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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 745451, 19 pages
http://dx.doi.org/10.1155/2011/745451
Research Article

Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

School of Science, University of Phayao, Phayao 56000, Thailand

Received 31 May 2011; Revised 26 August 2011; Accepted 30 August 2011

Academic Editor: Victor M.ย Perez Garcia

Copyright ยฉ 2011 Tanakit Thianwan. 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 study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach space ๐ธ which has a Frรฉchet differentiable norm or its dual ๐ธโˆ— has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Frรฉchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

1. Introduction and Preliminaries

Let ๐ถ be a nonempty closed convex subset of real normed linear space ๐‘‹. Let ๐‘‡โˆถ๐ถโ†’๐ถ be a mapping. A point ๐‘ฅโˆˆ๐ถ is called a fixed point of ๐‘‡ if and only if ๐‘‡๐‘ฅ=๐‘ฅ. The set of all fixed points of a mapping ๐‘‡ is denoted by ๐น(๐‘‡). A self-mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be nonexpansive if โ€–๐‘‡(๐‘ฅ)โˆ’๐‘‡(๐‘ฆ)โ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. A self-mapping ๐‘‡โˆถ๐ถโ†’๐ถ is called asymptotically nonexpansive if there exists a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž),๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž such that โ€–๐‘‡๐‘›(๐‘ฅ)โˆ’๐‘‡๐‘›(๐‘ฆ)โ€–โ‰ค๐‘˜๐‘›โ€–๐‘ฅโˆ’๐‘ฆโ€–(1.1) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1. A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be uniformly ๐ฟ-Lipschitzian if there exists a constant ๐ฟ>0 such that โ€–๐‘‡๐‘›(๐‘ฅ)โˆ’๐‘‡๐‘›(๐‘ฆ)โ€–โ‰ค๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–(1.2) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1. โ€‰๐‘‡ is uniformly Hรถlder continuous if there exist positive constants ๐ฟ and ๐›ผ such that โ€–๐‘‡๐‘›(๐‘ฅ)โˆ’๐‘‡๐‘›(๐‘ฆ)โ€–<๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–๐›ผ(1.3) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1. โ€‰๐‘‡ is termed as uniformly equicontinuous if, for any ๐œ€>0, there exists ๐›ฟ>0 such that โ€–๐‘‡๐‘›(๐‘ฅ)โˆ’๐‘‡๐‘›(๐‘ฆ)โ€–โ‰ค๐œ€(1.4) whenever โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ค๐›ฟ for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1 or, equivalently, ๐‘‡ is uniformly equicontinuous if and only if โ€–โ€–๐‘‡๐‘›๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘‡๐‘›๎€ท๐‘ฆ๐‘›๎€ธโ€–โ€–โŸถ0(1.5) whenever โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.

It is easy to see that if ๐‘‡ is an asymptotically nonexpansive, then it is uniformly ๐ฟ-Lipschitzian with the uniform Lipschitz constant ๐ฟ=sup{๐‘˜๐‘›โˆถ๐‘›โ‰ฅ1}.

Remark 1.1. It is clear that asymptotically nonexpansiveness โ‡’ uniformly L-Lipschitz โ‡’ uniformly Hรถlder continuous โ‡’ uniformly equicontinuous.

However, their converse fail in the presence of the following example.

Example 1.2 (see [1]). Define ๐‘‡โˆถ[0,1]โ†’[0,1] by ๐‘‡๐‘ฅ=(1โˆ’๐‘ฅ3/2)2/3 for all ๐‘ฅโˆˆ[0,1].

Fixed-point iteration process for nonexpansive self-mappings including Mann and Ishikawa iteration processes has been studied extensively by various authors [2โ€“8]. For nonexpansive nonself-mappings, some authors (see [9โ€“13]) have studied the strong and weak convergence theorems in Hilbert spaces or uniformly convex Banach spaces.

