Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

Abstract

Let 𝐶 be a nonempty bounded closed convex subset of a complete CAT(0) space 𝑋. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on 𝐶 is nonempty closed and convex. We also show that, under some suitable conditions, the sequence {𝑥_𝑘}^∞_𝑘=1 defined by 𝑥_𝑘+1=(1−𝑡_𝑚𝑘)𝑥_𝑘⊕𝑡_𝑚𝑘𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑦_{(𝑚−1)𝑘}=(1−𝑡_{(𝑚−1)𝑘})𝑥_𝑘⊕𝑡_{(𝑚−1)𝑘}𝑇^𝑛𝑘_𝑚−1𝑦_{(𝑚−2)𝑘},𝑦_{(𝑚−2)𝑘}=(1−𝑡_{(𝑚−2)𝑘})𝑥_𝑘⊕𝑡_{(𝑚−2)𝑘}𝑇^𝑛𝑘_𝑚−2𝑦_{(𝑚−3)𝑘},…,𝑦_2𝑘=(1−𝑡_2𝑘)𝑥_𝑘⊕𝑡_2𝑘𝑇^𝑛𝑘₂𝑦_1𝑘,𝑦_1𝑘=(1−𝑡_1𝑘)𝑥_𝑘⊕𝑡_1𝑘𝑇^𝑛𝑘₁𝑦_0𝑘,𝑦_0𝑘=𝑥_𝑘,𝑘∈ℕ, converges to a common fixed point of 𝑇₁,𝑇₂,…,𝑇_𝑚 where they are asymptotic pointwise nonexpansive mappings on 𝐶, {𝑡_𝑖𝑘}^∞_𝑘=1 are sequences in [0,1] for all 𝑖=1,2,…,𝑚, and {𝑛_𝑘} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

1. Introduction

A mapping 𝑇 on a subset 𝐶 of a Banach space 𝑋 is said to be asymptotic pointwise nonexpansive if there exists a sequence of mappings 𝛼_𝑛∶𝐶→[0,∞) such that𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝛼_𝑛(𝑥)𝑥−𝑦, (1.1) where limsup_𝑛→∞𝛼_𝑛(𝑥)≤1, for all 𝑥,𝑦∈𝐶. This class of mappings was introduced by Kirk and Xu [1], where it was shown that if 𝐶 is a bounded closed convex subset of a uniformly convex Banach space 𝑋, then every asymptotic pointwise nonexpansive mapping 𝑇∶𝐶→𝐶 always has a fixed point. In 2009, Hussain and Khamsi [2] extended Kirk-Xu's result to the case of metric spaces, specifically to the so-called CAT(0) spaces. Recently, Kozlowski [3] defined an iterative sequence for an asymptotic pointwise nonexpansive mapping 𝑇∶𝐶→𝐶 by 𝑥₁∈𝐶 and𝑥_𝑘+1=1−𝑡_𝑘𝑥_𝑘+𝑡_𝑘𝑇^𝑛𝑘𝑦_𝑘,𝑦_𝑘=1−𝑠_𝑘𝑥_𝑘+𝑠_𝑘𝑇^𝑛𝑘𝑥_𝑘,𝑘∈ℕ, (1.2) where {𝑡_𝑘} and {𝑠_𝑘} are sequences in [0,1] and {𝑛_𝑘} is an increasing sequence of natural numbers. He proved, under some suitable assumptions, that the sequence {𝑥_𝑘} defined by (1.2) converges weakly to a fixed point of 𝑇 where 𝑋 is a uniformly convex Banach space which satisfies the Opial condition and {𝑥_𝑘} converges strongly to a fixed point of 𝑇 provided 𝑇^𝑟 is a compact mapping for some 𝑟∈ℕ. On the other hand, Khan et al. [4] studied the iterative process defined by𝑥_𝑛+1=1−𝛼_𝑚𝑛𝑥_𝑛+𝛼_𝑚𝑛𝑇^𝑛_𝑚𝑦_{(𝑚−1)𝑛},𝑦_{(𝑚−1)𝑛}=1−𝛼_{(𝑚−1)𝑛}𝑥_𝑛+𝛼_{(𝑚−1)𝑛}𝑇^𝑛_𝑚−1𝑦_{(𝑚−2)𝑛},𝑦_{(𝑚−2)𝑛}=1−𝛼_{(𝑚−2)𝑛}𝑥_𝑛+𝛼_{(𝑚−2)𝑛}𝑇^𝑛_𝑚−2𝑦_{(𝑚−3)𝑛},⋮𝑦_2𝑛=1−𝛼_2𝑛𝑥_𝑛+𝛼_2𝑛𝑇^𝑛₂𝑦_1𝑛,𝑦_1𝑛=1−𝛼_1𝑛𝑥_𝑛+𝛼_1𝑛𝑇^𝑛₁𝑦_0𝑛,𝑦_0𝑛=𝑥_𝑛,𝑛∈ℕ, (1.3) where 𝑇₁,…,𝑇_𝑚 are asymptotically quasi-nonexpansive mappings on 𝐶 and {𝛼_𝑖𝑛}^∞_𝑛=1 are sequences in [0,1] for all 𝑖=1,2,…,𝑚.

In this paper, motivated by the results mentioned above, we ensure the existence of common fixed points for a family of asymptotic pointwise nonexpansive mappings in a CAT(0) space. Furthermore, we obtain ▵ and strong convergence theorems of a sequence defined by𝑥_𝑘+1=1−𝑡_𝑚𝑘𝑥_𝑘⊕𝑡_𝑚𝑘𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑦_{(𝑚−1)𝑘}=1−𝑡_{(𝑚−1)𝑘}𝑥_𝑘⊕𝑡_{(𝑚−1)𝑘}𝑇^𝑛𝑘_𝑚−1𝑦_{(𝑚−2)𝑘},𝑦_{(𝑚−2)𝑘}=1−𝑡_{(𝑚−2)𝑘}𝑥_𝑘⊕𝑡_{(𝑚−2)𝑘}𝑇^𝑛𝑘_𝑚−2𝑦_{(𝑚−3)𝑘},⋮𝑦_2𝑘=1−𝑡_2𝑘𝑥_𝑘⊕𝑡_2𝑘𝑇^𝑛𝑘₂𝑦_1𝑘,𝑦_1𝑘=1−𝑡_1𝑘𝑥_𝑘⊕𝑡_1𝑘𝑇^𝑛𝑘₁𝑦_0𝑘,𝑦_0𝑘=𝑥_𝑘,𝑘∈ℕ, (1.4) where 𝑇₁,…,𝑇_𝑚 are asymptotic pointwise nonexpansive mappings on a subset 𝐶 of a complete CAT(0) space and {𝑡_𝑖𝑘}^∞_𝑘=1 are sequences in [0,1] for all 𝑖=1,2,…,𝑚, and {𝑛_𝑘} is an increasing sequence of natural numbers. We also note that our method can be used to prove the analogous results for uniformly convex Banach spaces.

2. Preliminaries

A metric space 𝑋 is a CAT(0) space 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. It is well-known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [5]), ℝ-trees (see [6]), Euclidean buildings (see [7]), and the complex Hilbert ball with a hyperbolic metric (see [8]). For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [5].

Fixed point theory in CAT(0) spaces was first studied by Kirk (see [9, 10]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2, 11–22] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in ℝ-trees) can be applied to graph theory, biology, and computer science (see, e.g., [6, 23–26]).

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 map 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 ▵(𝑥₁,𝑥₂,x₃) in a geodesic space (𝑋,𝑑) consists of three points 𝑥₁,𝑥₂,𝑥₃ in 𝑋 (the vertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for geodesic triangle ▵(𝑥₁,𝑥₂,𝑥₃) in (𝑋,𝑑) is a triangle ▵(𝑥₁,𝑥₂,𝑥₃)∶=▵(𝑥₁,𝑥₂,𝑥₃) in the Euclidean plane 𝔼² such that 𝑑_𝔼2(𝑥_𝑖,𝑥_𝑗)=𝑑(𝑥_𝑖,𝑥_𝑗) for 𝑖,𝑗∈{1,2,3}.

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

CAT(0): Let ▵ be a geodesic triangle in 𝑋, and let ▵ be a comparison triangle for ▵. Then, ▵ is said to satisfy the CAT(0) inequality if for all 𝑥,𝑦∈▵ and all comparison points 𝑥,𝑦∈▵, 𝑑𝑥,𝑦≤𝑑_𝔼2𝑥,𝑦. (2.1)

Let 𝑥,𝑦∈𝑋, by Lemma  2.1(iv) of [14] for each 𝑡∈[0,1], there exists a unique point 𝑧∈[𝑥,𝑦] such that𝑑(𝑥,𝑧)=𝑡𝑑𝑥,𝑦,𝑑𝑦,𝑧=(1−𝑡)𝑑𝑥,𝑦. (2.2) We will use the notation (1−𝑡)𝑥⊕𝑡𝑦 for the unique point 𝑧 satisfying (2.2). We now collect some elementary facts about CAT(0) spaces.

Lemma 2.1. Let 𝑋 be a complete CAT(0) space.(i)[5, Proposition  2.4] If 𝐶 is a nonempty closed convex subset of 𝑋, then, for every 𝑥∈𝑋, there exists a unique point 𝑃(𝑥)∈𝐶 such that 𝑑(𝑥,𝑃(𝑥))=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐶}. Moreover, the map 𝑥↦𝑃(𝑥) is a nonexpansive retract from 𝑋 onto 𝐶.(ii)[14, Lemma  2.4] For 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], we have 𝑑(1−𝑡)𝑥⊕𝑡𝑦,𝑧≤(1−𝑡)𝑑(𝑥,𝑧)+𝑡𝑑𝑦,𝑧. (2.3)(iii)[14, Lemma  2.5] For 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], we have 𝑑(1−𝑡)𝑥⊕𝑡𝑦,𝑧²≤(1−𝑡)𝑑(𝑥,𝑧)²+𝑡𝑑𝑦,𝑧²−𝑡(1−𝑡)𝑑𝑥,𝑦². (2.4)

We now give the concept of Δ-convergence and collect some of its basic properties. Let {𝑥_𝑛} be a bounded sequence in a CAT(0) space 𝑋. For 𝑥∈𝑋, we set 𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛. (2.5) The asymptotic radius 𝑟({𝑥_𝑛}) of {𝑥_𝑛} is given by 𝑟𝑥_𝑛=inf𝑟𝑥,𝑥_𝑛∶𝑥∈𝑋, (2.6) and the asymptotic center 𝐴({𝑥_𝑛}) of {𝑥_𝑛} is the set 𝐴𝑥_𝑛=𝑥∈𝑋∶𝑟𝑥,𝑥_𝑛=𝑟𝑥_𝑛. (2.7)

It is known from Proposition  7 of [27] that, in a CAT(0) space, 𝐴({𝑥_𝑛}) consists of exactly one point.

Definition 2.2 (see [28, 29]). A sequence {𝑥_𝑛} in a CAT(0) space 𝑋 is said to Δ-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 Δ-lim_𝑛𝑥_𝑛=𝑥 and call 𝑥 the Δ-limit of {𝑥_𝑛}.

Lemma 2.3. Let 𝑋 be a complete CAT(0) space.(i)[28, page 3690] Every bounded sequence in 𝑋 has a Δ-convergent subsequence.(ii)[30, Proposition  2.1] If 𝐶 is a closed convex subset of a complete CAT(0) space and if {𝑥_𝑛} is a bounded sequence in 𝐶, then the asymptotic center of {𝑥_𝑛} is in 𝐶.(iii)[14, Lemma  2.8] If {𝑥_𝑛} is a bounded sequence in a complete CAT(0) space with 𝐴({𝑥_𝑛})={𝑥} and {𝑢_𝑛} is a subsequence of {𝑥_𝑛} with 𝐴({𝑢_𝑛})={𝑢} and the sequence {𝑑(𝑥_𝑛,𝑢)} converges, then 𝑥=𝑢.

Recall that a mapping 𝑇∶𝑋→𝑋 is said to be nonexpansive [31] if 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (2.8)where 𝑇 is called asymptotically nonexpansive [32] if there is a sequence {𝑘_𝑛} of positive numbers with the property lim_𝑛→∞𝑘_𝑛=1 and such that𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑘_𝑛𝑑𝑥,𝑦,∀𝑛≥1,𝑥,𝑦∈𝑋, (2.9)where 𝑇 is called an asymptotic pointwise nonexpansive mapping [1] if there exists a sequence of functions 𝛼_𝑛∶𝑋→[0,∞) such that𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝛼_𝑛(𝑥)𝑑𝑥,𝑦,∀𝑛≥1,𝑥,𝑦∈𝑋, (2.10) where limsup_𝑛→∞𝛼_𝑛(𝑥)≤1. The following implications hold. 𝑇isnonexpansive⟹𝑇isasymptoticallynonexpansive⟹𝑇isasymptoticpointwisenonexpansive. (2.11) A point 𝑥∈𝑋 is called a fixed point of 𝑇 if 𝑥=𝑇𝑥. We shall denote by 𝐹(𝑇) the set of fixed points of 𝑇. The existence of fixed points for asymptotic pointwise nonexpansive mappings in CAT(0) spaces was proved by Hussain and Khamsi [2] as the following result.

Theorem 2.4. Let 𝐶 be a nonempty bounded closed convex subset of a complete CAT(0) space 𝑋. Suppose that 𝑇∶𝐶→𝐶 is an asymptotic pointwise nonexpansive mapping. Then, 𝐹(𝑇) is nonempty closed and convex.

3. Existence Theorems

Let 𝑀 be a metric space and ℱ a family of subsets of 𝑀. Then, we say that ℱ defines a convexity structure on 𝑀 if it contains the closed balls and is stable by intersection.

Definition 3.1 (see [2]). Let ℱ be a convexity structure on 𝑀. We will say that ℱ is compact if any family {𝐴_𝛼}_𝛼∈Γ of elements of ℱ has a nonempty intersection provided ⋂_𝛼∈𝐹𝐴_𝛼≠∅ for any finite subset 𝐹⊂Γ.

Let 𝑋 be a complete CAT(0) space. We denote by 𝒞(𝑋) the family of all closed convex subsets of 𝑋. Then, 𝒞(𝑋) is a compact convexity structure on 𝑋 (see, e.g., [2]).

The following theorem is an extension of Theorem  4.3 in [33]. For an analog of this result in uniformly convex Banach spaces, see [34].

Theorem 3.2. Let 𝐶 be a nonempty bounded closed and convex subset of a complete CAT(0) space 𝑋. Then, for any commuting family 𝒮 of asymptotic pointwise nonexpansive mappings on 𝐶, the set ℱ(𝒮) of common fixed points of 𝒮 is nonempty closed and convex.

Proof. Let 𝒯 be the family of all finite intersections of the fixed point sets of mappings in the commutative family 𝒮. We first show that 𝒯 has the finite intersection property. Let 𝑇₁,𝑇₂,…,𝑇_𝑛∈𝒮. By Theorem 2.4, 𝐹(𝑇₁) is a nonempty closed and convex subset of 𝐶. We assume that 𝐴∶=⋂^𝑘−1_𝑗=1𝐹(𝑇_𝑗) is nonempty closed and convex for some 𝑘∈ℕ with 1<𝑘≤𝑛. For 𝑥∈𝐴 and 𝑗∈ℕ with 1≤𝑗<𝑘, we have 𝑇_𝑘(𝑥)=𝑇_𝑘∘𝑇_𝑗(𝑥)=𝑇_𝑗∘𝑇_𝑘(𝑥). (3.1) Thus, 𝑇_𝑘(𝑥) is a fixed point of 𝑇_𝑗, which implies that 𝑇_𝑘(𝑥)∈𝐴; therefore, 𝐴 is invariant under 𝑇_𝑘. Again, by Theorem 2.4, 𝑇_𝑘 has a fixed point in 𝐴, that is, ^𝑘_𝑗=1𝐹𝑇_𝑗=𝐹𝑇_𝑘𝐴≠∅. (3.2) By induction, ⋂^𝑛_𝑗=1𝐹(𝑇_𝑗)≠∅. Hence, 𝒯 has the finite intersection property. Since 𝒞(𝑋) is compact, ℱ(𝒮)=_𝑇∈𝒯𝑇≠∅. (3.3) Obviously, the set is closed and convex.

As a consequence of Lemma 2.1(i) and Theorem 3.2, we obtain an analog of Bruck's theorem [35].

Corollary 3.3. Let 𝐶 be a nonempty bounded closed and convex subset of a complete CAT(0) space 𝑋. Then, for any commuting family 𝒮 of nonexpansive mappings on 𝐶, the set ℱ(𝒮) of common fixed points of 𝒮 is a nonempty nonexpansive retract of 𝐶.

4. Convergence Theorems

Throughout this section, 𝑋 stands for a complete CAT(0) space. Let 𝐶 be a closed convex subset of 𝑋. We shall denote by 𝒯(𝐶) the class of all asymptotic pointwise nonexpansive mappings from 𝐶 into 𝐶. Let 𝑇₁,…,𝑇_𝑚∈𝒯(𝐶), without loss of generality, we can assume that there exists a sequence of mappings 𝛼_𝑛∶𝐶→[0,∞) such that for all 𝑥,𝑦∈𝐶,  𝑖=1,…,𝑚, and 𝑛∈ℕ, we have𝑑𝑇^𝑛_𝑖𝑥,𝑇^𝑛_𝑖𝑦≤𝛼_𝑛(𝑥)𝑑𝑥,𝑦,limsup_𝑛→∞𝛼_𝑛(𝑥)≤1. (4.1) Let 𝑎_𝑛(𝑥)=max{𝛼_𝑛(𝑥),1}. Again, without loss of generality, we can assume that𝑑𝑇^𝑛_𝑖𝑥,𝑇^𝑛_𝑖𝑦≤𝑎_𝑛(𝑥)𝑑𝑥,𝑦,lim_𝑛→∞𝑎_𝑛(𝑥)=1,𝑎_𝑛(𝑥)≥1, (4.2) for all 𝑥,𝑦∈𝐶,𝑖=1,…,𝑚, and 𝑛∈ℕ. We define 𝑏_𝑛(𝑥)=𝑎_𝑛(𝑥)−1, then, for each 𝑥∈𝐶, we have lim_𝑛→∞𝑏_𝑛(𝑥)=0.

The following definition is a mild modification of [3, Definition  2.3].

Definition 4.1. Define 𝒯_𝑟(𝐶) as a class of all 𝑇∈𝒯(𝐶) such that ^∞_𝑛=1sup_𝑥∈𝐶𝑏_𝑛(𝑥)<∞, (4.3)𝑎_𝑛isaboundedfunctionforevery𝑛∈ℕ.
Let 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶), and let {𝑡_𝑖𝑘}^∞_𝑘=1⊂(0,1) be bounded away from 0 and 1 for all 𝑖=1,2,…,𝑚, and {𝑛_𝑘} an increasing sequence of natural numbers. Let 𝑥₁∈𝐶, and define a sequence {𝑥_𝑘} in 𝐶 as

𝑥_𝑘+1=1−𝑡_𝑚𝑘𝑥_𝑘⊕𝑡_𝑚𝑘𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑦_{(𝑚−1)𝑘}=1−𝑡_{(𝑚−1)𝑘}𝑥_𝑘⊕𝑡_{(𝑚−1)𝑘}𝑇^𝑛𝑘_𝑚−1𝑦_{(𝑚−2)𝑘},𝑦_{(𝑚−2)𝑘}=1−𝑡_{(𝑚−2)𝑘}𝑥_𝑘⊕𝑡_{(𝑚−2)𝑘}𝑇^𝑛𝑘_𝑚−2𝑦_{(𝑚−3)𝑘},⋮𝑦_2𝑘=1−𝑡_2𝑘𝑥_𝑘⊕𝑡_2𝑘𝑇^𝑛𝑘₂𝑦_1𝑘,𝑦_1𝑘=1−𝑡_1𝑘𝑥_𝑘⊕𝑡_1𝑘𝑇^𝑛𝑘₁𝑦_0𝑘,𝑦_0𝑘=𝑥_𝑘,𝑘∈ℕ. (4.4) We say that the sequence {𝑥_𝑘} in (4.4) is well defined if limsup_𝑘→∞𝑎_𝑛𝑘(𝑥_𝑘)=1. As in [3], we observe that lim_𝑘→∞𝑎_𝑘(𝑥)=1 for every 𝑥∈𝐶. Hence, we can always choose a subsequence {𝑎_𝑛𝑘} which makes {𝑥_𝑘} well defined.

Lemma 4.2 (see [36, Lemma  2.2]). Let {𝑎_𝑛} and {𝑢_𝑛} be sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝑢_𝑛𝑎_𝑛,∀𝑛∈ℕ,^∞_𝑛=1𝑢_𝑛<∞. (4.5) Then, (i) lim_𝑛𝑎_𝑛 exists, (ii) if liminf_𝑛𝑎_𝑛=0, then lim_𝑛𝑎_𝑛=0.

Lemma 4.3 (see [37, 38]). Suppose {𝑡_𝑛} is a sequence in [𝑏,𝑐] for some 𝑏,𝑐∈(0,1) and {𝑢_𝑛}, {𝑣_𝑛} are sequences in 𝑋 such that limsup_𝑛𝑑(𝑢_𝑛,𝑤)≤𝑟,  limsup_𝑛𝑑(𝑣_𝑛,𝑤)≤𝑟, and lim_𝑛𝑑((1−𝑡_𝑛)𝑢_𝑛⊕𝑡_𝑛𝑣_𝑛,𝑤)=𝑟 for some 𝑟≥0. Then, lim_𝑛→∞𝑑𝑢_𝑛,𝑣_𝑛=0. (4.6)

Lemma 4.4. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶). Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝑎,𝑏]⊂(0,1) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Assume that 𝐹∶=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)≠∅. Then,(a)there exists a sequence {𝑣_𝑘} in [0,∞) such that ∑^∞_𝑘=1𝑣_𝑘<∞ and 𝑑(𝑥_𝑘+1,𝑝)≤(1+𝑣_𝑘)^𝑚𝑑(𝑥_𝑘,𝑝),   for all 𝑝∈𝐹 and all 𝑘∈ℕ,(b)there exists a constant 𝑀>0 such that 𝑑(𝑥_𝑘+𝑙,𝑝)≤𝑀𝑑(𝑥_𝑘,𝑝),   for all 𝑝∈𝐹 and 𝑘,𝑙∈ℕ.

Proof. (a) Let 𝑝∈𝐹 and 𝑣_𝑘=sup_𝑥∈𝐶𝑏_𝑛𝑘(𝑥) for all 𝑘∈ℕ. Since ∑^∞_𝑘=1sup_𝑥∈𝐶𝑏_𝑛𝑘(𝑥)<∞, we have ∑^∞_𝑘=1𝑣_𝑘<∞. Now, 𝑑𝑦_1𝑘,𝑝≤1−𝑡_1𝑘𝑑𝑥_𝑘,𝑝+𝑡_1𝑘𝑑𝑇^𝑛𝑘₁𝑥_𝑘≤,𝑝1−𝑡_1𝑘𝑑𝑥_𝑘,𝑝+𝑡_1𝑘1+𝑏_𝑛𝑘𝑝𝑑𝑥_𝑘=,𝑝1+𝑡_1𝑘𝑏_𝑛𝑘𝑝𝑑𝑥_𝑘≤,𝑝1+𝑣_𝑘𝑑𝑥_𝑘,𝑝. (4.7) Suppose that 𝑑(𝑦_𝑗𝑘,𝑝)≤(1+𝑣_𝑘)^𝑗𝑑(𝑥_𝑘,𝑝) holds for some 1≤𝑗≤𝑚−2. Then, 𝑑𝑦_(𝑗+1)𝑘,𝑝≤1−𝑡_(𝑗+1)𝑘𝑑𝑥_𝑘,𝑝+𝑡_(𝑗+1)𝑘𝑑𝑇^𝑛𝑘_𝑗+1𝑦_𝑗𝑘≤,𝑝1−𝑡_(𝑗+1)𝑘𝑑𝑥_𝑘,𝑝+𝑡_(𝑗+1)𝑘1+𝑏_𝑛𝑘𝑝𝑑𝑦_𝑗𝑘≤,𝑝1−𝑡_(𝑗+1)𝑘𝑑𝑥_𝑘,𝑝+𝑡_(𝑗+1)𝑘1+𝑣_𝑘^𝑗+1𝑑𝑥_𝑘=,𝑝1−𝑡_(𝑗+1)𝑘+𝑡_(𝑗+1)𝑘1+^𝑗+1_𝑟=1𝑗+1𝑗⋯𝑗+2−𝑟𝑣𝑟!^𝑟𝑘𝑑𝑥_𝑘=,𝑝1+𝑡_(𝑗+1)𝑘^𝑗+1_𝑟=1𝑗+1𝑗⋅⋅⋅𝑗+2−𝑟𝑣𝑟!^𝑟𝑘𝑑𝑥_𝑘≤,𝑝1+𝑣_𝑘^𝑗+1𝑑𝑥_𝑘,𝑝. (4.8) By induction, we have 𝑑𝑦_𝑖𝑘,𝑝≤1+𝑣_𝑘^𝑖𝑑𝑥_𝑘,𝑝,∀𝑖=1,2,…,𝑚−1. (4.9) This implies 𝑑𝑥_𝑘+1,𝑝≤1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘𝑑𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘}≤,𝑝1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘1+𝑏_𝑛𝑘𝑝𝑑𝑦_{(𝑚−1)𝑘}≤,𝑝1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘1+𝑣_𝑘1+𝑣_𝑘^𝑚−1𝑑𝑥_𝑘≤,𝑝1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘1+𝑣_𝑘^𝑚𝑑𝑥_𝑘=,𝑝1−𝑡_𝑚𝑘+𝑡_𝑚𝑘1+^𝑚_𝑟=1𝑚(𝑚−1)⋯(𝑚−𝑟+1)𝑣𝑟!^𝑟𝑘𝑑𝑥_𝑘=,𝑝1+𝑡^𝑚_{𝑚𝑘𝑟=1}𝑚(𝑚−1)⋅⋅⋅(𝑚−𝑟+1)𝑣𝑟!^𝑟𝑘𝑑𝑥_𝑘≤,𝑝1+𝑣_𝑘^𝑚𝑑𝑥_𝑘,𝑝. (4.10) This completes the proof of (a).
(b) We observe that (1+𝛼)^𝑛≤𝑒^𝑛𝛼 holds for all 𝑛∈ℕ and 𝛼≥0. Thus, by (a), for 𝑘,𝑙∈ℕ, we have 𝑑𝑥_𝑘+𝑙,𝑝≤1+𝑣_{𝑘+𝑙−1}^𝑚𝑑𝑥_{𝑘+𝑙−1},𝑝≤exp𝑚𝑣_{𝑘+𝑙−1}𝑑𝑥_{𝑘+𝑙−1},𝑝≤⋯≤exp𝑚^{𝑘+𝑙−1}_𝑖=1𝑣_𝑖𝑑𝑥_𝑘𝑚,𝑝≤exp^∞_𝑖=1𝑣_𝑖𝑑𝑥_𝑘,𝑝. (4.11) The proof is complete by setting 𝑀=exp{𝑚∑^∞_𝑖=1𝑣_𝑖}.

Theorem 4.5. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶). Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝑎,𝑏]⊂(0,1) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Assume that 𝐹≠∅. Then, {𝑥_𝑘} converges to some point in 𝐹 if and only if liminf_𝑘→∞𝑑(𝑥_𝑘,𝐹)=0, where 𝑑(𝑥,𝐹)=inf_𝑝∈𝐹𝑑(𝑥,𝑝).

Proof. The necessity is obvious. Now, we prove the sufficiency. From Lemma 4.4(a), we have 𝑑𝑥_𝑘+1,𝑝≤1+𝑣_𝑘^𝑚𝑑𝑥_𝑘,𝑝,∀𝑝∈𝐹,∀𝑘∈ℕ. (4.12) This implies 𝑑𝑥_𝑘+1,𝐹≤1+𝑣_𝑘^𝑚𝑑𝑥_𝑘,𝐹=1+^𝑚_𝑟=1𝑚(𝑚−1)⋅⋅⋅(𝑚−𝑟+1)𝑣𝑟!^𝑟𝑘𝑑𝑥_𝑘,𝐹. (4.13) Since ∑^∞_𝑘=1𝑣_𝑘<∞, then ∑^∞_𝑘=1∑^𝑚_𝑟=1(𝑚(𝑚−1)⋅⋅⋅(𝑚−𝑟+1)/𝑟!)𝑣^𝑟_𝑘<∞. By Lemma 4.2(ii), we get lim_𝑘→∞𝑑(𝑥_𝑘,𝐹)=0. Next, we show that {𝑥_𝑘} is a Cauchy sequence. From Lemma 4.4(b), there exists 𝑀>0 such that 𝑑𝑥_𝑘+𝑙,𝑝≤𝑀𝑑𝑥_𝑘,𝑝,∀𝑝∈𝐹,𝑘,𝑙∈ℕ. (4.14) Since lim_𝑘→∞𝑑(𝑥_𝑘,𝐹)=0, for each 𝜀>0, there exists 𝑘₁∈ℕ such that 𝑑𝑥_𝑘<𝜀,𝐹2𝑀,∀𝑘≥𝑘₁. (4.15) Hence, there exists 𝑧₁∈𝐹 such that 𝑑𝑥_𝑘1,𝑧₁<𝜀.2𝑀 (4.16) By (4.14) and (4.16), for 𝑘≥𝑘₁, we have 𝑑𝑥_𝑘+𝑙,𝑥_𝑘≤𝑑𝑥_𝑘+𝑙,𝑧₁+𝑑𝑥_𝑘,𝑧₁≤𝑀𝑑𝑥_𝑘1,𝑧₁+𝑀𝑑𝑥_𝑘1,𝑧₁𝜀<2𝑀2𝑀=𝜀. (4.17) This shows that {𝑥_𝑘} is a Cauchy sequence and so converges to some 𝑞∈𝐶. We next show that 𝑞∈𝐹. Let 𝐿=sup{𝑎₁(𝑥)∶𝑥∈𝐶}. Then, for each 𝜖>0, there exists 𝑘₂∈ℕ such that 𝑑𝑥_𝑘<𝜖,𝑞2(1+𝐿),∀𝑘≥𝑘₂. (4.18) Since lim_𝑘→∞𝑑(𝑥_𝑘,𝐹)=0, there exists 𝑘₃≥𝑘₂ such that 𝑑𝑥_𝑘<𝜖,𝐹2(1+𝐿),∀𝑘≥𝑘₃. (4.19) Thus, there exists 𝑧₂∈𝐹 such that 𝑑𝑥_𝑘3,𝑧₂<𝜖2().1+𝐿 (4.20) By (4.18) and (4.20), for each 𝑖=1,2,…,𝑚, we have 𝑑𝑇_𝑖𝑞,𝑞≤𝑑𝑇_𝑖𝑞,𝑇_𝑖𝑥_𝑘3+𝑑𝑇_𝑖𝑥_𝑘3,𝑧₂+𝑑𝑧₂,𝑥_𝑘3+𝑑𝑥_𝑘3𝑥,𝑞≤𝐿𝑑_𝑘3,𝑞+𝐿𝑑𝑥_𝑘3,𝑧₂+𝑑𝑥_𝑘3,𝑧₂+𝑑𝑥_𝑘3≤()𝑑𝑥,𝑞1+𝐿_𝑘3,𝑞+(1+𝐿)𝑑𝑥_𝑘3,𝑧₂<()𝜖1+𝐿2()+()𝜖1+𝐿1+𝐿2(1+𝐿)=𝜖. (4.21) Since 𝜖 is arbitrary, we have 𝑇_𝑖𝑞=𝑞 for all 𝑖=1,2,…,𝑚. Hence, 𝑞∈𝐹.

As an immediate consequence of Theorem 4.5, we obtain the following.

Corollary 4.6. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶). Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝑎,𝑏]⊂(0,1) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Assume that 𝐹≠∅. Then, {𝑥_𝑘} converges to a point 𝑝∈𝐹 if and only if there exists a subsequence {𝑥_𝑘𝑗} of {𝑥_𝑘} which converges to 𝑝.

Definition 4.7. A strictly increasing sequence {𝑛_𝑘}⊂ℕ is called quasiperiodic [39] if the sequence {𝑛_𝑘+1−𝑛_𝑘} is bounded or equivalently if there exists a number 𝑝∈ℕ such that any block of 𝑝 consecutive natural numbers must contain a term of the sequence {𝑛_𝑘}. The smallest of such numbers 𝑝 will be called a quasiperiod of {𝑛_𝑘}.

Lemma 4.8. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶). Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝛿,1−𝛿] for some 𝛿∈(0,1/2) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Then,(i)lim_𝑘→∞𝑑(𝑥_𝑘,𝑝) exists for all 𝑝∈𝐹,(ii)lim_𝑘→∞𝑑(𝑥_𝑘,𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘})=0,   for all 𝑗=1,2,…,𝑚,(iii) if the set 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=1+𝑛_𝑘} is quasiperiodic, then lim_𝑘→∞𝑑(𝑥_𝑘,𝑇_𝑗𝑥_𝑘)=0,   for all 𝑗=1,2,…,𝑚.

Proof. (i) Follows from Lemmas 4.2(i) and 4.4(a).
(ii) Let 𝑝∈𝐹, then, by (i), we have lim_𝑘→∞𝑑(𝑥_𝑘,𝑝) exists. Let lim_𝑘→∞𝑑𝑥_𝑘,𝑝=𝑐. (4.22) By (4.9) and (4.22), we get that limsup_𝑘→∞𝑑𝑦_𝑗𝑘,𝑝≤𝑐,for1≤𝑗≤𝑚−1. (4.23) Note that 𝑑𝑥_𝑘+1,𝑝≤1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘𝑑𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘}≤,𝑝1−𝑡_𝑚𝑘𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘1+𝑣_𝑘𝑑𝑦_{(𝑚−1)𝑘}⋮≤,𝑝1−𝑡_𝑚𝑘𝑡_{(𝑚−1)𝑘}⋅⋅⋅𝑡_(𝑗+1)𝑘1+𝑣_𝑘^𝑚−𝑗𝑑𝑥_𝑘,𝑝+𝑡_𝑚𝑘𝑡_{(𝑚−1)𝑘}⋅⋅⋅𝑡_(𝑗+1)𝑘1+𝑣_𝑘^𝑚−𝑗𝑑𝑦_𝑗𝑘,𝑝. (4.24) Thus, 𝑑𝑥_𝑘≤𝑑𝑥,𝑝^𝑘,𝑝𝛿^𝑚−𝑗−𝑑𝑥_𝑘+1,𝑝𝛿^𝑚−𝑗1+𝑣_𝑘^𝑚−𝑗+𝑑𝑦_𝑗𝑘,𝑝, (4.25) so that 𝑐≤liminf_𝑘→∞𝑑𝑦_𝑗𝑘,𝑝,for1≤𝑗≤𝑚−1. (4.26) From (4.23) and (4.26), we have lim_𝑘→∞𝑑𝑦_𝑗𝑘,𝑝=𝑐,foreach𝑗=1,2,…,𝑚−1. (4.27) That is lim_𝑘→∞𝑑1−𝑡_𝑗𝑘𝑥_𝑘⊕𝑡_𝑗𝑘𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑝=𝑐, (4.28) for each 𝑗=1,2,…,𝑚−1.
We also obtain from (4.23) that limsup_𝑘→∞𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑝≤𝑐,foreach𝑗=1,2,…,𝑚−1. (4.29) By Lemma 4.3, we get that lim_𝑘→∞𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑥_𝑘=0,foreach𝑗=1,2,…,𝑚−1. (4.30) For the case 𝑗=𝑚, by (4.1), we have 𝑑𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑝≤1+𝑏_𝑛𝑘𝑝𝑑𝑦_{(𝑚−1)𝑘},𝑝≤1+𝑏_𝑛𝑘𝑝1+𝑣_𝑛𝑘^𝑚−1𝑑𝑥_𝑘,𝑝. (4.31) But since lim_𝑘→∞𝑑(𝑥_𝑘,𝑝)=𝑐, then limsup_𝑘→∞𝑑𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑝≤𝑐. (4.32) Moreover, lim_𝑘→∞𝑑1−𝑡_𝑚𝑘𝑥_𝑘⊕𝑡_𝑚𝑘𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑝=lim_𝑘→∞𝑑𝑥_𝑘+1,𝑝=𝑐. (4.33) Again, by Lemma 4.3, we get that lim_𝑘→∞𝑑𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑥_𝑘=0. (4.34) Thus, (4.30) and (4.34) imply that lim_𝑘→∞𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑥_𝑘=0,foreach𝑗=1,2,…,𝑚. (4.35)(iii) For 𝑗=1, from (ii), we havelim_𝑘→∞𝑑𝑇^𝑛𝑘₁𝑥_𝑘,𝑥_𝑘=0. (4.36) If 𝑗=2,3,…,𝑚, then we have 𝑑𝑇^𝑛𝑘_𝑗𝑥_𝑘,𝑥_𝑘≤𝑑𝑇^𝑛𝑘_𝑗𝑥_𝑘,𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘}+𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑥_𝑘≤𝑎_𝑛𝑘𝑥_𝑘𝑑𝑥_𝑘,𝑦_{(𝑗−1)𝑘}+𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑥_𝑘≤𝑎_𝑛𝑘𝑥_𝑘𝑡_{(𝑗−1)𝑘}𝑑𝑥_𝑘,𝑇^𝑛𝑘_𝑗−1𝑦_{(𝑗−2)𝑘}+𝑑𝑇^𝑛𝑘_𝑗𝑦_{(𝑗−1)𝑘},𝑥_𝑘. (4.37) By (ii) and limsup_𝑘→∞𝑎_𝑛𝑘(𝑥_𝑘)=1, we get limsup_𝑘→∞𝑑𝑇^𝑛𝑘_𝑗𝑥_𝑘,𝑥_𝑘=0,for𝑗=2,3,…,𝑚. (4.38) By (4.36) and (4.38), we have lim_𝑘→∞𝑑𝑇^𝑛𝑘_𝑗𝑥_𝑘,𝑥_𝑘=0,∀𝑗=1,2,…,𝑚. (4.39) By the construction of the sequence {𝑥_𝑘}, we have from (4.35) that lim_𝑘→∞𝑑𝑥_𝑘+1,𝑥_𝑘=0. (4.40) Next, we show that lim_𝑘→∞𝑑𝑇_𝑗𝑥_𝑘,𝑥_𝑘=0,∀𝑗=1,2,…,𝑚. (4.41) It is enough to prove that 𝑑(𝑇_𝑗𝑥_𝑘,𝑥_𝑘)→0 as 𝑘→∞ though 𝒥. Indeed, let 𝑝 be a quasiperiod of 𝒥, and let 𝜀>0 be given. Then, there exists 𝑁₁∈ℕ such that lim_𝑘→∞𝑑𝑇_𝑗𝑥_𝑘,𝑥_𝑘<𝜀3,∀𝑘∈𝒥suchthat𝑘≥𝑁₁. (4.42) By the quasiperiodicity of 𝒥, for each 𝑙∈ℕ, there exists 𝑖_𝑙∈𝒥 such that |𝑙−𝑖_𝑙|≤𝑝. Without loss of generality, we can assume that 𝑙≤𝑖_𝑙≤𝑙+𝑝 (the proof for the other case is identical). Let 𝑀=sup{𝑎₁(𝑥)∶𝑥∈𝐶}. Then, 𝑀≥1. Since lim_𝑙→∞𝑑(𝑥_𝑙+1,𝑥_𝑙)=0 by (4.40), there exists 𝑁₂∈ℕ such that 𝑑𝑥_𝑙+1,𝑥_𝑙<𝜀3𝑝𝑀,∀𝑙≥𝑁₂. (4.43) This implies that 𝑑𝑥_𝑖𝑙,𝑥_𝑙≤𝑑𝑥_𝑖𝑙,𝑥_𝑖𝑙−1+⋯+𝑑𝑥_𝑙+1,𝑥_𝑙𝜀≤𝑝=𝜀3𝑝𝑀.3𝑀 (4.44) By the definition of 𝑇, we have 𝑑𝑇_𝑗𝑥_𝑖𝑙,𝑇_𝑗𝑥_𝑙≤𝑀𝑑𝑥_𝑖𝑙,𝑥_𝑙𝜀≤𝑀=𝜀3𝑀3. (4.45)
Let 𝑁=max{𝑁₁,𝑁₂}. Then, for 𝑙≥𝑁, we have from (4.42), (4.44), and (4.45) that 𝑑𝑥_𝑙,𝑇_𝑗𝑥_𝑙≤𝑑𝑥_𝑙,𝑥_𝑖𝑙+𝑑𝑥_𝑖𝑙,𝑇_𝑗𝑥_𝑖𝑙+𝑑𝑇_𝑗𝑥_𝑖𝑙,𝑇_𝑗𝑥_𝑙<𝜀+𝜀3𝑀3+𝜀3≤𝜀. (4.46) To prove that 𝑑(𝑇_𝑗𝑥_𝑘,𝑥_𝑘)→0 as 𝑘→∞ though 𝒥. Since 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=𝑛_𝑘+1} is quasiperiodic, for each 𝑘∈𝒥, we have 𝑑𝑥_𝑘,𝑇_𝑗𝑥_𝑘≤𝑑𝑥_𝑘,𝑥_𝑘+1+𝑑𝑥_𝑘+1,𝑇^𝑛𝑘+1_𝑗𝑥_𝑘+1+𝑑𝑇^𝑛𝑘+1_𝑗𝑥_𝑘+1,𝑇^𝑛𝑘+1_𝑗𝑥_𝑘+𝑑𝑇^𝑛𝑘_𝑗⁺¹𝑥_𝑘,𝑇_𝑗𝑥_𝑘≤𝑑𝑥_𝑘,𝑥_𝑘+1+𝑑𝑥_𝑘+1,𝑇^𝑛𝑘+1_𝑗𝑥_𝑘+1+𝑎_𝑛𝑘+1𝑥_𝑘+1𝑑𝑥_𝑘+1,𝑥_𝑘+𝑎₁𝑥_𝑘𝑑𝑇^𝑛𝑘_𝑗𝑥_𝑘,𝑥_𝑘. (4.47) From this, together with (4.39) and (4.40), we can obtain that 𝑑(𝑇_𝑗𝑥_𝑘,𝑥_𝑘)→0 as 𝑘→∞ through 𝒥.

The following lemmas can be found in [3] (see also [2]).

Lemma 4.9. Let 𝐶 be a nonempty closed convex subset of 𝑋, and let 𝑇∈𝒯_𝑟(𝐶). If lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑇^𝑙𝑥_𝑛)=0 for every 𝑙∈ℕ.

Lemma 4.10. Let 𝐶 be a nonempty closed convex subset of 𝑋, and let 𝑇∈𝒯_𝑟(𝐶). Suppose {𝑥_𝑛} is a bounded sequence in 𝐶 such that lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0 and Δ-lim_𝑛𝑥_𝑛=𝑤. Then, 𝑇𝑤=𝑤.

By using Lemmas 2.3 and 4.10, we can obtain the following result. We omit the proof because it is similar to the one given in [38].

Lemma 4.11. Let 𝐶 be a closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be an asymptotic pointwise nonexpansive mapping. Suppose {𝑥_𝑛} is a bounded sequence in 𝐶 such that lim_𝑛𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))=0 and 𝑑(𝑥_𝑛,𝑣) converges for each 𝑣∈𝐹(𝑇), then 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑇). Here, 𝜔_𝑤(𝑥_𝑛)=⋃𝐴({𝑢_𝑛}) where the union is taken over all subsequences {𝑢_𝑛} of {𝑥_𝑛}. Moreover, 𝜔_𝑤(𝑥_𝑛) consists of exactly one point.

Now, we are ready to prove our Δ-convergence theorem.

Theorem 4.12. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶). Let {𝑡_𝑖𝑘}^∞_k=1⊂[𝛿,1−𝛿] for some 𝛿∈(0,1/2) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Suppose that 𝐹∶=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)≠∅ and the set 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=1+𝑛_𝑘} is quasiperiodic. Then, {𝑥_𝑘}Δ-converges to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑚}.

Proof. Let 𝑝∈𝐹, by Lemma 4.8,  lim_𝑘→∞𝑑(𝑥_𝑘,𝑝) existsm and hence {𝑥_𝑘} is bounded. Since lim_𝑘→∞𝑑(𝑥_𝑘,𝑇_𝑗𝑥_𝑘)=0 for all 𝑗=1,2,…,𝑚, then by Lemma 4.11  𝜔_𝑤(𝑥_𝑘)⊂𝐹(𝑇_𝑗) for all 𝑗=1,2,…,𝑚, and hence 𝜔_𝑤(𝑥_𝑘)⊂⋂^𝑚_𝑗=1𝐹(𝑇_𝑗)=𝐹. Since 𝜔_𝑤(𝑥_𝑛) consists of exactly one point, then {𝑥_𝑘}Δ-converges to an element of 𝐹.

Before proving our strong convergence theorem, we recall that a mapping 𝑇∶𝐶→𝐶 is said to be semicompact if 𝐶 is closed and, for any bounded sequence {𝑥_𝑛} in 𝐶 with lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} and 𝑥∈𝐶 such that lim_𝑘→∞𝑥_𝑛𝑘=𝑥.

Theorem 4.13. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶) such that 𝑇^𝑙_𝑖 is semicompact for some 𝑖∈{1,…,𝑚} and 𝑙∈ℕ. Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝛿,1−𝛿] for some 𝛿∈(0,1/2) and {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (4.4) is well defined. Suppose that 𝐹∶=⋂^m_𝑖=1𝐹(𝑇_𝑖)≠∅ and the set 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=1+𝑛_𝑘} is quasiperiodic. Then, {𝑥_𝑘} converges to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑚}.

Proof. By Lemma 4.8, we have lim_𝑘→∞𝑑𝑥_𝑘,𝑇_𝑖𝑥_𝑘=0,for𝑖=1,…,𝑚. (4.48) Let 𝑖∈{1,…,𝑚} be such that 𝑇^𝑙_𝑖 is semicompact. Thus, by Lemma 4.9, lim_𝑘→∞𝑑𝑥_𝑘,𝑇^𝑙_𝑖𝑥_𝑘=0. (4.49) We can also find a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑘} such that lim_𝑗→∞𝑥_𝑘𝑗=𝑞∈𝐶. Hence, from (4.48), we have 𝑑𝑞,𝑇_𝑖𝑞=lim_𝑗→∞𝑑𝑥_𝑘𝑗,𝑇_𝑖𝑥_𝑘𝑗=0,∀𝑖=1,…,𝑚. (4.50) Thus, 𝑞∈𝐹, and, by Corollary 4.6, {𝑥_𝑘} converges to 𝑞. This completes the proof.

5. Concluding Remarks

One may observe that our method can be used to obtain the analogous results for uniformly convex Banach spaces. Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋 and fix 𝑥₁∈𝐶. Define a sequence {𝑥_𝑘} in 𝐶 as𝑥_𝑘+1=1−𝑡_𝑚𝑘𝑥_𝑘+𝑡_𝑚𝑘𝑇^𝑛𝑘_𝑚𝑦_{(𝑚−1)𝑘},𝑦_{(𝑚−1)𝑘}=1−𝑡_{(𝑚−1)𝑘}𝑥_𝑘+𝑡_{(𝑚−1)𝑘}𝑇^𝑛𝑘_𝑚−1𝑦_{(𝑚−2)𝑘},𝑦_{(𝑚−2)𝑘}=1−𝑡_{(𝑚−2)𝑘}𝑥_𝑘+𝑡_{(𝑚−2)𝑘}𝑇^𝑛𝑘_𝑚−2𝑦_{(𝑚−3)𝑘},⋮𝑦_2𝑘=1−𝑡_2𝑘𝑥_𝑘+𝑡_2𝑘𝑇^𝑛𝑘₂𝑦_1𝑘,𝑦_1𝑘=1−𝑡_1𝑘𝑥_𝑘+𝑡_1𝑘𝑇^𝑛𝑘₁𝑦_0𝑘,𝑦_0𝑘=𝑥_𝑘,𝑘∈ℕ, (5.1) where 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶),  {𝑡_𝑖𝑘}^∞_𝑘=1 are sequences in [0,1] for all 𝑖=1,2,…,𝑚, and {𝑛_𝑘} is an increasing sequence of natural numbers.

Theorem 5.1. Let 𝑋 be a uniformly convex Banach space with the Opial property, and let 𝐶 be a nonempty closed convex subset of 𝑋. Let 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶),  {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝛿,1−𝛿] for some 𝛿∈(0,1/2), and let {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (5.1) is well defined. Suppose that 𝐹∶=⋂^𝑚_𝑖=1𝐹(T_𝑖)≠∅ and the set 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=1+𝑛_𝑘} is quasiperiodic. Then, {𝑥_𝑘} converges weakly to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑚}.

Theorem 5.2. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 and 𝑇₁,…,𝑇_𝑚∈𝒯_𝑟(𝐶) such that 𝑇^𝑙_𝑖 is semicompact for some 𝑖∈{1,…,𝑚} and 𝑙∈ℕ. Let {𝑡_𝑖𝑘}^∞_𝑘=1⊂[𝛿,1−𝛿] for some 𝛿∈(0,1/2), and let {𝑛_𝑘}⊂ℕ be such that {𝑥_𝑘} in (5.1) is well defined. Suppose that 𝐹∶=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)≠∅ and the set 𝒥={𝑘∈ℕ∶𝑛_𝑘+1=1+𝑛_𝑘} is quasiperiodic. Then, {𝑥_𝑘} converges strongly to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑚}.

Acknowledgments

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author also thanks the Graduate School of Chiang Mai University, Thailand.