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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 929037, 18 pages
http://dx.doi.org/10.1155/2011/929037
Research Article

Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Materials Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 3 February 2011; Accepted 20 June 2011

Academic Editor: Stefanย Siegmund

Copyright ยฉ 2011 Withun Phuengrattana and Suthep Suantai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {๐‘‡๐‘›} in convex metric spaces. We prove that the sequence {๐‘ฅ๐‘›} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {๐‘‡๐‘›} satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept of W-mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept of W-mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT-condition. Our results generalize and refine many known results in the current literature.

1. Introduction

Let ๐ถ be a nonempty closed convex subset of a metric space (๐‘‹,๐‘‘), and let ๐‘‡ be a mapping of ๐ถ into itself. A mapping ๐‘‡ is called nonexpansive if ๐‘‘(๐‘‡๐‘ฅ,๐‘‡๐‘ฆ)โ‰ค๐‘‘(๐‘ฅ,๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. The set of all fixed points of ๐‘‡ is denoted by ๐น(๐‘‡), that is, ๐น(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘ฅ=๐‘‡๐‘ฅ}.

In 1967, Halpern [1] introduced the following iterative scheme in Hilbert spaces which was referred to as Halpern iteration for approximating a fixed point of ๐‘‡:๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›โˆ€๐‘›โˆˆโ„•,(1.1) where ๐‘ฅ1,๐‘ขโˆˆ๐ถ are arbitrarily chosen, and {๐›ผ๐‘›} is a sequence in [0,1]. Wittmann [2] studied the iterative scheme (1.1) in a Hilbert space and obtained the strong convergence of the iteration. Reich [3] and Shioji and Takahashi [4] extended Wittmann's result to a real Banach space.

The modified version of Halpern iteration was investigated widely by many mathematicians. For instance, Kim and Xu [5] studied the sequence {๐‘ฅ๐‘›} generated as follows:๐‘ฆ๐‘›=๐›ผ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐›ฝ๐‘›๎€ท๐‘ข+1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฆ๐‘›โˆ€๐‘›โˆˆโ„•,(1.2) where ๐‘ฅ1,๐‘ขโˆˆ๐ถ are arbitrarily chosen and {๐›ผ๐‘›}, {๐›ฝ๐‘›} are two sequences in [0,1]. They proved the strong convergence of iterative scheme (1.2) in the framework of a uniformly smooth Banach space. In 2007, Aoyama et al. [6] introduced a Halpern iteration for finding a common fixed point of a countable infinite family of nonexpansive mappings in a Banach space as follows:๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›โˆ€๐‘›โˆˆโ„•,(1.3) where ๐‘ฅ1,๐‘ขโˆˆ๐ถ are arbitrarily chosen, {๐›ผ๐‘›} is a sequence in [0,1], and {๐‘‡๐‘›} is a sequence of nonexpansive mappings with some conditions. They proved that the sequence {๐‘ฅ๐‘›} generated by (1.3) converges strongly to a common fixed point of {๐‘‡๐‘›}. In 2010, Saejung [7] extended the results of Halpern [1], Wittmann [2], Reich [3], Shioji and Takahashi [4], and Aoyama et al. [6] to the case of a CAT(0) space which is an example of a convex metric space. Recently, Cuntavepanit and Panyanak [8] extended the result of Kim and Xu [5] to a CAT(0) space.

Takahashi [9] introduced the concept of convex metric spaces by using the convex structure as follows. Let (๐‘‹,๐‘‘) be a metric space. A mapping ๐‘Šโˆถ๐‘‹ร—๐‘‹ร—[0,1]โ†’๐‘‹ is said to be a convex structure on ๐‘‹ if for each ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐œ†โˆˆ[0,1], ๐‘‘(๐‘ง,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))โ‰ค๐œ†๐‘‘(๐‘ง,๐‘ฅ)+(1โˆ’๐œ†)๐‘‘(๐‘ง,๐‘ฆ),(1.4) for all ๐‘งโˆˆ๐‘‹. A metric space (๐‘‹,๐‘‘) together with a convex structure ๐‘Š is called a convex metric space which will be denoted by (๐‘‹,๐‘‘,๐‘Š). A nonempty subset ๐ถ of ๐‘‹ is said to be convex if ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)โˆˆ๐ถ for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐œ†โˆˆ[0,1]. Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold.

Motivated by the above results, we introduce a new iterative scheme for finding a common fixed point of a countable infinite family of nonexpansive mappings {๐‘‡๐‘›} of ๐ถ into itself in a convex metric space as follows:๐‘ฆ๐‘›๎€ท=๐‘Š๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›,๐›ผ๐‘›๎€ธ,๐‘ฅ๐‘›+1๎€ท๐‘ฆ=๐‘Š๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›,๐›ฝ๐‘›๎€ธโˆ€๐‘›โˆˆโ„•,(1.5) where ๐‘ฅ1,๐‘ขโˆˆ๐ถ are arbitrarily chosen, and {๐›ผ๐‘›}, {๐›ฝ๐‘›} are two sequences in [0,1]. The main propose of this paper is to prove the convergence theorem of the sequence {๐‘ฅn} generated by (1.5) to a common fixed point of a countable infinite family of nonexpansive mappings in convex metric spaces and CAT(0) spaces under certain suitable conditions.

2. Preliminaries

We recall some definitions and useful lemmas used in the main results.

Lemma 2.1 (see [9, 10]). Let (๐‘‹,๐‘‘,๐‘Š) be a convex metric space. For each ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐œ†,๐œ†1,๐œ†2โˆˆ[0,1], we have the following. (i)๐‘Š(๐‘ฅ,๐‘ฅ,๐œ†)=๐‘ฅ,๐‘Š(๐‘ฅ,๐‘ฆ,0)=๐‘ฆ and ๐‘Š(๐‘ฅ,๐‘ฆ,1)=๐‘ฅ.(ii)๐‘‘(๐‘ฅ,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))=(1โˆ’๐œ†)๐‘‘(๐‘ฅ,๐‘ฆ) and ๐‘‘(๐‘ฆ,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))=๐œ†๐‘‘(๐‘ฅ,๐‘ฆ). (iii)๐‘‘(๐‘ฅ,๐‘ฆ)=๐‘‘(๐‘ฅ,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))+๐‘‘(๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†),๐‘ฆ). (iv)|๐œ†1โˆ’๐œ†2|๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐‘‘(๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†1),๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†2)).

We say that a convex metric space (๐‘‹,๐‘‘,๐‘Š) has the property: (C)if ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)=๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐œ†โˆˆ[0,1], (I)if ๐‘‘(๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†1),๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†2))โ‰ค|๐œ†1โˆ’๐œ†2|๐‘‘(๐‘ฅ,๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐œ†1,๐œ†2โˆˆ[0,1], (H)if ๐‘‘(๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†),๐‘Š(๐‘ฅ,๐‘ง,๐œ†))โ‰ค(1โˆ’๐œ†)๐‘‘(๐‘ฆ,๐‘ง) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹ and ๐œ†โˆˆ[0,1], (S)if ๐‘‘(๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†),๐‘Š(๐‘ง,๐‘ค,๐œ†))โ‰ค๐œ†๐‘‘(๐‘ฅ,๐‘ง)+(1โˆ’๐œ†)๐‘‘(๐‘ฆ,๐‘ค) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘คโˆˆ๐‘‹ and ๐œ†โˆˆ[0,1].

From the above properties, it is obvious that the property (C) and (H) imply continuity of a convex structure ๐‘Šโˆถ๐‘‹ร—๐‘‹ร—[0,1]โ†’๐‘‹. Clearly, the property (S) implies the property (H). In [10], Aoyama et al. showed that a convex metric space with the property (C) and (H) has the property (S).

In 1996, Shimizu and Takahashi [11] introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space (๐‘‹,๐‘‘,๐‘Š) is said to be uniformly convex if for any ๐œ€>0, there exists ๐›ฟ=๐›ฟ(๐œ€)>0 such that for all ๐‘Ÿ>0 and ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹ with ๐‘‘(๐‘ง,๐‘ฅ)โ‰ค๐‘Ÿ, ๐‘‘(๐‘ง,๐‘ฆ)โ‰ค๐‘Ÿ and ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘Ÿ๐œ€ imply that ๐‘‘(๐‘ง,๐‘Š(๐‘ฅ,๐‘ฆ,1/2))โ‰ค(1โˆ’๐›ฟ)๐‘Ÿ. Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. In fact, the property (I) holds in uniformly convex metric spaces, see [12].

Lemma 2.2. Property (C) holds in uniformly convex metric spaces.

Proof. Suppose that (๐‘‹,๐‘‘,๐‘Š) is a uniformly convex metric space. Let ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐œ†โˆˆ[0,1]. It is obvious that the conclusion holds if ๐œ†=0 or ๐œ†=1. So, suppose ๐œ†โˆˆ(0,1). By Lemma 2.1(ii), we have ๐‘‘๐‘‘(๐‘ฅ,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))=(1โˆ’๐œ†)๐‘‘(๐‘ฅ,๐‘ฆ),๐‘‘(๐‘ฆ,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))=๐œ†๐‘‘(๐‘ฅ,๐‘ฆ),(๐‘ฅ,๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†))=(1โˆ’๐œ†)๐‘‘(๐‘ฅ,๐‘ฆ),๐‘‘(๐‘ฆ,๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†))=๐œ†๐‘‘(๐‘ฅ,๐‘ฆ).(2.1)
We will show that ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)=๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†). To show this, suppose not. Put ๐‘ง1=๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†) and ๐‘ง2=๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†). Let ๐‘Ÿ1=(1โˆ’๐œ†)๐‘‘(๐‘ฅ,๐‘ฆ)>0, ๐‘Ÿ2=๐œ†๐‘‘(๐‘ฅ,๐‘ฆ)>0, ๐œ€1=๐‘‘(๐‘ง1,๐‘ง2)/๐‘Ÿ1, and ๐œ€2=๐‘‘(๐‘ง1,๐‘ง2)/๐‘Ÿ2. It is easy to see that ๐œ€1,๐œ€2>0. Since (๐‘‹,๐‘‘,๐‘Š) is uniformly convex, we have ๐‘‘๎‚€๎‚€๐‘ง๐‘ฅ,๐‘Š1,๐‘ง2,12๎‚๎‚โ‰ค๐‘Ÿ1๎€ท๎€ท๐œ€1โˆ’๐›ฟ1๎‚€๎‚€๐‘ง๎€ธ๎€ธ,๐‘‘๐‘ฆ,๐‘Š1,๐‘ง2,12๎‚๎‚โ‰ค๐‘Ÿ2๎€ท๎€ท๐œ€1โˆ’๐›ฟ2๎€ธ๎€ธ.(2.2) By ๐œ†โˆˆ(0,1), we get ๐‘ฅโ‰ ๐‘ฆ. Since ๐›ฟ(๐œ€1)>0 and ๐›ฟ(๐œ€2)>0, then ๎‚€๎‚€๐‘ง๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐‘‘๐‘ฅ,๐‘Š1,๐‘ง2,12๎‚€๎‚€๐‘ง๎‚๎‚+๐‘‘๐‘ฆ,๐‘Š1,๐‘ง2,12๎‚๎‚โ‰ค๐‘Ÿ1๎€ท๎€ท๐œ€1โˆ’๐›ฟ1๎€ธ๎€ธ+๐‘Ÿ2๎€ท๎€ท๐œ€1โˆ’๐›ฟ2๎€ธ๎€ธ<๐‘Ÿ1+๐‘Ÿ2=๐‘‘(๐‘ฅ,๐‘ฆ).(2.3) This is a contradiction. Hence, ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)=๐‘Š(๐‘ฆ,๐‘ฅ,1โˆ’๐œ†).

By Lemma 2.2, it is clear that a uniformly convex metric space (๐‘‹,๐‘‘,๐‘Š) with the property (H) has the property (S), and the convex structure ๐‘Š is also continuous.

Next, we recall the special space of convex metric spaces, namely, CAT(0) spaces. Let (๐‘‹,๐‘‘) be a metric space. A geodesic path joining ๐‘ฅโˆˆ๐‘‹ to ๐‘ฆโˆˆ๐‘‹ (or, more briefly, a geodesic from ๐‘ฅ to ๐‘ฆ) is a map ๐‘ from a closed interval [0,๐‘™]โŠ‚โ„ to ๐‘‹ such that ๐‘(0)=๐‘ฅ,๐‘(๐‘™)=๐‘ฆ and ๐‘‘(๐‘(๐‘ก1),๐‘(๐‘ก2))=|๐‘ก1โˆ’๐‘ก2| for all ๐‘ก1,๐‘ก2โˆˆ[0,๐‘™]. In particular, ๐‘ is an isometry and ๐‘‘(๐‘ฅ,๐‘ฆ)=๐‘™. The image ๐›ผ of ๐‘ is called a geodesic (or metric) segment joining ๐‘ฅ and ๐‘ฆ. When unique, this geodesic is denoted [๐‘ฅ,๐‘ฆ]. The space (๐‘‹,๐‘‘) is said to be a geodesic metric space if every two points of ๐‘‹ are joined by a geodesic, and ๐‘‹ is said to be uniquely geodesic if there is exactly one geodesic joining ๐‘ฅ and ๐‘ฆ for each ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. A subset ๐‘Œ of ๐‘‹ is said to be convex if ๐‘Œ includes every geodesic segment joining any two of its points.

A geodesic triangle โ–ต(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3) in a geodesic metric space (๐‘‹,๐‘‘) consists of three points ๐‘ฅ1,๐‘ฅ2,๐‘ฅ3 in ๐‘‹ (the vertices of โ–ต) and a geodesic segment between each pair of vertices (the edges of โ–ต). A comparison triangle for geodesic triangle โ–ต(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3) in (๐‘‹,๐‘‘) is a triangle โ–ต(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3)โˆถ=โ–ต(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3) in the Euclidean plane ๐”ผ2 such that ๐‘‘๐”ผ2(๐‘ฅ๐‘–,๐‘ฅ๐‘—)=๐‘‘(๐‘ฅ๐‘–,๐‘ฅ๐‘—) for ๐‘–,๐‘—โˆˆ{1,2,3}.

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. Let โ–ต be a geodesic triangle in ๐‘‹, and let โ–ต be a comparison triangle for โ–ต. Then โ–ต is said to satisfy the CAT(0) inequality if for all ๐‘ฅ,๐‘ฆโˆˆโ–ต and all comparison points ๐‘ฅ,๐‘ฆโˆˆโ–ต, ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐‘‘๐”ผ2(๐‘ฅ,๐‘ฆ).

If ๐‘ง,๐‘ฅ,๐‘ฆ are points in a CAT(0) space and if ๐‘š is the midpoint of the segment [๐‘ฅ,๐‘ฆ], then the CAT(0) inequality implies ๐‘‘(๐‘ง,๐‘š)2โ‰ค12๐‘‘(๐‘ง,๐‘ฅ)2+12๐‘‘(๐‘ง,๐‘ฆ)2โˆ’14๐‘‘(๐‘ฅ,๐‘ฆ)2.(CN) This is the (CN) inequality of Bruhat and Tits [13], which is equivalent to ๐‘‘(๐‘ง,๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆ)2โ‰ค๐œ†๐‘‘(๐‘ง,๐‘ฅ)2+(1โˆ’๐œ†)๐‘‘(๐‘ง,๐‘ฆ)2โˆ’๐œ†(1โˆ’๐œ†)๐‘‘(๐‘ฅ,๐‘ฆ)2,(CNโˆ—) for any ๐œ†โˆˆ[0,1], where ๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆ denotes the unique point in [๐‘ฅ,๐‘ฆ]. The (CN*) inequality has appeared in [14]. By using the (CN) inequality, it is easy to see that the CAT(0) spaces are uniformly convex. In fact [15], a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality. Moreover, if ๐‘‹ is CAT(0) space and ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, then for any ๐œ†โˆˆ[0,1], there exists a unique point ๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆโˆˆ[๐‘ฅ,๐‘ฆ] such that๐‘‘(๐‘ง,๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆ)โ‰ค๐œ†๐‘‘(๐‘ง,๐‘ฅ)+(1โˆ’๐œ†)๐‘‘(๐‘ง,๐‘ฆ),(2.4) for any ๐‘งโˆˆ๐‘‹. It follows that CAT(0) spaces have convex structure ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)=๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆ. It is clear that the properties (C), (I), and (S) are satisfied for CAT(0) spaces, see [15, 16]. This is also true for Banach spaces.