In [14], Tan and Xu introduced a modified Ishikawa iteration process: ๐‘ฅ๐‘›+1=๎€ท1โˆ’๐‘๐‘›๎€ธ๐‘ฅ๐‘›+๐‘๐‘›๐‘‡๎€ท๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘ฅ๐‘›+๐›พ๐‘›๐‘‡๐‘ฅ๐‘›๎€ธ,๐‘›โ‰ฅ1,(1.6) to approximate fixed points of nonexpansive self-mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space ๐‘‹. The mapping ๐‘‡ remains self-mapping of a nonempty closed convex subset ๐ถ of a uniformly convex Banach space. If, however, the domain ๐ถ of ๐‘‡ is a proper subset of ๐‘‹ (and this is the case in several applications) and ๐‘‡ maps ๐ถ into ๐‘‹ then, the sequence {๐‘ฅ๐‘›} generated by (1.6) may not be well defined. More precisely, Tan and Xu [14] proved weak convergence of the sequences generated by (1.6) to some fixed point of ๐‘‡ in a uniformly convex Banach space which satisfies Opial's condition or has a Frรฉchet differentiable norm.

Note that each ๐‘™๐‘(1โ‰ค๐‘<โˆž) satisfies Opial's condition, while all ๐ฟ๐‘ do not have the property unless ๐‘=2 and the dual of reflexive Banach spaces with a Frรฉchet differentiable norm has the Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Frรฉchet differentiable norm nor Opial property; however, their dual does have the Kadec-Klee property (see [15, 16]).

In 2005, Shahzad [11] extended Tan and Xu's result [14] to the case of nonexpansive nonself-mapping in a uniformly convex Banach space. He studied weak convergence of the modified Ishikawa type iteration process: ๐‘ฅ๐‘›+1=๐‘ƒ๎€ท๎€ท1โˆ’๐‘๐‘›๎€ธ๐‘ฅ๐‘›+๐‘๐‘›๐‘‡๐‘ƒ๎€ท๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘ฅ๐‘›+๐›พ๐‘›๐‘‡๐‘ฅ๐‘›๎€ธ๎€ธ,๐‘›โ‰ฅ1,(1.7) in a uniformly convex Banach space whose dual has the Kadec-Klee property. The result applies not only to ๐ฟ๐‘ spaces with (1โ‰ค๐‘<โˆž) but also to other spaces which do not satisfy Opial's condition or have a Frรฉchet differentiable norm. Meanwhile, the results of [11] generalized the results of [14].

The class of asymptotically nonexpansive self-mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [17] proved that if ๐ถ is a nonempty closed convex and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

In 1991, the modified Mann iteration which was introduced by Schu [18] generates a sequence {๐‘ฅ๐‘›} in the following manner: ๐‘ฅ๐‘›+1=๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ผ๐‘›๐‘‡๐‘›๐‘ฅ๐‘›,๐‘›โ‰ฅ1,(1.8) where {๐›ผ๐‘›} is a sequence in the interval (0,1) and ๐‘‡โˆถ๐ถโ†’๐ถ is an asymptotically nonexpansive mapping. To be more precise, Schu [18] obtained the following weak convergence result for an asymptotically nonexpansive mapping in a uniformly convex Banach space which satisfies Opial's condition.

Theorem 1.3 (see [18]). Let ๐‘‹ be a uniformly convex Banach space satisfying Opialโ€™s condition, โˆ…โ‰ ๐ถโŠ‚๐‘‹ closed bounded and convex, and ๐‘‡โˆถ๐ถโ†’๐ถ asymptotically nonexpansive with sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) for which โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž and {๐›ผ๐‘›}โˆˆ[0,1] is bounded away. Let {๐‘ฅ๐‘›} be a sequence generated in (1.8). Then, the sequence {๐‘ฅ๐‘›} converges weakly to some fixed point ofโ€‰ ๐‘‡.

Since then, Schu's iteration process has been widely used to approximate fixed points of asymptotically nonexpansive self-mappings in Hilbert space or Banach spaces (see [6, 14, 19, 20]).

