝜀≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝜀>0 and 𝑏∈(0,(1+𝐿²−1)/𝐿²), then {𝑥_𝑛} converges strongly to some fixed point of 𝑇.

Proof. By Definition 2, 𝑇 is an asymptotically pseudocontractive mapping, then 𝑇 is an asymptotically hemicontractive mapping. Since 𝑎_𝑛∈[1,∞), we have 𝑎²_𝑛−1≥𝑎_𝑛−1≥0. Obviously, ∑^∞_𝑛=1(𝑎_𝑛−1)≤∑^∞_𝑛=1(𝑎²_𝑛−1)<∞. Therefore, Corollary 12 can be proved by using Theorem 11.

4. Some Remarks and Open Problems

Let 𝑆 be a semigroup. We denote by 𝐵(𝑆) the space of all bounded real-valued functions defined on 𝑆 with supremum norm. For each 𝑠∈𝑆, we define the left and right translation operators 𝑙_𝑠 and 𝑟_𝑠 on 𝐵(𝑆) by 𝑙_𝑠𝑓(𝑡)=𝑓(𝑠𝑡),𝑟_𝑠𝑓(𝑡)=𝑓(𝑡𝑠), (72) for each 𝑡∈𝑆 and 𝑓∈𝐵(𝑆), respectively. Let 𝑋 be a subspace of 𝐵(𝑆) containing 1. An element 𝜇 in the dual space 𝑋^∗ of 𝑋 is said to be a mean on 𝑋 if ‖𝜇‖=𝜇(1)=1. For 𝑠∈𝑆, we can define a point evaluation 𝛿_𝑠 by 𝛿_𝑠(𝑓)=𝑓(𝑠) for each 𝑓∈𝑋. It is well known that 𝜇 is mean on 𝑋 if and only if inf_𝑠∈𝑆𝑓(𝑠)≤𝜇𝑓≤sup_𝑠∈𝑆𝑓(𝑠) (73) for each 𝑓∈𝑋. Each mean on 𝑋 is the weak*-limit of convex combination of point evaluations.

Let 𝑋 be a translation invariant subspace of 𝐵(𝑆) (i.e., 𝑙_𝑠𝑋⊂𝑋 and 𝑟_𝑠𝑋⊂𝑋 for each 𝑠∈𝑆) containing 1. Then a mean 𝜇 on 𝑋 is said to be left invariant (resp., right invariant) if 𝜇𝑙_𝑠𝑓=𝜇𝑓resp.,𝜇𝑟_𝑠𝑓=𝜇𝑓, (74) for each 𝑠∈𝑆 and 𝑓∈𝑋. A mean 𝜇 on 𝑋 is said to be invariant if 𝜇 is both left and right invariant ([30–34]). 𝑋 is said to be left (resp., right) amenable if 𝑋 has a left (resp., right) invariant mean. 𝑋 is amenable if 𝑋 is left and right amenable. In this case, we say that the semigroup 𝑆 is an amenable semigroup (see [35, 36]). Moreover, 𝐵(𝑆) is amenable when 𝑆 is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable.

A net {𝜇_𝛼} of means on 𝑋 is said to be asymptotically left (resp., right) invariant if lim_𝛼𝜇_𝛼𝑙_𝑠𝑓−𝜇_𝛼𝑓=0resp.,lim_𝛼𝜇_𝛼𝑟_𝑠𝑓−𝜇_𝛼𝑓=0, (75) for each 𝑓∈𝑋 and 𝑠∈𝑆, and it is said to be left (resp., right) strongly asymptotically invariant (or strong regular) if lim_𝛼𝑙^∗_𝑠𝜇_𝛼−𝜇_𝛼=0resp.,lim_𝛼𝑟^∗_𝑠𝜇_𝛼−𝜇_𝛼=0, (76) for each 𝑠∈𝑆, where 𝑙^∗_𝑠 and 𝑟^∗_𝑠 are the adjoint operators of 𝑙_𝑠 and 𝑟_𝑠, respectively. Such nets were first studied by Day in [35] where they were called weak* invariant and norm invariant, respectively.

It is easy to see that if a semigroup 𝑆 is left (resp., right) amenable, then the semigroup 𝑆^′=𝑆∪{𝑒}, where 𝑒𝑠^′=𝑠^′𝑒=𝑠^′ for all 𝑠^′∈𝑆 is also left (resp., right) amenable and conversely.

A semigroup 𝑆 is called left reversible if any two right ideals of 𝑆 have nonvoid intersection, that is, 𝑎𝑆∩𝑏𝑆≠∅ for 𝑎,𝑏∈𝑆. In this case, (𝑆,⪯) is a directed system when the binary relation “⪯” on 𝑆 is defined by 𝑎⪯𝑏 if and only if {𝑎}∪𝑎𝑆⊇{𝑏}∪𝑏𝑆 for 𝑎,𝑏∈𝑆. It is easy to see that 𝑡⪯𝑡𝑠 for all 𝑡,𝑠∈𝑆. Further, if 𝑡⪯𝑠, then 𝑝𝑡⪯𝑝𝑠 for all 𝑝∈𝑆. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup 𝑆 is left amenable, then 𝑆 is left reversible. But the converse is not true ([31, 37–41]).

Let 𝑆 be a semigroup and 𝐹(𝑇) denote the fixed point set of 𝑇. Then ℑ={𝑇_𝑠∶𝑠∈𝑆} is called a representation of 𝑆 if 𝑇_𝑒=𝐼 and 𝑇_𝑠𝑡=𝑇_𝑠𝑇_𝑡 for each 𝑠,𝑡∈𝑆. We denote by 𝐹(ℑ) the set of common fixed points of {𝑇_𝑠∶𝑠∈𝑆}, that is, 𝐹(ℑ)=_𝑠∈𝑆𝐹𝑇_𝑠=_𝑠∈𝑆𝑥∈𝐶∶𝑇_𝑠𝑥=𝑥. (77)

Open Problem 1. It will be interesting to obtain a generalization of both Theorems 7 and 11 to commutative, amenable, and reversible semigroups as in the case of Hilbert spaces or some Banach spaces (cf. [8, 30, 32, 42–45]).
For a real number 𝜅, a 𝐶𝐴𝑇(𝜅) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding triangle in a model space with curvature 𝜅.
For 𝜅=0, the 2-dimensional model space 𝑀²_𝜅=𝑀²₀ is the Euclidean space ℝ² with the metric induced from the Euclidean norm. For 𝜅>0,  𝑀²_𝜅 is the 2-dimensional sphere (1/√𝜅)𝕊² whose metric is length of a minimal great arc joining each two points. For 𝜅<0, 𝑀²_𝜅 is the 2-dimensional hyperbolic space (1/√−𝜅)ℍ² with the metric defined by a usual hyperbolic distance. For more details about the properties of 𝐶𝐴𝑇(𝜅) spaces, see [4, 46–48].

Open Problem 2. It will be interesting to obtain a generalization of both Theorems 7 and 11 to 𝐶𝐴𝑇(𝜅) space.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank Professor A. T.-M Lau and Professor J. K. Kim for their helpful suggestions. Also, special thanks are due to the referees for their deep insight which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).