Let ๐œ‡ be a continuous linear functional on ๐‘™โˆž, the Banach space of bounded real sequences, and let (๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. We write ๐œ‡๐‘›(๐‘Ž๐‘›) instead of ๐œ‡((๐‘Ž1,๐‘Ž2,โ€ฆ)). We call ๐œ‡ a Banach limit if ๐œ‡ satisfies โ€–๐œ‡โ€–=๐œ‡(1,1,โ€ฆ)=1 and ๐œ‡๐‘›(๐‘Ž๐‘›)=๐œ‡๐‘›(๐‘Ž๐‘›+1) for each (๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. For a Banach limit ๐œ‡, we know that liminf๐‘›โ†’โˆž๐‘Ž๐‘›โ‰ค๐œ‡๐‘›(๐‘Ž๐‘›)โ‰คlimsup๐‘›โ†’โˆž๐‘Ž๐‘› for all (๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž. So if (๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž with lim๐‘›โ†’โˆž๐‘Ž๐‘›=๐‘, then ๐œ‡๐‘›(๐‘Ž๐‘›)=๐‘, see also [17].

Lemma 2.3 ([4], Proposition 2). Let (๐‘Ž1,๐‘Ž2,โ€ฆ)โˆˆ๐‘™โˆž be such that ๐œ‡๐‘›(๐‘Ž๐‘›)โ‰ค0 for all Banach limit ๐œ‡. If limsup๐‘›โ†’โˆž(๐‘Ž๐‘›+1โˆ’๐‘Ž๐‘›)โ‰ค0, then limsup๐‘›โ†’โˆž๐‘Ž๐‘›โ‰ค0.

Lemma 2.4 ([6], Lemma 2.3). Let {๐‘ ๐‘›} be a sequence of nonnegative real numbers, let {๐›ผ๐‘›} be a sequence of real numbers in [0,1] with โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, let {๐›ฟ๐‘›} be a sequence of nonnegative real numbers with โˆ‘โˆž๐‘›=1๐›ฟ๐‘›<โˆž, and let {๐›พ๐‘›}be a sequence of real numbers with limsup๐‘›โ†’โˆž๐›พ๐‘›โ‰ค0. Suppose that ๐‘ ๐‘›+1โ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ ๐‘›+๐›ผ๐‘›๐›พ๐‘›+๐›ฟ๐‘›โˆ€๐‘›โˆˆโ„•.(2.5) Then lim๐‘›โ†’โˆž๐‘ ๐‘›=0.

Lemma 2.5 ([18], Lemma 1). Let (๐‘‹,๐‘‘,๐‘Š) be a uniformly convex metric space with a continuous convex structure ๐‘Šโˆถ๐‘‹ร—๐‘‹ร—[0,1]โ†’๐‘‹. Then for arbitrary positive number ๐œ€ and ๐‘Ÿ, there exists ๐œ‚=๐œ‚(๐œ€)>0 such that ๐‘‘(๐‘ง,๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†))โ‰ค๐‘Ÿ(1โˆ’2min{๐œ†,1โˆ’๐œ†}๐œ‚),(2.6) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹, ๐‘‘(๐‘ง,๐‘ฅ)โ‰ค๐‘Ÿ, ๐‘‘(๐‘ง,๐‘ฆ)โ‰ค๐‘Ÿ, ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘Ÿ๐œ€, and ๐œ†โˆˆ[0,1].

Remark 2.6. The above lemma also holds for a uniformly convex metric space with the property (H).

3. Main Results

The following condition was introduced by Aoyama et al. [6]. Let ๐ถ be a subset of a complete convex metric space (๐‘‹,๐‘‘,๐‘Š), and let {๐‘‡๐‘›} be a countable infinite family of mappings from ๐ถ into itself. We say that {๐‘‡๐‘›} satisfies AKTT-condition ifโˆž๎“๐‘›=1๎€ฝ๐‘‘๎€ท๐‘‡sup๐‘›+1๐‘ง,๐‘‡๐‘›๐‘ง๎€ธ๎€พโˆถ๐‘งโˆˆ๐ต<โˆž,(3.1) for each bounded subset ๐ต of ๐ถ. If ๐ถ is a closed subset and {๐‘‡๐‘›} satisfies AKTT-condition, then we can define a mapping ๐‘‡โˆถ๐ถโ†’๐ถ such that ๐‘‡๐‘ฅ=lim๐‘›โ†’โˆž๐‘‡๐‘›๐‘ฅ for all ๐‘ฅโˆˆ๐ถ. In this case, we also say that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition. By using the same argument as in [6, Lemma 3.2], we have the following lemma.

Lemma 3.1. If ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition, then lim๐‘›โ†’โˆžsup{๐‘‘(๐‘‡๐‘ง,๐‘‡๐‘›๐‘ง)โˆถ๐‘งโˆˆ๐ต}=0 for all bounded subsets ๐ต of ๐ถ.

Theorem 3.2. Let ๐ถ be a nonempty closed convex subset of a complete convex metric space (๐‘‹,๐‘‘,๐‘Š) with the properties (I) and (S). Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. Suppose that {๐‘ฅ๐‘›} is a sequence of ๐ถ generated by (1.5), and let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be sequences in [0,1] which satisfy the conditions: (C1)0<๐›ผ๐‘›<1, lim๐‘›โ†’โˆž๐›ผ๐‘›=0, โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž and โˆ‘โˆž๐‘›=1|๐›ผ๐‘›+1โˆ’๐›ผ๐‘›|<โˆž,(C2)๐›ฝ๐‘›โˆˆ(๐‘,1] for some ๐‘โˆˆ(0,1) and โˆ‘โˆž๐‘›=1|๐›ฝ๐‘›+1โˆ’๐›ฝ๐‘›|<โˆž. Suppose that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition. Then lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0 and lim๐‘›โ†’โˆž๐‘‘(๐‘‡๐‘ฅ๐‘›,๐‘ฅ๐‘›)=0.

Proof. Let โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). By the definition of {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›}, we have ๐‘‘๎€ท๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘Š๎€ท๐‘ฆ,๐‘=๐‘‘๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›,๐›ฝ๐‘›๎€ธ๎€ธ,๐‘โ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›๎€ธ+๎€ท,๐‘1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘โ‰ค๐‘‘๐‘›๎€ธ๎€ท๐‘Š๎€ท,๐‘=๐‘‘๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›,๐›ผ๐‘›๎€ธ๎€ธ,๐‘โ‰ค๐›ผ๐‘›๐‘‘๎€ท(๐‘ข,๐‘)+1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ,๐‘โ‰ค๐›ผ๐‘›๎€ท๐‘‘(๐‘ข,๐‘)+1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›๎€ธ๎€ฝ๎€ท๐‘ฅ,๐‘โ‰คmax๐‘‘(๐‘ข,๐‘),๐‘‘๐‘›.,๐‘๎€ธ๎€พ(3.2) By induction on ๐‘›, we obtain that ๐‘‘(๐‘ฅ๐‘›,๐‘)โ‰คmax{๐‘‘(๐‘ข,๐‘),๐‘‘(๐‘ฅ1,๐‘)} for all ๐‘›โˆˆโ„• and all โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). Hence, the sequence {๐‘ฅ๐‘›} is bounded and so {๐‘ฆ๐‘›}, {๐‘‡๐‘›๐‘ฅ๐‘›}, {๐‘‡๐‘›๐‘ฆ๐‘›} are bounded.
It follows by condition (๐ถ1) that ๐‘‘๎€ท๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘Š๎€ท=๐‘‘๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›,๐›ผ๐‘›๎€ธ,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ=๐›ผ๐‘›๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธโŸถ0.(3.3) By the definition of {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›}, we have ๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘Š๎€ท=๐‘‘๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›,๐›ผ๐‘›๎€ธ๎€ท,๐‘Š๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›โˆ’1๎€ท๐‘Š๎€ท๎€ธ๎€ธโ‰ค๐‘‘๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›,๐›ผ๐‘›๎€ธ๎€ท,๐‘Š๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›๎€ท๐‘Š๎€ท๎€ธ๎€ธ+๐‘‘๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›๎€ธ๎€ท,๐‘Š๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›๎€ท๐‘Š๎€ท๎€ธ๎€ธ+๐‘‘๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›๎€ธ๎€ท,๐‘Š๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1,๐›ผ๐‘›โˆ’1โ‰ค๎€ท๎€ธ๎€ธ1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘‘๎€ท๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธโ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘‘๎€ท๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธโ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘‘๎€ท๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ,๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘Š๎€ท๐‘ฆ=๐‘‘๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›,๐›ฝ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘Š๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1,๐›ฝ๐‘›โˆ’1๎€ท๐‘Š๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค๐‘‘๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›,๐›ฝ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘Š๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1,๐›ฝ๐‘›๎€ท๐‘Š๎€ท๐‘ฆ๎€ธ๎€ธ+๐‘‘๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1,๐›ฝ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘Š๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1,๐›ฝ๐‘›โˆ’1๎€ธ๎€ธโ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฆ๐‘›,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธ+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘‘๎€ท๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธโ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๐‘‘๎€ท๐‘‡๎€ธ๎€ท๐‘›๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1+||๐›ฝ๎€ธ๎€ธ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘‘๎€ท๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธโ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๎€ธ๎€ท๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1+||๐›ฝ๎€ธ๎€ธ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘‘๎€ท๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆโ‰ค๐‘‘๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธ+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘‘๎€ท๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธโ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘‘๎€ท๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธ+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘‘๎€ท๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธโ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๎€ธ๐‘€๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1๎€ธ,(3.4) where ๐‘€=max{sup๐‘›๐‘‘(๐‘ข,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1),sup๐‘›๐‘‘(๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1)}.
Putting ๐›ฟ๐‘›=(|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|+|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|)๐‘€+๐‘‘(๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1)+๐‘‘(๐‘‡๐‘›๐‘ฆ๐‘›โˆ’1,๐‘‡๐‘›โˆ’1๐‘ฆ๐‘›โˆ’1), we have โˆž๎“๐‘›=2๐›ฟ๐‘›โ‰ค๐‘€โˆž๎“๐‘›=2๎€ท||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๎€ธ+โˆž๎“๐‘›=2๎€ฝ๐‘‘๎€ท๐‘‡sup๐‘›๐‘ง,๐‘‡๐‘›โˆ’1๐‘ง๎€ธ๎€ฝ๐‘ฅโˆถ๐‘งโˆˆ๐‘˜+๎€พ๎€พโˆž๎“๐‘›=2๎€ฝ๐‘‘๎€ท๐‘‡sup๐‘›๐‘ง,๐‘‡๐‘›โˆ’1๐‘ง๎€ธ๎€ฝ๐‘ฆโˆถ๐‘งโˆˆ๐‘˜.๎€พ๎€พ(3.5) Hence, it follows from conditions (๐ถ1), (๐ถ2), AKTT-condition, and Lemma 2.4 that lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ=0.(3.6) Now, observe that ๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘ฆ๐‘›๎€ธ๎€ท๐‘Š๎€ท๐‘ฆ=๐‘‘๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›,๐›ฝ๐‘›๎€ธ,๐‘ฆ๐‘›๎€ธ=๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฆ๐‘›๎€ธโ‰ค๎€ท๐‘‘๎€ท๐‘ฆ(1โˆ’๐‘)๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›+1,๐‘‡๐‘›๐‘ฆ๐‘›๎€ท๐‘‘๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค(1โˆ’๐‘)๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›+1,๐‘ฆ๐‘›.๎€ธ๎€ธ(3.7) We obtain ๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘ฆ๐‘›๎€ธโ‰ค1โˆ’๐‘๐‘๎€ท๐‘‘๎€ท๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ธ.(3.8) This implies by (3.3) and (3.6) that lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฆ๐‘›)=0. Therefore, we have ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโ‰ค๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›+1,๐‘ฆ๐‘›๎€ธโŸถ0.(3.9) Since ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘‡โ‰ค๐‘‘๐‘›๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ+๐‘‘๐‘›,๐‘ฅ๐‘›๎€ธ,(3.10) it follows by (3.3) and (3.9) that lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ=0.(3.11) By (3.11) and Lemma 3.1, we get ๐‘‘๎€ท๐‘‡๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ๎€ทโ‰ค๐‘‘๐‘‡๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธ๎€ฝ๐‘‘๎€ทโ‰คsup๐‘‡๐‘ง,๐‘‡๐‘›๐‘ง๎€ธ๎€ฝ๐‘ฅโˆถ๐‘งโˆˆ๐‘˜๎€ท๐‘‡๎€พ๎€พ+๐‘‘๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธโŸถ0.(3.12)

Next, we consider a convergence theorem in CAT(0) spaces. The following two lemmas obtained by Saejung [7] are useful for our main results.

Lemma 3.3. Let ๐ถ be a closed convex subset of a complete CAT(0) space ๐‘‹, and let ๐‘‡โˆถ๐ถโ†’๐ถ be a nonexpansive mapping. Let ๐‘ขโˆˆ๐ถ be fixed. For each ๐‘กโˆˆ(0,1), the mapping ๐‘†๐‘กโˆถ๐ถโ†’๐ถ defined by ๐‘†๐‘ก๐‘ฅ=๐‘ก๐‘ขโŠ•(1โˆ’๐‘ก)๐‘‡๐‘ฅ for ๐‘ฅโˆˆ๐ถ has a unique fixed point ๐‘ฅ๐‘กโˆˆ๐ถ, that is, ๐‘ฅ๐‘ก=๐‘†๐‘ก๐‘ฅ๐‘ก=๐‘ก๐‘ขโŠ•(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘ก.

Lemma 3.4. Let ๐ถ, ๐‘‡ be as the preceding lemma. Then ๐น(๐‘‡)โ‰ โˆ… if and only if {๐‘ฅ๐‘ก} remains bounded as ๐‘กโ†’0. In this case, the following statements hold: (i){๐‘ฅ๐‘ก} converges to the unique fixed point ๐‘ง of ๐‘‡ which is nearest to ๐‘ข; (ii)๐‘‘(๐‘ข,๐‘ง)2โ‰ค๐œ‡๐‘›๐‘‘(๐‘ข,๐‘ฅ๐‘›)2 for all Banach limit ๐œ‡ and all bounded sequences {๐‘ฅ๐‘›} with lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›,๐‘‡๐‘ฅ๐‘›)=0.

Previously, we know that CAT(0) spaces have convex structure ๐‘Š(๐‘ฅ,๐‘ฆ,๐œ†)=๐œ†๐‘ฅโŠ•(1โˆ’๐œ†)๐‘ฆ and also have the properties (C), (I), and (S). Thus, we have the following result.

Theorem 3.5. Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. Suppose that ๐‘ข,๐‘ฅ1โˆˆ๐ถ are arbitrarily chosen and {๐‘ฅ๐‘›} is a sequence of ๐ถ generated by ๐‘ฆ๐‘›=๐›ผ๐‘›๎€ท๐‘ขโŠ•1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฆ๐‘›โŠ•๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๐‘›๐‘ฆ๐‘›โˆ€๐‘›โˆˆโ„•,(3.13) where {๐›ผ๐‘›} and {๐›ฝ๐‘›} are sequences in [0,1] which satisfy the conditions (๐ถ1) and (๐ถ2) as in Theorem 3.2. Suppose that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition. Then lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0 and lim๐‘›โ†’โˆž๐‘‘(๐‘‡๐‘ฅ๐‘›,๐‘ฅ๐‘›)=0.

Theorem 3.6. Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. Suppose that {๐‘ฅ๐‘›} is a sequence of ๐ถ generated by (3.13), and let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be sequences in [0,1] which satisfy the conditions (๐ถ1) and (๐ถ2) as in Theorem 3.2. Suppose that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition and โ‹‚๐น(๐‘‡)=โˆž๐‘›=1๐น(๐‘‡๐‘›). Then {๐‘ฅ๐‘›} converges strongly to a common fixed point of {๐‘‡๐‘›} which is nearest to ๐‘ข.

Proof. By Theorem 3.5, we have lim๐‘›โ†’โˆž๐‘‘(๐‘‡๐‘ฅ๐‘›,๐‘ฅ๐‘›)=0. For each ๐‘กโˆˆ(0,1), let ๐‘ง๐‘ก be a unique point of ๐ถ such that ๐‘ง๐‘ก=๐‘ก๐‘ขโŠ•(1โˆ’๐‘ก)๐‘‡๐‘ง๐‘ก. It follows from Lemma 3.4 that {๐‘ง๐‘ก} converges to a point ๐‘งโˆˆ๐น(๐‘‡) which is nearest to ๐‘ข, and ๐‘‘(๐‘ข,๐‘ง)2โ‰ค๐œ‡๐‘›๐‘‘๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ2forallBanachlimits๐œ‡,(3.14) that is, ๐œ‡๐‘›(๐‘‘(๐‘ข,๐‘ง)2โˆ’๐‘‘(๐‘ข,๐‘ฅ๐‘›)2)โ‰ค0. Moreover, by Theorem 3.5, we get lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0. It follows that limsup๐‘›โ†’โˆž๎‚€๎‚€๐‘‘(๐‘ข,๐‘ง)2๎€ทโˆ’๐‘‘๐‘ข,๐‘ฅ๐‘›+1๎€ธ2๎‚โˆ’๎‚€๐‘‘(๐‘ข,๐‘ง)2๎€ทโˆ’๐‘‘๐‘ข,๐‘ฅ๐‘›๎€ธ2๎‚๎‚=0.(3.15) By lim๐‘›โ†’โˆž๐‘‘(๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›)=0 and Lemma 2.3, we obtain limsup๐‘›โ†’โˆž๎‚€๐‘‘(๐‘ข,๐‘ง)2โˆ’๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ2๎‚=limsup๐‘›โ†’โˆž๎‚€๐‘‘(๐‘ข,๐‘ง)2๎€ทโˆ’๐‘‘๐‘ข,๐‘ฅ๐‘›๎€ธ2๎‚โ‰ค0.(3.16) Finally, we show that lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›,๐‘ง)=0. By the definition of {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›}, we have ๐‘‘๎€ท๐‘ฅ๐‘›+1๎€ธ,๐‘ง2๎€ท๐›ฝ=๐‘‘๐‘›๐‘ฆ๐‘›โŠ•๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๐‘›๐‘ฆ๐‘›๎€ธ,๐‘ง2โ‰ค๎€ท๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›๎€ธ+๎€ท,๐‘ง1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฆ๐‘›,๐‘ง๎€ธ๎€ธ2๎€ท๐‘ฆโ‰ค๐‘‘๐‘›๎€ธ,๐‘ง2๎€ท๐›ผ=๐‘‘๐‘›๎€ท๐‘ขโŠ•1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ,๐‘ง2โ‰ค๐›ผ๐‘›๐‘‘(๐‘ข,๐‘ง)2+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ,๐‘ง2โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ2โ‰ค๐›ผ๐‘›๐‘‘(๐‘ข,๐‘ง)2+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›๎€ธ,๐‘ง2โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ2=๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ฅ๐‘›๎€ธ,๐‘ง2+๐›ผ๐‘›๎‚€๐‘‘(๐‘ข,๐‘ง)2โˆ’๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ2๎‚.(3.17) This implies by โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, inequality (3.16), and Lemma 2.4 that lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›,๐‘ง)2=0. Hence, {๐‘ฅ๐‘›} converges to โ‹‚๐‘งโˆˆ๐น(๐‘‡)=โˆž๐‘›=1๐น(๐‘‡๐‘›) which is nearest to ๐‘ข.

Corollary 3.7 (see [7], Theorem 8). Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. Suppose that ๐‘ข,๐‘ฅ1โˆˆ๐ถ are arbitrarily chosen and {๐‘ฅ๐‘›} is a sequence of ๐ถ generated by ๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ขโŠ•1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›โˆ€๐‘›โˆˆโ„•,(3.18) where {๐›ผ๐‘›} is a sequence in [0,1] which satisfies the condition (๐ถ1) as in Theorem 3.2. Suppose that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition and โ‹‚๐น(๐‘‡)=โˆž๐‘›=1๐น(๐‘‡๐‘›). Then {๐‘ฅ๐‘›} converges strongly to a common fixed point of {๐‘‡๐‘›} which is nearest to ๐‘ข.

Proof. By putting ๐›ฝ๐‘›=1 for all ๐‘›โˆˆโ„• in Theorem 3.6, we obtain the desired result.

In 2009, Song and Zheng [19] introduced a condition in Banach spaces for a countable infinite family of nonexpansive mappings which is different from AKTT-condition and also give some examples of a family of mappings that satisfies this condition. Now, we state this condition in CAT(0) spaces, and it is referred as SZ-condition as follows. Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Suppose that {๐‘‡๐‘›} is a family of nonexpansive mappings from ๐ถ into itself with โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. We say that {๐‘‡๐‘›} satisfies SZ-condition if, for any bounded subset ๐พ of ๐ถ, there exists a nonexpansive mapping ๐‘‡ of ๐ถ into itself such thatlim๐‘›โ†’โˆž๎€ฝ๐‘‘๎€ท๐‘‡๎€ท๐‘‡sup๐‘›๐‘ฅ๎€ธ,๐‘‡๐‘›๐‘ฅ๎€ธ๎€พโˆถ๐‘ฅโˆˆ๐พ=0,๐น(๐‘‡)=โˆž๎™๐‘›=1๐น๎€ท๐‘‡๐‘›๎€ธ.(3.19)

Theorem 3.8. Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ… and satisfies SZ-condition. Suppose that {๐‘ฅ๐‘›} is a sequence of ๐ถ defined by (3.13) with lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0. Let {๐›ผ๐‘›} and {๐›ฝ๐‘›} be sequences in [0,1] which satisfy the following conditions: (C3)0<๐›ผ๐‘›<1, lim๐‘›โ†’โˆž๐›ผ๐‘›=0, and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C4)lim๐‘›โ†’โˆž๐›ฝ๐‘›=1. Then {๐‘ฅ๐‘›} converges strongly to a common fixed point of {๐‘‡๐‘›} which is nearest to ๐‘ข.

Proof. As in the proof of Theorem 3.2, we have that {๐‘ฅ๐‘›} and {๐‘‡๐‘›๐‘ฅ๐‘›} are bounded. Since {๐‘‡๐‘›} satisfies SZ-condition, there exists a nonexpansive mapping ๐‘‡ of ๐ถ into itself such that lim๐‘›โ†’โˆžsup{๐‘‘(๐‘‡(๐‘‡๐‘›๐‘ฅ),๐‘‡๐‘›๐‘ฅ)โˆถ๐‘ฅโˆˆ{๐‘ฅ๐‘˜}}=0 and โ‹‚๐น(๐‘‡)=โˆž๐‘›=1๐น(๐‘‡๐‘›). By the definition of {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›}, we have ๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐›ฝ=๐‘‘๐‘›๐‘ฆ๐‘›โŠ•๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‡๐‘›๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธโ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธโ‰ค๐›ฝ๐‘›๐‘‘๎€ท๐‘ฆ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฅ๐‘›๎€ธ=๐›ฝ๐‘›๐‘‘๎€ท๐›ผ๐‘›๎€ท๐‘ขโŠ•1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘‘๎€ท๐›ผ๐‘›๎€ท๐‘ขโŠ•1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›๎€ธโ‰ค๐›ฝ๐‘›๐›ผ๐‘›๐‘‘๎€ท๐‘ข,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๐›ผ๎€ธ๎€ท๐‘›๐‘‘๎€ท๐‘ข,๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ๐‘›,๐‘ฅ๐‘›.๎€ธ๎€ธ(3.20) It follows from condition (๐ถ3) and (๐ถ4) that lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ=0.(3.21) Since ๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘‡๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅโ‰ค๐‘‘๐‘›+1,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘‡+๐‘‘๐‘›๐‘ฅ๐‘›๎€ท๐‘‡,๐‘‡๐‘›๐‘ฅ๐‘›๎€ท๐‘‡๎€ท๐‘‡๎€ธ๎€ธ+๐‘‘๐‘›๐‘ฅ๐‘›๎€ธ,๐‘‡๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅโ‰ค2๐‘‘๐‘›+1,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ฝ๐‘‘๎€ท๐‘‡๎€ท๐‘‡+sup๐‘›๐‘ฅ๎€ธ,๐‘‡๐‘›๐‘ฅ๎€ธ๎€ฝ๐‘ฅโˆถ๐‘ฅโˆˆ๐‘˜,๎€พ๎€พ(3.22) this implies by (3.21) and SZ-condition, we have lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘ฅ๐‘›,๐‘‡๐‘ฅ๐‘›๎€ธ=0.(3.23) From lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0 and ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโ‰ค๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›+1,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ,(3.24) it follows that lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘ฅ๐‘›,๐‘‡๐‘›๐‘ฅ๐‘›๎€ธ=0.(3.25) By using the same arguments and techniques as those of Theorem 3.6, we can show that {๐‘ฅ๐‘›} converges to a common fixed point of {๐‘‡๐‘›} which is nearest to ๐‘ข.

Corollary 3.9. Let ๐ถ be a nonempty closed convex subset of a complete CAT(0) space ๐‘‹. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ… and satisfies SZ-condition. Suppose that {๐‘ฅ๐‘›} is a sequence of ๐ถ defined by (3.18) with lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›+1,๐‘ฅ๐‘›)=0. Let {๐›ผ๐‘›} be a sequence in [0,1] which satisfies the condition (๐ถ3) as in Theorem 3.8. Then {๐‘ฅ๐‘›} converges strongly to a common fixed point of {๐‘‡๐‘›} which is nearest to ๐‘ข.

Proof. By putting ๐›ฝ๐‘›=1 for all ๐‘›โˆˆโ„• in Theorem 3.8, we obtain the desired result.

4. W-Mapping in Convex Metric Spaces

In Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, to obtain a convergence result, we have to assume that ({๐‘‡๐‘›},๐‘‡) satisfies AKTT-condition. In general, one cannot apply these results for a sequence of nonexpansive mappings. However, we give an example of a sequence {๐‘‡๐‘›} of nonexpansive mappings satisfying AKTT-condition.

Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself, where ๐ถ is a convex subset of a convex metric space (๐‘‹,๐‘‘,๐‘Š). We now define mappings ๐‘ˆ๐‘›;1,๐‘ˆ๐‘›;2,โ€ฆ,๐‘ˆ๐‘›;๐‘› and ๐‘†๐‘› as follows. For {๐œ†๐‘›} a sequence in [0,1] and ๐‘ฅโˆˆ๐‘‹, ๐‘ˆ๐‘›;๐‘›๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ,๐‘ˆ๐‘›;๐‘›โˆ’1๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘›โˆ’1๐‘ˆ๐‘›;๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›โˆ’1๎€ธ,๐‘ˆ๐‘›;๐‘›โˆ’2๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘›โˆ’2๐‘ˆ๐‘›;๐‘›โˆ’1๐‘ฅ,๐‘ฅ,๐œ†๐‘›โˆ’2๎€ธ,โ‹ฎ๐‘ˆ๐‘›;๐‘˜๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜๎€ธ,๐‘ˆ๐‘›;๐‘˜โˆ’1๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘˜โˆ’1๐‘ˆ๐‘›;๐‘˜๐‘ฅ,๐‘ฅ,๐œ†๐‘˜โˆ’1๎€ธ,โ‹ฎ๐‘ˆ๐‘›;2๎€ท๐‘‡๐‘ฅ=๐‘Š2๐‘ˆ๐‘›;3๐‘ฅ,๐‘ฅ,๐œ†2๎€ธ,๐‘†๐‘›x=๐‘ˆ๐‘›;1๎€ท๐‘‡๐‘ฅ=๐‘Š1๐‘ˆ๐‘›;2๐‘ฅ,๐‘ฅ,๐œ†1๎€ธ.(4.1) Such a mapping ๐‘†๐‘› is called the ๐‘Š-mapping generated by ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘› and ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›.

In 2007, Shimizu [18] generalized ๐‘Š-mapping which was introduced by Takahashi [20] from Banach spaces to convex metric spaces. Then, the following result is obtained by using the same proof as in of [18, Lemma 2].

Lemma 4.1. Let ๐ถ be a nonempty closed convex subset of a uniformly convex metric space (๐‘‹,๐‘‘,๐‘Š) with a continuous convex structure ๐‘Šโˆถ๐‘‹ร—๐‘‹ร—[0,1]โ†’๐‘‹. Let ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘ be nonexpansive mappings of ๐ถ into itself such that โ‹‚๐‘๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ… and let ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘ be real numbers such that 0<๐œ†๐‘›<1 for every ๐‘›=1,2,โ€ฆ,๐‘. Let ๐‘†๐‘ be the ๐‘Š-mapping of ๐ถ into itself generated by ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘ and ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘. Then ๐น(๐‘†๐‘โ‹‚)=๐‘๐‘›=1๐น(๐‘‡๐‘›).

Next, we consider the ๐‘Š-mapping given by a countable infinite family of nonexpansive mappings in a uniformly convex metric space.

Lemma 4.2. Let ๐ถ be a nonempty closed convex subset of a complete uniformly convex metric space (๐‘‹,๐‘‘,๐‘Š) with the property (H). Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…, and let ๐œ†1,๐œ†2,โ€ฆ be real numbers such that 0<๐œ†๐‘›โ‰ค๐‘<1 for every ๐‘›โˆˆโ„•. Then for every ๐‘ฅโˆˆ๐ถ, and ๐‘˜โˆˆโ„•, lim๐‘›โ†’โˆž๐‘ˆ๐‘›;๐‘˜๐‘ฅ exists.

Proof. Let ๐‘ฅโˆˆ๐ถ and โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). Fix ๐‘˜โˆˆโ„•. Then for any ๐‘›โˆˆโ„• with ๐‘›>๐‘˜, we have ๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘˜๐‘ฅ,๐‘ˆ๐‘›;๐‘˜๐‘ฅ๎€ธ๎€ท๐‘Š๎€ท๐‘‡=๐‘‘๐‘˜๐‘ˆ๐‘›+1;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜๎€ธ๎€ท๐‘‡,๐‘Š๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜๎€ธ๎€ธโ‰ค๐œ†๐‘˜๐‘‘๎€ท๐‘‡๐‘˜๐‘ˆ๐‘›+1;๐‘˜+1๐‘ฅ,๐‘‡๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ๎€ธโ‰ค๐œ†๐‘˜๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘˜+1๐‘ฅ,๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ๎€ธ=๐œ†๐‘˜๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘˜+1๐‘ˆ๐‘›+1;๐‘˜+2๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+1๎€ธ๎€ท๐‘‡,๐‘Š๐‘˜+1๐‘ˆ๐‘›;๐‘˜+2๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+1๎€ธ๎€ธโ‰ค๐œ†๐‘˜๐œ†๐‘˜+1๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘˜+2๐‘ฅ,๐‘ˆ๐‘›;๐‘˜+2๐‘ฅ๎€ธโ‹ฎโ‰ค๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘›๐‘ฅ,๐‘ˆ๐‘›;๐‘›๐‘ฅ๎€ธ=๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘›๐‘ˆ๐‘›+1;๐‘›+1๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ๎€ท๐‘‡,๐‘Š๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ๎€ธโ‰ค๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘‡๐‘›๐‘ˆ๐‘›+1;๐‘›+1๐‘ฅ,๐‘‡๐‘›๐‘ฅ๎€ธโ‰ค๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘›+1๎€ธ๐‘ฅ,๐‘ฅ=๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘›+1๐‘ฅ,๐‘ฅ,๐œ†๐‘›+1๎€ธ๎€ธ,๐‘ฅ=๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›+1๐‘‘๎€ท๐‘‡๐‘›+1๎€ธ๐‘ฅ,๐‘ฅโ‰ค๐œ†๐‘˜๐œ†๐‘˜+1โ‹ฏ๐œ†๐‘›+1๎€ท๐‘‘๎€ท๐‘‡๐‘›+1๎€ธ๎€ธ๐‘ฅ,๐‘+๐‘‘(๐‘,๐‘ฅ)โ‰ค2๐‘‘(๐‘,๐‘ฅ)๐‘๐‘›โˆ’๐‘˜+2.(4.2) Thus for ๐‘š>๐‘›, ๐‘‘๎€ท๐‘ˆ๐‘š;๐‘˜๐‘ฅ,๐‘ˆ๐‘›;๐‘˜๐‘ฅ๎€ธ๎€ท๐‘ˆโ‰ค๐‘‘๐‘š;๐‘˜๐‘ฅ,๐‘ˆ๐‘šโˆ’1;๐‘˜๐‘ฅ๎€ธ๎€ท๐‘ˆ+๐‘‘๐‘šโˆ’1;๐‘˜๐‘ฅ,๐‘ˆ๐‘šโˆ’2;๐‘˜๐‘ฅ๎€ธ๎€ท๐‘ˆ+โ‹ฏ+๐‘‘๐‘›+1;๐‘˜๐‘ฅ,๐‘ˆ๐‘›;๐‘˜๐‘ฅ๎€ธโ‰ค2๐‘‘(๐‘,๐‘ฅ)๐‘(๐‘šโˆ’1)โˆ’๐‘˜+2+2๐‘‘(๐‘,๐‘ฅ)๐‘(๐‘šโˆ’2)โˆ’๐‘˜+2+โ‹ฏ+2๐‘‘(๐‘,๐‘ฅ)๐‘๐‘›โˆ’๐‘˜+2=2๐‘‘(๐‘,๐‘ฅ)๐‘šโˆ’1๎“๐‘—=๐‘›๐‘๐‘—โˆ’๐‘˜+2.(4.3) It follows that {๐‘ˆ๐‘›;๐‘˜๐‘ฅ} is a Cauchy sequence. Hence, lim๐‘›โ†’โˆž๐‘ˆ๐‘›;๐‘˜๐‘ฅ exists.

Using the above lemma, one can define mappings ๐‘ˆโˆž;๐‘˜ and ๐‘† of ๐ถ into itself as๐‘ˆโˆž;๐‘˜๐‘ฅ=lim๐‘›โ†’โˆž๐‘ˆ๐‘›;๐‘˜๐‘ฅ,๐‘†๐‘ฅ=lim๐‘›โ†’โˆž๐‘†๐‘›๐‘ฅ=lim๐‘›โ†’โˆž๐‘ˆ๐‘›;1๐‘ฅ,(4.4) for every ๐‘ฅโˆˆ๐ถ. Such a mapping ๐‘† is called the ๐‘Š-mapping generated by ๐‘‡1,๐‘‡2,โ€ฆ and ๐œ†1,๐œ†2,โ€ฆ.

Lemma 4.3. Let ๐ถ be a nonempty closed convex subset of a complete uniformly convex metric space (๐‘‹,๐‘‘,๐‘Š) with the property (H). Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ถ into itself such that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…, and let ๐œ†1,๐œ†2,โ€ฆ be real numbers such that 0<๐œ†๐‘›โ‰ค๐‘<1 for every ๐‘›โˆˆโ„•. Let ๐‘† be the ๐‘Š-mapping generated by ๐‘‡1,๐‘‡2,โ€ฆ and ๐œ†1,๐œ†2,โ€ฆ. Then, ๐‘† is a nonexpansive mapping and โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘‡๐‘›).

Proof. First, we show that ๐‘† is a nonexpansive mapping. For ๐‘ฅ,๐‘ฆโˆˆ๐ถ, we have ๐‘‘๎€ท๐‘†๐‘›๐‘ฅ,๐‘†๐‘›๐‘ฆ๎€ธ๎€ท๐‘Š๎€ท๐‘‡=๐‘‘1๐‘ˆ๐‘›;2๐‘ฅ,๐‘ฅ,๐œ†1๎€ธ๎€ท๐‘‡,๐‘Š1๐‘ˆ๐‘›;2๐‘ฆ,๐‘ฆ,๐œ†1๎€ธ๎€ธโ‰ค๐œ†1๐‘‘๎€ท๐‘‡1๐‘ˆ๐‘›;2๐‘ฅ,๐‘‡1๐‘ˆ๐‘›;2๐‘ฆ๎€ธ+๎€ท1โˆ’๐œ†1๎€ธ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐œ†1๐‘‘๎€ท๐‘ˆ๐‘›;2๐‘ฅ,๐‘ˆ๐‘›;2๐‘ฆ๎€ธ+๎€ท1โˆ’๐œ†1๎€ธ๐‘‘โ‹ฎ(๐‘ฅ,๐‘ฆ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘ˆ๐‘›;๐‘›๐‘ฅ,๐‘ˆ๐‘›;๐‘›๐‘ฆ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘ฅ,๐‘ฆ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ๎€ท๐‘‡,๐‘Š๐‘›๐‘ฆ,๐‘ฆ,๐œ†๐‘›+๎€ท๎€ธ๎€ธ1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐œ†๐‘›๐‘‘๎€ท๐‘‡๐‘›๐‘ฅ,๐‘‡๐‘›๐‘ฆ๎€ธ+๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ท1โˆ’๐œ†๐‘›๎€ธ+๎€ท๐‘‘(๐‘ฅ,๐‘ฆ)1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ค๐‘‘(๐‘ฅ,๐‘ฆ).(4.5) This implies that ๐‘†๐‘› is a nonexpansive mapping, and we have ๐‘‘(๐‘†๐‘ฅ,๐‘†๐‘ฆ)=lim๐‘›โ†’โˆž๐‘‘(๐‘†๐‘›๐‘ฅ,๐‘†๐‘›๐‘ฆ)โ‰ค๐‘‘(๐‘ฅ,๐‘ฆ). Thus, ๐‘† is also a nonexpansive mapping.
Finally, we show that โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘‡๐‘›). Let โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). Then, it is obvious that ๐‘ˆ๐‘›;๐‘˜๐‘=๐‘ for all ๐‘›,๐‘˜โˆˆโ„• with ๐‘›>๐‘˜. So we have ๐‘ˆโˆž;๐‘˜๐‘=๐‘ for all ๐‘˜โˆˆโ„•. Therefore, we have ๐‘†๐‘=๐‘ˆโˆž;1๐‘=๐‘, and hence, โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โŠ†๐น(๐‘†). We now show that โ‹‚๐น(๐‘†)โŠ†โˆž๐‘›=1๐น(๐‘‡๐‘›). Let ๐‘ฅโˆˆ๐น(๐‘†) and let โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). Then we have ๐‘‘๎€ท๐‘†๐‘›๐‘,๐‘†๐‘›๐‘ฅ๎€ธ๎€ท๐‘ˆ=๐‘‘๐‘›;1๐‘,๐‘ˆ๐‘›;1๐‘ฅ๎€ธ๎€ท๎€ท๐‘‡=๐‘‘๐‘,๐‘Š1๐‘ˆ๐‘›;2๐‘ฅ,๐‘ฅ,๐œ†1๎€ธ๎€ธโ‰ค๐œ†1๐‘‘๎€ท๐‘,๐‘‡1๐‘ˆ๐‘›;2๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐‘‘๎€ท๐‘,๐‘ˆ๐‘›;2๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๎€ธโ‹ฎ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๐‘‘๎€ท๐‘,๐‘ˆ๐‘›;๐‘˜๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๐‘‘๎€ท๎€ท๐‘‡๐‘,๐‘Š๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+๎€ท๎€ธ๎€ธ1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๐œ†๐‘˜๐‘‘๎€ท๐‘,๐‘‡๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ๎€ธ+๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ท1โˆ’๐œ†๐‘˜๎€ธ๐‘‘+๎€ท(๐‘,๐‘ฅ)1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๐‘‘๎€ท๐‘,๐‘‡๐‘˜๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๐‘‘๎€ท๐‘,๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๎€ธโ‹ฎ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘,๐‘ˆ๐‘›;๐‘›๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๎€ท๐‘‡๐‘,๐‘Š๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›+๎€ท๎€ธ๎€ธ1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐œ†๐‘›๐‘‘๎€ท๐‘,๐‘‡๐‘›๐‘ฅ๎€ธ+๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ท1โˆ’๐œ†๐‘›๎€ธ+๎€ท๐‘‘(๐‘,๐‘ฅ)1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘,๐‘‡๐‘›๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐‘‘(๐‘,๐‘ฅ).(4.6) Taking ๐‘›โ†’โˆž, we obtain ๐‘‘(๐‘†๐‘,๐‘†๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๐‘‘๎€ท๎€ท๐‘‡๐‘,๐‘Š๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+๎€ท๎€ธ๎€ธ1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๐œ†๐‘˜๐‘‘๎€ท๐‘,๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ๎€ธ+๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ท1โˆ’๐œ†๐‘˜๎€ธ+๎€ท๐‘‘(๐‘,๐‘ฅ)1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜โˆ’1๎€ธ๐‘‘(๐‘,๐‘ฅ)=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๐‘‘๎€ท๐‘,๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ๎€ธ+๎€ท1โˆ’๐œ†1๐œ†2โ‹ฏ๐œ†๐‘˜๎€ธ๐‘‘(๐‘,๐‘ฅ)โ‰ค๐‘‘(๐‘,๐‘ฅ).(4.7) Since โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›)โŠ†๐น(๐‘†), we have ๐‘‘(๐‘†๐‘,๐‘†๐‘ฅ)=๐‘‘(๐‘,๐‘ฅ). Then, for ๐œ†๐‘›โˆˆ(0,1), ๐‘›โˆˆโ„•, we have ๐‘‘๎€ท๐‘,๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ๎€ธ๎€ท๎€ท๐‘‡=๐‘‘(๐‘,๐‘ฅ),๐‘‘๐‘,๐‘Š๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜๎€ธ๎€ธ=๐‘‘(๐‘,๐‘ฅ),(4.8) for every ๐‘˜โˆˆโ„•. Suppose that ๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅโ‰ ๐‘ฅ. Then ๐‘‘(๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ,๐‘ฅ)>0. It follows by Lemma 2.5, we have ๐‘‘๎€ท๎€ท๐‘‡๐‘,๐‘Š๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ,๐‘ฅ,๐œ†๐‘˜๎€ธ๎€ธ<๐‘‘(๐‘,๐‘ฅ).(4.9) This is a contradiction. Hence, ๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ=๐‘ฅ. Since ๐‘ˆ๐‘›;๐‘˜+1๐‘ฅ=๐‘Š(๐‘‡๐‘˜+1๐‘ˆ๐‘›;๐‘˜+2๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+1), we have ๐‘ˆโˆž;๐‘˜+1๐‘ฅ=lim๐‘›โ†’โˆž๐‘ˆ๐‘›;๐‘˜+1๎€ท๐‘‡๐‘ฅ=๐‘Š๐‘˜+1๐‘ˆโˆž;๐‘˜+2๐‘ฅ,๐‘ฅ,๐œ†๐‘˜+1๎€ธ=๐‘ฅ.(4.10) So, we have ๐‘ฅ=๐‘‡๐‘˜๐‘ˆโˆž;๐‘˜+1๐‘ฅ=๐‘‡๐‘˜๐‘ฅ for every ๐‘˜โˆˆโ„•. This implies that โ‹‚๐‘ฅโˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›). Therefore, we have โ‹‚๐น(๐‘†)โŠ†โˆž๐‘›=1๐น(๐‘‡๐‘›).

Lemma 4.4. Suppose that ๐‘‹,๐ถ,{๐‘‡๐‘›},{๐œ†๐‘›} are as in Lemma 4.3. Let ๐‘†๐‘› and ๐‘† be the ๐‘Š-mappings generated by ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘› and ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›, and ๐‘‡1,๐‘‡2,โ€ฆ and ๐œ†1,๐œ†2,โ€ฆ, respectively. Then ({๐‘†๐‘›},๐‘†) satisfies AKTT-condition, and โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘†๐‘›).

Proof. Let ๐ต be a bounded subset of ๐ถ and ๐‘ฅโˆˆ๐ต. For โ‹‚๐‘โˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›), we have ๐‘‘๎€ท๐‘†๐‘›+1๐‘ฅ,๐‘†๐‘›๐‘ฅ๎€ธ๎€ท๐‘ˆ=๐‘‘๐‘›+1;1๐‘ฅ,๐‘ˆ๐‘›;1๐‘ฅ๎€ธ๎€ท๐‘Š๎€ท๐‘‡=๐‘‘1๐‘ˆ๐‘›+1;2๐‘ฅ,๐‘ฅ,๐œ†1๎€ธ๎€ท๐‘‡,๐‘Š1๐‘ˆ๐‘›;2๐‘ฅ,๐‘ฅ,๐œ†1๎€ธ๎€ธโ‰ค๐œ†1๐‘‘๎€ท๐‘‡1๐‘ˆ๐‘›+1;2๐‘ฅ,๐‘‡1๐‘ˆ๐‘›;2๐‘ฅ๎€ธโ‰ค๐œ†1๐‘‘๎€ท๐‘ˆ๐‘›+1;2๐‘ฅ,๐‘ˆ๐‘›;2๐‘ฅ๎€ธโ‹ฎโ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘›๐‘ฅ,๐‘ˆ๐‘›;๐‘›๐‘ฅ๎€ธ=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›โˆ’1๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘›๐‘ˆ๐‘›+1;๐‘›+1๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ๎€ท๐‘‡,๐‘Š๐‘›๐‘ฅ,๐‘ฅ,๐œ†๐‘›๎€ธ๎€ธโ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘ˆ๐‘›+1;๐‘›+1๎€ธ๐‘ฅ,๐‘ฅ=๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›๐‘‘๎€ท๐‘Š๎€ท๐‘‡๐‘›+1๐‘ฅ,๐‘ฅ,๐œ†๐‘›+1๎€ธ๎€ธ,๐‘ฅโ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›+1๐‘‘๎€ท๐‘‡๐‘›+1๎€ธ๐‘ฅ,๐‘ฅโ‰ค๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›+1๎€ท๐‘‘๎€ท๐‘‡๐‘›+1๎€ธ๎€ธ๐‘ฅ,๐‘+๐‘‘(๐‘,๐‘ฅ)โ‰ค2๐œ†1๐œ†2โ‹ฏ๐œ†๐‘›+1๐‘‘(๐‘,๐‘ฅ)โ‰ค2๐‘๐‘›+1๐‘‘(๐‘,๐‘ฅ).(4.11) This implies โˆž๎“๐‘›=1๎€ฝ๐‘‘๎€ท๐‘†sup๐‘›+1๐‘ฅ,๐‘†๐‘›๐‘ฅ๎€ธ๎€พโˆถ๐‘ฅโˆˆ๐ต<โˆž.(4.12) Thus, ({๐‘†๐‘›},๐‘†) satisfies AKTT-condition. Moreover, from Lemmas 4.1โ€“4.3, we obtain that โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘†๐‘›).

Remark 4.5. Lemmas 4.2 and 4.3 were proved in Banach spaces by Shimoji and Takahashi [21], and Lemma 4.4 was proved in Banach spaces by Peng and Yao [22].

Remark 4.6. Suppose that ๐‘‹,๐ถ,{๐‘‡๐‘›},{๐œ†๐‘›} are as in Lemma 4.3. Let ๐‘†๐‘› and ๐‘† be the ๐‘Š-mappings generated by ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘› and ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›, and ๐‘‡1,๐‘‡2,โ€ฆ and ๐œ†1,๐œ†2,โ€ฆ, respectively. By Lemma 4.4, we know that ({๐‘†๐‘›},๐‘†) satisfies the AKTT-condition and โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘†๐‘›). Therefore, in Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, the mapping ๐‘‡๐‘› can be also replaced by ๐‘†๐‘› without assuming the AKTT-condition and โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘†๐‘›).

Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and the National Research University Project under Thailand's Office of the Higher Education Commission, the Commission on Higher Education for financial support. The first author is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand.

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