In 1994, Tan and Xu [21] obtained the following results.

Theorem 1.4 (see [21]). Let ๐‘‹ be a uniformly convex Banach space whose norm is Frรฉchet differentiable, ๐ถ a nonempty closed and convex subset of ๐‘‹, and ๐‘‡โˆถ๐ถโ†’๐ถ an asymptotically nonexpansive mapping with a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž such that ๐น(๐‘‡) is nonempty. Let {๐‘ฅ๐‘›} be sequence generated in (1.8), where {๐›ผ๐‘›} is a real sequence bounded away from 0 and 1. Then, the sequence {๐‘ฅ๐‘›} converges weakly to some point in ๐น(๐‘‡).

In 2001, Khan and Takahashi [22] constructed and studied the following Ishikawa iteration process:๐‘ฅ๐‘›+1=๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ผ๐‘›๐‘‡๐‘›1๐‘ฆ๐‘›,๐‘ฆ๐‘›=๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ฝ๐‘›๐‘‡๐‘›2๐‘ฅ๐‘›,๐‘›โ‰ฅ1,(1.9) whereโ€‰ ๐‘‡1, ๐‘‡2 โ€‰are asymptotically nonexpansive self-mappings on ๐ถ withโ€‰ โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž (rate of convergence) and 0โ‰ค๐›ผ๐‘›, ๐›ฝ๐‘›โ‰ค1.

Note that the rate of convergence condition, namely, โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž has remained in extensive use to prove both weak and strong convergence theorems to approximate fixed points of asymptotically nonexpansive maps. The conditions like Opial's condition, Kadec-Klee property, or Frรฉchet differentiable norm have remained key to prove weak convergence theorems.

In 2010, Khan and Fukhar-Ud-Din [23] established weak convergence of Ishikawa iterates of two asymptotically nonexpansive self-mappings without any condition on the rate of convergence associated with the two mappings. They got that the following new weak convergence theorem does not require any of Opialโ€™s condition, Kadec-Klee property or Frรฉchet differentiable norm.

Theorem 1.5 (see [23]). Let ๐ถ be a nonempty bounded closed convex subset of a uniformly convex Banach space ๐‘‹. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐ถ be asymptotically nonexpansive maps with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that lim๐‘›โ†’โˆž๐‘˜๐‘›=1, lim๐‘›โ†’โˆž๐‘™๐‘›=1, respectively. Let the sequence {๐‘ฅ๐‘›} be as in (1.9) with ๐›ฟโ‰ค๐›ผ๐‘›,๐›ฝ๐‘›โ‰ค1โˆ’๐›ฟ. for some ๐›ฟโˆˆ(0,1/2). If ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…, then {๐‘ฅ๐‘›} converges weakly to a common fixed point of ๐‘‡1 and ๐‘‡2.

The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [24] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition 1.6 (see [24]). Let ๐ถ be a nonempty subset of a real normed linear space ๐‘‹. Let ๐‘ƒโˆถ๐‘‹โ†’๐ถ be a nonexpansive retraction of ๐‘‹ onto ๐ถ. A nonself-mapping ๐‘‡โˆถ๐ถโ†’๐‘‹ is called asymptotically nonexpansive if there exists a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž), ๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž such that โ€–โ€–๐‘‡(๐‘ƒ๐‘‡)๐‘›โˆ’1๐‘ฅโˆ’๐‘‡(๐‘ƒ๐‘‡)๐‘›โˆ’1๐‘ฆโ€–โ€–โ‰ค๐‘˜๐‘›โ€–๐‘ฅโˆ’๐‘ฆโ€–(1.10) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1. โ€‰๐‘‡ is said to be uniformly L-Lipschitzian if there exists a constant ๐ฟ>0 such that โ€–โ€–๐‘‡(๐‘ƒ๐‘‡)๐‘›โˆ’1๐‘ฅโˆ’๐‘‡(๐‘ƒ๐‘‡)๐‘›โˆ’1๐‘ฆโ€–โ€–โ‰ค๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–(1.11) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘›โ‰ฅ1.

By studying the following iteration process: ๐‘ฅ1โˆˆ๐ถ,๐‘ฅ๐‘›+1=๐‘ƒ๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ผ๐‘›๐‘‡(๐‘ƒ๐‘‡)๐‘›โˆ’1๐‘ฅ๐‘›๎€ธ,(1.12)Chidume et al. [24] got the following weak convergence theorem for asymptotically nonexpansive nonself-mapping.

Theorem 1.7 (see [24]). Let ๐‘‹ be a real uniformly convex Banach space which has a Frรฉchet differentiable norm and ๐ถ a nonempty closed convex subset of ๐‘‹. Let ๐‘‡โˆถ๐ถโ†’๐‘‹ be an asymptotically nonexpansive map with sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜2๐‘›โˆ’1)<โˆž and ๐น(๐‘‡)โ‰ โˆ…. Let {๐›ผ๐‘›}โŠ‚(0,1) be such that ๐œ–โ‰ค1โˆ’๐›ผ๐‘›โ‰ค1โˆ’๐œ–, for all ๐‘›โ‰ฅ1 and some ๐œ–>0. From an arbitrary ๐‘ฅ1โˆˆ๐ถ, define the sequence {๐‘ฅ๐‘›} by (1.12). Then, {๐‘ฅ๐‘›} converges weakly to some fixed point of ๐‘‡.

If ๐‘‡ is a self-mapping, then ๐‘ƒ becomes the identity mapping so that (1.10) and (1.11) reduce to (1.1) and (1.2), respectively. Equation (1.12) reduces to (1.8).

In 2006, Wang [25] generalizes the iteration process (1.12) as follows: ๐‘ฅ1โˆˆ๐ถ,๐‘ฅ๐‘›+1๎‚€๎€ท=๐‘ƒ1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚,๐‘ฆ๐‘›๎‚€๎€ท=๐‘ƒ1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›๎‚,๐‘›โ‰ฅ1,(1.13) where ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ are asymptotically nonexpansive nonself-mappings and {๐›ผ๐‘›}, {๐›ฝ๐‘›} are real sequences in [0,1). He studied the strong and weak convergence of the iterative scheme (1.13) under proper conditions. Meanwhile, the results of [25] generalized the results of [24].

Recently, an iterative scheme which is called the projection type Ishikawa iteration for two asymptotically nonexpansive nonself-mappings was defined and constructed by Thianwan [26]. It is given as follows:๐‘ฅ๐‘›+1๎‚€๎€ท=๐‘ƒ1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚,๐‘ฆ๐‘›๎‚€๎€ท=๐‘ƒ1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›๎‚,๐‘›โ‰ฅ1,(1.14) where {๐›ผ๐‘›} and {๐›ฝ๐‘›} are appropriate real sequences in [0,1).

In [26], Thianwan gave the following weak convergence theorem.

Theorem 1.8. Let ๐‘‹ be a uniformly convex Banach space which satisfies Opial's condition and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž, โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be the sequences defined by (1.14). Then, {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} converge weakly to a common fixed point of ๐‘‡1 and ๐‘‡2.

The iterative schemes (1.14) and (1.13) are independent: neither reduces to the other. If ๐‘‡1=๐‘‡2 and ๐›ฝ๐‘›=0 for all ๐‘›โ‰ฅ1, then (1.14) reduces to (1.12). It also can be reduces to Schu process (1.8).

Inspired and motivated by the recent works, we prove some new weak convergence theorems of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Now, we recall some well-known concepts and results.

Let ๐‘‹ be a Banach space with dimension ๐‘‹โ‰ฅ2. The modulus of ๐‘‹ is the function ๐›ฟ๐‘‹โˆถ(0,2]โ†’[0,1] defined by ๐›ฟ๐‘‹๎‚†โ€–โ€–โ€–1(๐œ–)=inf1โˆ’2โ€–โ€–โ€–โˆถ๎‚‡(๐‘ฅ+๐‘ฆ)โ€–๐‘ฅโ€–=1,โ€–๐‘ฆโ€–=1,๐œ–=โ€–๐‘ฅโˆ’๐‘ฆโ€–.(1.15) Banach space ๐‘‹ is uniformly convex if and only if ๐›ฟ๐‘‹(๐œ–)>0 for all ๐œ–โˆˆ(0,2]. It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that a Banach space ๐‘‹ is said to satisfy Opial's condition [27] if ๐‘ฅ๐‘›โ†’๐‘ฅ weakly as ๐‘›โ†’โˆž and ๐‘ฅโ‰ ๐‘ฆ implying that limsup๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฅ<limsup๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ฆ.(1.16) The norm of ๐‘‹ is said to be Frรฉchet differentiable if for each ๐‘ฅโˆˆ๐‘‹ with โ€–๐‘ฅโ€–=1 the limit lim๐‘กโ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(1.17) exists and is attained uniformly for ๐‘ฆ, with โ€–๐‘ฆโ€–=1. In the case of Frรฉchet differentiable norm, it has been obtained in [21] that1โŸจโ„Ž,๐ฝ(๐‘ฅ)โŸฉ+2โ€–๐‘ฅโ€–2โ‰ค12โ€–๐‘ฅ+โ„Žโ€–21โ‰คโŸจโ„Ž,๐ฝ(๐‘ฅ)โŸฉ+2โ€–๐‘ฅโ€–2+๐‘(โ€–โ„Žโ€–)(1.18) for all ๐‘ฅ, โ„Ž in ๐ธ, where ๐ฝ is the normalized duality map from ๐ธ to ๐ธโˆ— defined by๎‚†๐‘ฅ๐ฝ(๐‘ฅ)=โˆ—โˆˆ๐ธโˆ—โˆถโŸจ๐‘ฅ,๐‘ โˆ—โŸฉ=โ€–๐‘ฅโ€–2=โ€–๐‘ฅโˆ—โ€–2๎‚‡,(1.19)

โŸจโ‹…,โ‹…โŸฉ is the duality pairing between ๐ธ and ๐ธโˆ— and ๐‘ is an increasing function defined on [0,โˆž) such that lim๐‘กโ†“0๐‘(๐‘ก)/๐‘ก=0.

A subset ๐ถ of ๐‘‹ is said to be retract if there exists continuous mapping ๐‘ƒโˆถ๐‘‹โ†’๐ถ such that ๐‘ƒ๐‘ฅ=๐‘ฅ for all ๐‘ฅโˆˆ๐ถ. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping ๐‘ƒโˆถ๐‘‹โ†’๐‘‹ is said to be a retraction if ๐‘ƒ2=๐‘ƒ. If a mapping ๐‘ƒ is a retraction, then ๐‘ƒ๐‘ง=๐‘ง for every ๐‘งโˆˆ๐‘…(๐‘ƒ), range of ๐‘ƒ. A set ๐ถ is optimal if each point outside ๐ถ can be moved to be closer to all points of ๐ถ. It is well known (see [28]) that(1)if ๐‘‹ is a separable, strictly convex, smooth, reflexive Banach space, and if ๐ถโŠ‚๐‘‹ is an optimal set with interior, then ๐ถ is a nonexpansive retract of ๐‘‹;(2)a subset of ๐‘™๐‘, with 1<๐‘<โˆž, is a nonexpansive retract if and only if it is optimal.

Note that every nonexpansive retract is optimal. In strictly convex Banach spaces, optimal sets are closed and convex. Moreover, every closed convex subset of a Hilbert space is optimal and also a nonexpansive retract.

Recall that weak convergence is defined in terms of bounded linear functionals on ๐‘‹ as follows.

A sequence {๐‘ฅ๐‘›} in a normed space ๐‘‹ is said to be weakly convergent if there is an ๐‘ฅโˆˆ๐‘‹ such that lim๐‘›โ†’โˆž๐‘“(๐‘ฅ๐‘›)=๐‘“(๐‘ฅ) for every bounded linear functional ๐‘“ on ๐‘‹. The element ๐‘ฅ is called the weak limit of {๐‘ฅ๐‘›}, and we say that {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ. In this paper, we use โ†’ and โ‡€ to denote the strong convergence and weak convergence, respectively.

A Banach space ๐‘‹ is said to have the Kadec-Klee property if, for every sequence {๐‘ฅ๐‘›} in ๐‘‹, ๐‘ฅ๐‘›โ‡€๐‘ฅ and โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘ฅโ€– together imply โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโ€–โ†’0; for more details on Kadec-Klee property, the reader is referred to [29, 30] and the references therein.

In the sequel, the following lemmas are needed to prove our main results.

Lemma 1.9 (see [31]). Let ๐‘>1, ๐‘Ÿ>0 be two fixed numbers. Then, a Banach space ๐‘‹ is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function ๐‘”โˆถ[0,โˆž)โ†’[0,โˆž), ๐‘”(0)=0 such that โ€–๐œ†๐‘ฅ+(1โˆ’๐œ†)๐‘ฆโ€–๐‘โ‰ค๐œ†โ€–๐‘ฅโ€–๐‘+(1โˆ’๐œ†)โ€–๐‘ฆโ€–๐‘โˆ’๐‘ค๐‘(๐œ†)๐‘”(โ€–๐‘ฅโˆ’๐‘ฆโ€–)(1.20) for all ๐‘ฅ, ๐‘ฆ in ๐ต๐‘Ÿ={๐‘ฅโˆˆ๐‘‹โˆถโ€–๐‘ฅโ€–โ‰ค๐‘Ÿ}, ๐œ†โˆˆ[0,1], where ๐‘ค๐‘(๐œ†)=๐œ†(1โˆ’๐œ†)๐‘+๐œ†๐‘(1โˆ’๐œ†).(1.21)

Lemma 1.10 (see [24]). Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a nonempty closed convex subset of ๐‘‹, and let ๐‘‡โˆถ๐ถโ†’๐‘‹ be an asymptotically nonexpansive mapping with a sequence {๐‘˜๐‘›}โŠ‚[1,โˆž) and ๐‘˜๐‘›โ†’1 as ๐‘›โ†’โˆž. Then, ๐ผโˆ’๐‘‡ is demiclosed at zero; that is, if ๐‘ฅ๐‘›โ†’๐‘ฅ weakly and ๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ†’0 strongly, then ๐‘ฅโˆˆ๐น(๐‘‡).

Lemma 1.11 (see [26]). Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž, โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [0,1). From an arbitrary ๐‘ฅ1โˆˆ๐ถ, define the sequence {๐‘ฅ๐‘›} by (1.14). If ๐‘žโˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2), then lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘žโ€– exists.

Lemma 1.12 (see [26]). Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž, โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). From an arbitrary ๐‘ฅ1โˆˆ๐ถ, define the sequence {๐‘ฅ๐‘›} by (1.14). Then, lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡1๐‘ฅ๐‘›โ€–=lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡2๐‘ฅ๐‘›โ€–=0.

Lemma 1.13 (see [16]). Let ๐‘‹ be a real reflexive Banach space such that its dual ๐‘‹โˆ— has the Kadec-Klee property. Let {๐‘ฅ๐‘›} be a bounded sequence in ๐‘‹ and ๐‘ฅโˆ—,๐‘ฆโˆ—โˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›), where ๐œ”๐‘ค(๐‘ฅ๐‘›) denotes the set of all weak subsequential limits of {๐‘ฅ๐‘›}. Suppose that lim๐‘›โ†’โˆžโ€–๐‘ก๐‘ฅ๐‘›+(1โˆ’๐‘ก)๐‘ฅโˆ—โˆ’๐‘ฆโˆ—โ€– exists for all ