Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces

Abstract

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the Δ-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

1. Introduction

Let (𝑋,𝑑) be a metric space and 𝐾 be a subset of 𝑋. A mapping 𝑇∶𝐾→𝑋 is said to be nonexpansive if 𝑑(𝑇𝑥,𝑇𝑦)≤𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝐾. We denote the set of all nonnegative elements in ℝ by ℝ⁺ and denote the set of all fixed points of 𝑇 by 𝐹(𝑇), that is, 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}. (1) For each 𝑛∈ℕ, let 𝑇_𝑛∶𝐾→𝐾 be nonexpansive mappings and denote the common fixed points set of the family {𝑇_𝑛} by ⋂^∞_𝑛=1𝐹(𝑇_𝑛). A family of mappings {𝑇_𝑛} is said to be uniformly asymptotically regular if, for any bounded subset 𝐵 of 𝐾, lim_𝑛→∞sup_𝑧∈𝐵𝑑𝑇_𝑛𝑧,𝑇_𝑖𝑇_𝑛𝑧=0, (2) for all 𝑖∈ℕ.

A nonexpansive semigroup is a family, Γ∶={𝑇(𝑡)∶𝑡≥0}, (3) of mappings 𝑇(𝑡) on 𝐾 such that (1)𝑇(𝑠+𝑡)𝑥=𝑇(𝑠)(𝑇(𝑡)𝑥) for all 𝑥∈𝐾 and 𝑠,𝑡≥0; (2)𝑇(𝑡)∶𝐾→𝐾 is nonexpansive for each 𝑡≥0; (3)for each 𝑥∈𝐾, the mapping 𝑇(⋅)𝑥 from ℝ⁺ to 𝐾 is continuous.

We denote by 𝐹(Γ) the common fixed points set of nonexpansive semigroup Γ, that is, 𝐹(Γ)=_𝑡∈ℝ+𝐹(𝑇(𝑡))={𝑥∈𝑋∶𝑇(𝑡)𝑥=𝑥foreach𝑡≥0}. (4) Note that, if 𝐾 is compact, then 𝐹(Γ) is nonempty (see [1, 2, 28]).

A geodesic from 𝑥 to 𝑦 in 𝑋 is a mapping Ψ from a closed interval [0,𝑙]⊂ℝ to 𝑋 such that Ψ(0)=𝑥, Ψ(𝑙)=𝑦, and 𝑑(Ψ(𝑡),Ψ(𝑡^′))=|𝑡−𝑡^′| for all 𝑡, 𝑡^′∈[0,𝑙]. In particular, Ψ is an isometry and 𝑑(𝑥,𝑦)=𝑙. The image Θ of Ψ is called a geodesic (or metric) segment joining 𝑥 and 𝑦. The space (𝑋,𝑑) is said to be a geodesic space if any 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 any 𝑥,𝑦∈𝑋, which is denoted by [𝑥,𝑦] and is called the segment joining 𝑥 and 𝑦. A subset 𝐾 of a geodesic space 𝑋 is said to be convex if for any 𝑥,𝑦∈𝐾, [𝑥,𝑦]⊂𝐾.

A geodesic triangle Δ(𝑥₁,𝑥₂,𝑥₃) in a geodesic metric space (𝑋,𝑑) consists of three points 𝑥₁, 𝑥₂, 𝑥₃ in 𝑋 (the vertices of Δ) and 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 all 𝑖,𝑗∈{1,2,3}. It is known that such a triangle always exists (see [3]). A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom (CA).

Let Δ be a geodesic triangle in (𝑋,𝑑), 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)

The complete CAT(0) spaces are often called, Hadamard spaces (see [4]). For any 𝑥,𝑦∈𝑋, we denote by 𝛼𝑥⊕(1−𝛼)𝑦 the unique point 𝑧∈[𝑥,𝑦] which satisfies 𝑑()𝑦=()𝑑,𝑑()𝑦.𝑥,𝛼𝑥⊕1−𝛼1−𝛼𝑥,𝑦𝑦,𝛼𝑥⊕1−𝛼=𝛼𝑑𝑥,𝑦 (6) It is known that if (𝑋,𝑑) is a CAT(0) space and 𝑥,𝑦∈𝑋, then for any 𝛽∈[0,1], there exists a unique point 𝛽𝑥⊕(1−𝛽)𝑦∈[𝑥,𝑦]. For any 𝑧∈𝑋, the following inequality holds: 𝑑𝑧,𝛽𝑥⊕1−𝛽𝑦≤𝛽𝑑(𝑧,𝑥)+1−𝛽𝑑𝑧,𝑦, (7) where 𝛽𝑥⊕(1−𝛽)𝑦∈[𝑥,𝑦] (for metric spaces of hyperbolic type, see [5]).

Recently, Cho et al. [6] proved the strong convergence of an explicit iterative sequence {𝑧_𝑛} in a CAT(0) space, where {𝑧_𝑛} is generated by the following iterative scheme for a nonexpansive semigroup Γ: 𝑧₁∈𝐾,𝑧_𝑛+1=𝛼𝑧_𝑛⊕(1−𝛼)𝑇𝑡_𝑛𝑧_𝑛,∀𝑛≥1, (8) where 𝛼∈(0,1) and {𝑡_𝑛}⊂ℝ⁺. The existence of fixed points, an invariant approximation, and convergence theorems for several mappings in CAT(0) spaces have been studied by many authors (see [7–18]).

On the other hand, Thong [19] considered an implicit Mann iteration process for a nonexpansive semigroup Γ={𝑇(𝑡)∶𝑡∈ℝ⁺} on a closed convex subset 𝐶 of a Banach space as follows: 𝑥₁∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛+1,𝑛≥1. (9) Under different conditions, Thong [19] proved the weak convergence and strong convergence results of the implicit Mann iteration scheme (9) for nonexpansive semigroups in certain Banach spaces. Many authors have studied the convergence of implicit iteration sequences for nonexpansive mappings, nonexpansive semigroups and pseudocontractive semigroups in the Banach spaces (see [20–23]). Readers may consult [24, 25] for the convergence of the Ishikawa iteration sequences for nonexpansive mappings and nonexpansive semigroups in certain Banach spaces. However, to our best knowledge, there is no paper to study the convergence of the implicit Ishikawa type iteration processes for nonexpansive semigroups in CAT(0) spaces. Therefore, it is of interest to investigate the convergence of implicit Ishikawa type iteration processes for nonexpansive semigroups in CAT(0) spaces under some suitable conditions.

Motivated and inspired by the work mentioned previously, we consider the following implicit Ishikawa iteration scheme for a family of nonexpansive mappings in a CAT(0) space: 𝑥₁𝑥∈𝐾,_𝑛+1=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1,∀𝑛≥1, (10) where {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] are given sequences. We prove that {𝑥_𝑛} generated by (10) is Δ-convergent to some point in ⋂^∞_𝑛=1𝐹(𝑇_𝑛) under appropriate conditions. We also consider the following implicit Ishikawa iteration process for a nonexpansive semigroup Γ={𝑇(𝑡)∶𝑡∈ℝ⁺} in a CAT(0) space: 𝑥₁𝑥∈𝐾,_𝑛+1=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇𝑡_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇𝑡_𝑛𝑥_𝑛+1,∀𝑛≥1, (11) where {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] are given sequences. Under various and appropriate conditions, we obtain that {𝑥_𝑛} generated by (11) converges strongly to a common fixed point of Γ. The results presented in this paper extend and generalize some previous results in [6, 19].

2. Definitions and Lemmas

Let {𝑥_𝑛} be a bounded sequence in a CAT(0) space (𝑋,𝑑). For any 𝑥∈𝑋, denote 𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛. (12)

Consider the following:(i)𝑟({𝑥_𝑛})=inf{𝑟(𝑥,𝑥_𝑛)∶𝑥∈𝑋} is called the asymptotic radius of {𝑥_𝑛};(ii)𝑟_𝐾({𝑥_𝑛})=inf{𝑟(𝑥,𝑥_𝑛)∶𝑥∈𝐾} is called the asymptotic radius of {𝑥_𝑛} with respect to 𝐾;(iii)the set 𝐴({𝑥_𝑛})={𝑥∈𝑋∶𝑟(𝑥,{𝑥_𝑛})=𝑟({𝑥_𝑛})} is called the asymptotic center of {𝑥_𝑛};(iv)the set 𝐴_𝐾({𝑥_𝑛})={𝑥∈𝐾∶𝑟(𝑥,{𝑥_𝑛})=𝑟_𝐾({𝑥_𝑛})} is called the asymptotic center of {𝑥_𝑛} with respect to 𝐾.

Definition 1 (see [12, 26]). A sequence {𝑥_𝑛} in a CAT(0) space 𝑋 is said to be Δ-convergent to a point 𝑥 in 𝑋, if 𝑥 is the unique asymptotic center of {𝑥_𝑛𝑗} for all subsequences {𝑥_𝑛𝑗}⊆{𝑥_𝑛}. In this case, we write Δ-lim_𝑛→∞𝑥_𝑛=𝑥, and 𝑥 is called the Δ-limit of {𝑥_𝑛}.

For the sake of convenience, we restate the following lemmas that shall be used.

Lemma 2 (see [10]). Let (𝑋,𝑑) be a CAT(0) space. Then, 𝑑(1−𝑡)𝑥⊕𝑡𝑦,𝑧≤(1−𝑡)𝑑(𝑥,𝑧)+𝑡𝑑𝑦,𝑧, (13) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1].

Lemma 3 (see [10]). Let (𝑋,𝑑) be a CAT(0) space. Then, 𝑑(1−𝑡)𝑥⊕𝑡𝑦,𝑧²≤(1−𝑡)[𝑑(𝑥,𝑧)]²+𝑡𝑑𝑦,𝑧²−𝑡(1−𝑡)𝑑𝑥,𝑦², (14) for all 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1].

Lemma 4 (see [10]). Let 𝐾 be a closed convex subset of a complete CAT(0) space and 𝑇∶𝐾→𝐾 be a nonexpansive mapping. Suppose that {𝑥_𝑛} is a bounded sequence in 𝐾 such that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0 and {𝑑(𝑥_𝑛,𝑝)} converges for all 𝑝∈𝐹(𝑇). Then, 𝜔_𝑤(𝑥_𝑛)=⋃𝐴({𝑥_𝑛𝑗})⊂𝐹(𝑇), where the union is taken over all subsequences {𝑥_𝑛𝑗} of {𝑥_𝑛}. Moreover, 𝜔_𝑤(𝑥_𝑛) consists of exactly one point.

Lemma 5 (see [6]). Let {𝑧_𝑛} and {𝑤_𝑛} be bounded sequences in a CAT(0) space 𝑋. Let {𝛼_𝑛} be a sequence in [0,1] such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Define 𝑧_𝑛+1=𝛼_𝑛𝑧_𝑛⊕(1−𝛼_𝑛)𝑤_𝑛 for all 𝑛∈ℕ and suppose that limsup_𝑛→∞𝑑𝑤_𝑛+1,𝑤_𝑛−𝑑𝑧_𝑛+1,𝑧_𝑛≤0. (15) Then, lim_𝑛→∞𝑑(𝑤_𝑛,𝑧_𝑛)=0.

3. Main Results

It is necessary for us to show that the implicit Ishikawa iteration sequences generated by schemes (10) and (11) are well defined, before providing the main results of this present paper.

Lemma 6. Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be nonexpansive mappings. Suppose that {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] are given parameter sequences. Then, the sequence {𝑥_𝑛} generated by the implicit Ishikawa iteration process (10) is well defined.

Proof. For each 𝑛∈ℕ and any given 𝑢∈𝐾, define a mapping 𝑆_𝑛+1∶𝐾→𝐾 by 𝑆_𝑛+1𝑥∶=𝛼_𝑛𝑢⊕1−𝛼_𝑛1−𝜃_𝑛𝑇_𝑛𝑢⊕𝜃_𝑛𝑇_𝑛𝑥,∀𝑛≥1. (16) It can be verified that for any fixed 𝑛∈ℕ, 𝑆_𝑛+1 is a contractive mapping. Indeed, if setting 𝑝_𝑛=(1−𝜃_𝑛)𝑇_𝑛𝑢⊕𝜃_𝑛𝑇_𝑛𝑥 and 𝑞_𝑛=(1−𝜃_𝑛)𝑇_𝑛𝑢⊕𝜃_𝑛𝑇_𝑛𝑦, then we have 𝑆_𝑛+1𝑥=𝛼_𝑛𝑢⊕(1−𝛼_𝑛)𝑝_𝑛 and 𝑆_𝑛+1𝑦=𝛼_𝑛𝑢⊕(1−𝛼_𝑛)𝑞_𝑛. It follows from Lemmas 3 and 2 that 𝑑𝑝_𝑛,𝑞_𝑛²=𝑑𝑝_𝑛,1−𝜃_𝑛𝑇_𝑛𝑢⊕𝜃_𝑛𝑇_𝑛𝑦²≤𝜃_𝑛𝑑𝑝_𝑛,𝑇_𝑛𝑦²+1−𝜃_𝑛𝑑𝑝_𝑛,𝑇_𝑛𝑢²−𝜃_𝑛1−𝜃_𝑛𝑑𝑇_𝑛𝑢,𝑇_𝑛𝑦²≤𝜃_𝑛1−𝜃_𝑛𝑑𝑇_𝑛𝑢,𝑇_𝑛𝑦²+𝜃_𝑛𝑑𝑇_𝑛𝑥,𝑇_𝑛𝑦²−𝜃_𝑛1−𝜃_𝑛𝑑𝑇_𝑛𝑢,𝑇_𝑛𝑥²+1−𝜃_𝑛𝜃²_𝑛𝑑𝑇_𝑛𝑢,𝑇_𝑛𝑥²−𝜃_𝑛1−𝜃_𝑛𝑑𝑇_𝑛𝑢,𝑇_𝑛𝑦²=𝜃²_𝑛𝑑𝑇_𝑛𝑥,𝑇_𝑛𝑦². (17) Consequently, 𝑑(𝑝_𝑛,𝑞_𝑛)≤𝜃_𝑛𝑑(𝑇_𝑛𝑥,𝑇_𝑛𝑦)≤𝜃_𝑛𝑑(𝑥,𝑦), and thus, 𝑑𝑆_𝑛+1𝑥,𝑆_𝑛+1𝑦≤1−𝛼_𝑛𝑑𝑝_𝑛,𝑞_𝑛≤1−𝛼_𝑛𝜃_𝑛𝑑𝑥,𝑦, (18) which shows that for each 𝑛∈ℕ, 𝑆_𝑛+1 is a contractive mapping. By induction, Banach’s fixed theorem yields that the sequence {𝑥_𝑛} generated by (10) is well defined. This completes the proof.

We need the following lemma for our main results. The analogs of [6, Lemma 3.1] and [27, Lemma 2.2] are given in what follows. We sketch the proof here for the convenience of the reader.

Lemma 7. Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be nonexpansive mappings. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Suppose that {𝑥_𝑛} generated by (10) is bounded and either lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1=0𝑜𝑟lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛,𝑇_𝑛𝑥_𝑛=0 (19) holds. If lim_𝑛→∞𝜃_𝑛=0, then lim_𝑛→∞𝑑(𝑇_𝑛𝑥_𝑛,𝑥_𝑛)=0.

Proof. First, we show that the boundedness of {𝑥_𝑛} implies the boundedness of {𝑇_𝑛𝑥_𝑛}. If {𝑥_𝑛} is bounded, then set 𝑀=sup𝑑𝑥_𝑛,𝑥∶𝑛∈ℕ<+∞, (20) for some given point 𝑥∈𝑋 and 𝛼=liminf_𝑛→∞𝛼_𝑛>0, 𝛽=limsup_𝑛→∞𝛼_𝑛<1. With 0<𝑎<𝛼≤𝛽<𝑏<1, there exists 𝑛₁∈ℕ such that for all 𝑛≥𝑛₁, 0<𝑎<𝛼_𝑛<𝑏<1. (21) It follows from Lemma 2 that 𝑑𝑇_𝑛𝑥_𝑛𝑇,𝑥≤𝑑_𝑛𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1𝑇,𝑥=𝑑_𝑛𝑥_𝑛,𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥≤𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥≤𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛+1−𝛼_𝑛𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥≤𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥+𝛼_𝑛𝑑𝑥_𝑛,𝑥+1−𝛼_𝑛𝜃_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥≤𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥+𝛼_𝑛+1−𝛼_𝑛𝜃_𝑛𝑑𝑥_𝑛+,𝑥1+1−𝛼_𝑛𝜃_𝑛𝑑𝑥_𝑛+1,𝑥. (22) Hence, for all 𝑛≥𝑛₁, from (21), we have 𝑑𝑇_𝑛𝑥_𝑛≤1,𝑥1−𝛼_𝑛2𝑑𝑥_𝑛,𝑥+2𝑑𝑥_𝑛+1≤4,𝑥1−𝑏𝑀, (23) which means that {𝑇_𝑛𝑥_𝑛} is bounded.
Next, we prove the conclusion of Lemma 7. If lim_𝑛→∞𝑑(𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1)=0, then we have limsup_𝑛→∞𝑑1−𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+1⊕𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+2,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝑑1−𝜃_𝑛+1×𝑇_𝑛+1𝑥_𝑛+1⊕𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+2,𝑇_𝑛+1𝑥_𝑛+1+𝑑𝑇_𝑛+1𝑥_𝑛+1,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝜃_𝑛+1𝑑𝑇_𝑛+1𝑥_𝑛+2,𝑇_𝑛+1𝑥_𝑛+1+𝑑𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1+𝑑𝑇_𝑛𝑥_𝑛+1,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝜃_𝑛+1𝑑𝑥_𝑛+2,𝑥_𝑛+1+1−𝜃_𝑛𝑑𝑥_𝑛+1,𝑥_𝑛−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝜃_𝑛+1+𝜃_𝑛𝑑𝑥_𝑛+2,𝑥+2𝑑𝑥_𝑛+1𝑥,𝑥+𝑑_𝑛,𝑥≤limsup_𝑛→∞4𝑀𝜃_𝑛+1+𝜃_𝑛=0. (24) Similarly, if lim_𝑛→∞𝑑(𝑇_𝑛+1𝑥_𝑛,𝑇_𝑛𝑥_𝑛)=0, then we have limsup_𝑛→∞𝑑1−𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+1⊕𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+2,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝑑1−𝜃_𝑛+1×𝑇_𝑛+1𝑥_𝑛+1⊕𝜃_𝑛+1𝑇_𝑛+1𝑥_𝑛+2,𝑇_𝑛+1𝑥_𝑛+1+𝑑𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛+1𝑥_𝑛+𝑑𝑇_𝑛+1𝑥_𝑛,𝑇_𝑛𝑥_𝑛+𝑑𝑇_𝑛𝑥_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1−𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝜃_𝑛+1𝑑𝑇_𝑛+1𝑥_𝑛+2,𝑇_𝑛+1𝑥_𝑛+1+𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥_𝑛+1≤limsup_𝑛→∞𝜃_𝑛+1𝑑𝑥_𝑛+2,𝑥_𝑛+1+𝜃_𝑛𝑑𝑥_𝑛+1,𝑥_𝑛≤limsup_𝑛→∞𝜃_𝑛+1+𝜃_𝑛𝑑𝑥_𝑛+2,𝑥+2𝑑𝑥_𝑛+1𝑥,𝑥+𝑑_𝑛,𝑥≤limsup_𝑛→∞4𝑀𝜃_𝑛+1+𝜃_𝑛=0. (25) It follows from Lemma 5 that lim_𝑛→∞𝑑((1−𝜃_𝑛)𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1,𝑥_𝑛)=0. Since 𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+𝑑1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+𝜃_𝑛𝑑𝑥_𝑛+1,𝑥+𝑑𝑥_𝑛𝑥,𝑥≤𝑑_𝑛,1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1+2𝑀𝜃_𝑛⟶0, (26) we obtain that lim_𝑛→∞𝑑(𝑇_𝑛𝑥_𝑛,𝑥_𝑛)=0. This completes the proof.

As a direct consequence of Lemma 7, the following lemma is immediate.

Lemma 8. Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be nonexpansive mappings. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Suppose that {𝑥_𝑛} generated by (10) is bounded and lim_𝑛→∞sup_𝑥∈𝐾𝑑𝑇_𝑛+1𝑥,𝑇_𝑛𝑥=0 (27) holds. If lim_𝑛→∞𝜃_𝑛=0, then lim_𝑛→∞𝑑(𝑇_𝑛𝑥_𝑛,𝑥_𝑛)=0.

We now present our main results in this paper. The following theorem discusses the Δ-convergence of the implicit Ishikawa iteration sequence (10) for a family of nonexpansive mappings in CAT(0) spaces.

Theorem 9. Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be uniformly asymptotically regular and nonexpansive mappings such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Then, the sequence {𝑥_𝑛} generated by (10) is well defined. Suppose that either lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛,𝑇_𝑛𝑥_𝑛=0𝑜𝑟lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1=0 (28) holds. If lim_𝑛→∞𝜃_𝑛=0, then {𝑥_𝑛}Δ-converges to some point in ⋂^∞_𝑛=1𝐹(𝑇_𝑛).

Proof. By Lemma 6, we know that the sequence {𝑥_𝑛} generated by (10) is well defined. For any 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛), from (10) and Lemma 2, we have 𝑑𝑥_𝑛+1𝛼,𝑝=𝑑_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1,𝑝≤𝛼_𝑛𝑑𝑥_𝑛,𝑝+1−𝛼_𝑛×1−𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑝+𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛+1,𝑝≤𝛼_𝑛𝑑𝑥_𝑛,𝑝+1−𝛼_𝑛1−𝜃_𝑛𝑑𝑥_𝑛+,𝑝1−𝛼_𝑛𝜃_𝑛𝑑𝑥_𝑛+1,𝑝. (29) Since 𝛼_𝑛+(1−𝛼_𝑛)(1−𝜃_𝑛)=1−(1−𝛼_𝑛)𝜃_𝑛, it follows that 𝑑𝑥_𝑛+1,𝑝≤𝑑𝑥_𝑛,𝑝. (30) Consequently, {𝑑(𝑥_𝑛,𝑝)} converges, and {𝑥_𝑛} is thus bounded.
It follows from (10) and (21) that for sufficiently large 𝑛∈ℕ, 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇_𝑛𝑥_𝑛+1,𝑥_𝑛≤1−𝛼_𝑛1−𝜃_𝑛𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛+1−𝛼_𝑛𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛+1,𝑥_𝑛≤1−𝛼_𝑛𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1−𝛼_𝑛𝜃_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛+𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛≤(1−𝑎)𝑑𝑥_𝑛+1,𝑥_𝑛+2𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛. (31) Applying Lemma 7, we have lim_𝑛→∞𝑑(𝑥_𝑛,𝑇_𝑛𝑥_𝑛)=0. Hence, 𝑑𝑥_𝑛+1,𝑥_𝑛≤2𝑎𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛⟶0, (32) and thus, 𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛+1,𝑥_𝑛+𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛⟶0. (33)
We prove that for each 𝑖∈ℕ, lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑇_𝑖𝑥_𝑛+1)=0. Since 𝑑𝑇_𝑖𝑥_𝑛+1,𝑇_𝑖𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛, (34) we know that lim_𝑛→∞𝑑(𝑇_𝑖𝑥_𝑛+1,𝑇_𝑖(𝑇_𝑛𝑥_𝑛))=0. Because the family of nonexpansive mappings {𝑇_𝑖} is uniformly asymptotically regular, we have 𝑑𝑥_𝑛+1,𝑇_𝑖𝑥_𝑛+1≤𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+𝑑𝑇_𝑛𝑥_𝑛,𝑇_𝑖𝑇_𝑛𝑥_𝑛+𝑑𝑇_𝑖𝑇_𝑛𝑥_𝑛,𝑇_𝑖𝑥_𝑛+1≤2𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+sup_{𝑥𝑧∈𝑛}𝑑𝑇_𝑛𝑧,𝑇_𝑖𝑇_𝑛𝑧⟶0. (35)
Since {𝑑(𝑥_𝑛,𝑝)} converges for any 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛), an application of Lemma 4 yields that 𝜔_𝑤(𝑥_𝑛) consists of exactly one point and is contained in 𝐹(𝑇_𝑖), for all 𝑖∈ℕ. This shows that {𝑥_𝑛}Δ-converges to some point in ⋂^∞_𝑛=1𝐹(𝑇_𝑛). This completes the proof.

In the special case where 𝜃_𝑛≡0, from Theorem 9, we have the following corollary.

Corollary 10 (see [6, Theorem 3.4]). Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be uniformly asymptotically regular and nonexpansive mappings such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Let {𝛼_𝑛}⊂(0,1] be a given sequence of real numbers such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Let {𝑥_𝑛} be a sequence defined by 𝑥₁∈𝐾,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,∀𝑛≥1. (36) Suppose that either lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛,𝑇_𝑛𝑥_𝑛=0𝑜𝑟lim_𝑛→∞𝑑𝑇_𝑛+1𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+1=0 (37) holds. Then, {𝑥_𝑛}Δ-converges to some point in ⋂^∞_𝑛=1𝐹(𝑇_𝑛).

Remark 11. Theorem 9 extends and improves [6, Theorem 3.4] from the explicit Mann iteration schemes to the implicit Ishikawa iteration schemes.

By Lemma 8 and Theorem 9, the following theorem holds trivially.

Theorem 12. Let 𝐾 be a nonempty, closed, and convex subset of a complete CAT(0) space 𝑋 and 𝑇_𝑛∶𝐾→𝐾 be uniformly asymptotically regular and nonexpansive mappings such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Then, the sequence {𝑥_𝑛} generated by (10) is well defined. Suppose that lim_𝑛→∞sup_𝑥∈𝐾𝑑𝑇_𝑛+1𝑥,𝑇_𝑛𝑥=0 (38) holds. If lim_𝑛→∞𝜃_𝑛=0, then {𝑥_𝑛}Δ-converges to some point in ⋂^∞_𝑛=1𝐹(𝑇_𝑛).

Finally, we study the strong convergence of the implicit Ishikawa iteration sequence (11) for nonexpansive semigroups in CAT(0) spaces, under various and appropriate conditions.

Theorem 13. Let 𝐶 be a compact convex subset of a complete CAT(0) space and Γ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup on 𝐶. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences of real numbers such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. Then, the sequence {𝑥_𝑛} generated by the implicit Ishikawa iteration process (11) is well defined. Suppose that {𝑡_𝑛} is a sequence in ℝ⁺ such that liminf_𝑛→∞𝑡_𝑛<limsup_𝑛→∞𝑡_𝑛,lim_𝑛→∞𝑡_𝑛+1−𝑡_𝑛=0. (39) If lim_𝑛→∞𝜃_𝑛=0, then {𝑥_𝑛} converges strongly to some point in 𝐹(Γ).

Proof. It is known that 𝐹(Γ) is nonempty (see [1, 2, 28]). From Lemma 6, we know that the sequence {𝑥_𝑛} generated by (11) is well defined. Then, we show that lim_𝑛→∞sup_𝑥∈𝐶𝑑𝑇𝑡_𝑛+1𝑥,𝑇𝑡_𝑛𝑥=0. (40) Assume for the contrary that (40) does not hold. There exist a subsequence {𝑡_𝑛𝑘}⊂{𝑡_𝑛}, a sequence {𝑦_𝑘}⊂𝐶, and an 𝜂>0 such that for all 𝑘∈ℕ, 𝑑𝑇𝑡_𝑛𝑘+1𝑦_𝑘,𝑇𝑡_𝑛𝑘𝑦_𝑘≥𝜂. (41) Since 𝐶 is compact, there exists a convergent subsequence contained in {𝑦_𝑘}. Without loss of generality, we assume that lim_𝑘→∞𝑦_𝑘=𝑦 with 𝑦∈𝐶. Consequently, 0<𝜂≤limsup_𝑘→∞𝑑𝑇𝑡_𝑛𝑘+1𝑦_𝑘,𝑇𝑡_𝑛𝑘𝑦_𝑘≤limsup_𝑘→∞𝑑𝑇𝑡_𝑛𝑘+1−𝑡_𝑛𝑘𝑦_𝑘,𝑇(0)𝑦_𝑘≤limsup_𝑘→∞𝑑𝑇𝑡_𝑛𝑘+1−𝑡_𝑛𝑘𝑦_𝑘,𝑇𝑡_𝑛𝑘+1−𝑡_𝑛𝑘𝑦+𝑑𝑇𝑡_𝑛𝑘+1−𝑡_𝑛𝑘(0)𝑦𝑇(0)(0)𝑦𝑦,𝑇+𝑑𝑦,𝑇_𝑘≤limsup_𝑘→∞2𝑑𝑦_𝑘,𝑦+𝑑𝑇𝑡_𝑛𝑘+1−𝑡_𝑛𝑘𝑦,𝑇(0)𝑦=0, (42) which is a contradiction. Formula (40) follows readily. Now, Lemma 8 yields that lim_𝑛→∞𝑑𝑇𝑡_𝑛𝑥_𝑛,𝑥_𝑛=0. (43)
Similar to the proof of [6, Theorem 3.5], it is easy to see that there exists a subsequence {𝑥_𝑛𝑗} which converges to 𝑥^∗, where 𝑥^∗ is a common fixed point in 𝐹(Γ). Since 𝑥^∗ is a cluster of {𝑥_𝑛}, we have liminf_𝑛→∞𝑑(𝑥_𝑛,𝑥^∗)=0. It follows from (11) and (30) that lim_𝑛→∞𝑑(𝑥_𝑛,𝑥^∗) exists. Hence, we obtain lim_𝑛→∞𝑑(𝑥_𝑛,𝑥^∗)=0, which completes the proof.

Remark 14. The proof of Theorem 13 is an analog of [6, Theorem 3.5]. If 𝛼_𝑛≡𝜆∈(0,1) and 𝜃_𝑛≡0, then Theorem 13 reduces to [6, Theorem 3.5]. Therefore, Theorem 13 extends and generalizes [6, Theorem 3.5] from the explicit Mann iteration processes to the implicit Ishikawa iteration processes.

We prove another strong convergence theorem which differs from Theorem 13.

Theorem 15. Let 𝐶 be a compact convex subset of a complete CAT(0) space and Γ={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup on 𝐶. Let {𝛼_𝑛}⊂(0,1] and {𝜃_𝑛}⊂[0,1] be given sequences. Then, the sequence {𝑥_𝑛} generated by the implicit Ishikawa iteration process (11) is well defined. Moreover, if lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝛼_𝑛+1−𝜃_𝑛𝑡_𝑛=0, (44) then {𝑥_𝑛} converges strongly to a common fixed point 𝑥^∗ of Γ.

Proof. It is known that 𝐹(Γ) is nonempty (see [1, 2, 28]). From Lemma 6, we know that {𝑥_𝑛} generated by (11) is well defined.
Claim 1. If {𝑟_𝑛} is a sequence of nonnegative real numbers such that lim_𝑛→∞𝑟_𝑛=0, then lim_𝑛→∞sup_𝑥∈𝐶𝑑𝑇𝑟_𝑛𝑥,𝑇(0)𝑥=0. (45) Assume for the contrary that (45) does not hold. There exist a subsequence {𝑟_𝑛𝑘}⊂{𝑟_𝑛}, a sequence {𝑦_𝑘}⊂𝐶, and an 𝜂>0 such that for all 𝑘∈ℕ, 𝑑𝑇𝑟_𝑛𝑘𝑦_𝑘,𝑇(0)𝑦_𝑘≥𝜂. (46) Since 𝐶 is compact, there exists a convergent subsequence of {𝑦_𝑘}. Without loss of generality, we assume that lim_𝑛→∞𝑦_𝑘=𝑦 with 𝑦∈𝐶. Consequently, 0<𝜂≤limsup_𝑘→∞𝑑𝑇𝑟_𝑛𝑘𝑦_𝑘,𝑇(0)𝑦_𝑘≤limsup_𝑘→∞𝑑𝑇𝑟_𝑛𝑘𝑦_𝑘,𝑇𝑟_𝑛𝑘𝑦+𝑑𝑇𝑟_𝑛𝑘𝑦,𝑇(0)𝑦+𝑑𝑇(0)𝑦,𝑇(0)𝑦_𝑘≤limsup_𝑘→∞2𝑑𝑦_𝑘,𝑦+𝑑𝑇𝑟_𝑛𝑘𝑦,𝑇(0)𝑦=0, (47) which is a contradiction. Formula (45) follows readily.
Claim 2. Consider that lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑇(𝑡)𝑥_𝑛+1)=0. Since 𝐶 is a compact convex subset of 𝑋, there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑥^∗ as 𝑗→∞. It follows from (11) and Lemma 2 that 𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1=𝑑𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛1−𝜃_𝑛𝑇𝑡_𝑛𝑥_𝑛⊕𝜃_𝑛𝑇𝑡_𝑛𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1≤𝛼_𝑛𝑑𝑥_𝑛,𝑇𝑡_𝑛𝑥_𝑛+1+1−𝛼_𝑛1−𝜃_𝑛𝑑𝑇𝑡_𝑛𝑥_𝑛,𝑇𝑡_𝑛𝑥_𝑛+1≤𝛼_𝑛𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1+𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1+1−𝜃_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1. (48) Hence, we have 𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1≤𝛼_𝑛+1−𝜃_𝑛1−𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1. (49) For any given 𝑥∈𝑋, let 𝑀=sup{𝑑(𝑥_𝑛,𝑥)∶𝑛∈ℕ}. Since 𝐶 is compact and lim_𝑛→∞𝛼_𝑛=0, we know that 𝑀<+∞ and 𝛼_𝑛≤2/3 for sufficiently large 𝑛∈ℕ. Consequently, 𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1≤3𝛼_𝑛+1−𝜃_𝑛𝑑𝑥_𝑛,𝑥+𝑑𝑥_𝑛+1𝛼,𝑥≤6𝑀_𝑛+1−𝜃_𝑛⟶0. (50) Thus, from (45) we get limsup_𝑛→∞𝑑𝑇(0)𝑥_𝑛+1,𝑥_𝑛+1≤limsup_𝑛→∞𝑑𝑇(0)𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1+𝑑𝑇𝑡_𝑛𝑥_𝑛+1,𝑥_𝑛+1≤lim_𝑛→∞sup_𝑥∈𝐶𝑑𝑇(0)𝑥,𝑇𝑡_𝑛𝑥+limsup_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1=0. (51) For any given 𝑡>0, it follows from (50) that 𝑑𝑇(0)𝑥_𝑛+1,𝑇(𝑡)𝑥_𝑛+1≤^{[𝑡/𝑡𝑛]−1}_𝑘=0𝑑𝑇𝑘𝑡_𝑛𝑥_𝑛+1,𝑇(𝑘+1)𝑡_𝑛𝑥_𝑛+1𝑇𝑡+𝑑𝑡_𝑛𝑡_𝑛𝑥_𝑛+1,𝑇(𝑡)𝑥_𝑛+1≤𝑡𝑡_𝑛𝑑𝑥_𝑛+1,𝑇𝑡_𝑛𝑥_𝑛+1𝑇𝑡+𝑑𝑡−𝑡_𝑛𝑡_𝑛𝑥_𝑛+1,𝑥_𝑛+1𝛼≤6𝑀𝑡^𝑛+1−𝜃_𝑛𝑡_𝑛+max𝑑𝑇(𝑠)𝑥_𝑛+1,𝑥_𝑛+1∶0≤𝑠≤𝑡_𝑛𝛼=6𝑀𝑡^𝑛+1−𝜃_𝑛𝑡_𝑛+𝑑𝑇𝑠_𝑛𝑥_𝑛+1,𝑥_𝑛+1say𝑠_𝑛∈0,𝑡_𝑛𝛼≤6𝑀𝑡^𝑛+1−𝜃_𝑛𝑡_𝑛+𝑑𝑇𝑠_𝑛𝑥_𝑛+1,𝑇(0)𝑥_𝑛+1+𝑑𝑇(0)𝑥_𝑛+1,𝑥_𝑛+1𝛼≤6𝑀𝑡^𝑛+1−𝜃_𝑛𝑡_𝑛+𝑑𝑇(0)𝑥_𝑛+1,𝑥_𝑛+1+sup_𝑥∈𝐶𝑑𝑇𝑠_𝑛𝑥,𝑇(0)𝑥, (52) where [𝑡/𝑡_𝑛] is the integer part of 𝑡/𝑡_𝑛. Since lim_𝑛→∞((𝛼_𝑛+1−𝜃_𝑛)/𝑡_𝑛)=0, it follows from (45) and (51) that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇(𝑡)𝑥_𝑛+1=0. (53) Therefore, from the previous formula, we know that 𝑥^∗∈𝐹(Γ). Iteration process (11) and inequality (30) imply that {𝑥_𝑛} converges strongly to 𝑥^∗. This completes the proof.

Remark 16. If 𝑋 is a Banach space, the notation “𝛽𝑥⊕(1−𝛽)𝑦” with 𝛽∈[0,1] is replaced by “𝛽𝑥+(1−𝛽)𝑦” and 𝜃_𝑛≡1, then Theorem 15 reduces to [19, Theorem 2.3]. Therefore, Theorem 15 extends and generalizes [19, Theorem 2.3] from the implicit Mann iteration processes in the Banach spaces to the implicit Ishikawa iteration processes in CAT(0) spaces.

Remark 17. The results presented in this paper can be immediately applied to any CAT(𝑘) space with 𝑘≤0, because any CAT(𝑘) space is a CAT(𝑘^′) space for any 𝑘^′>𝑘 (see [3, 6]).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11101069 and 11026063) and the Open Fund (PLN1104 and PLN1102) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).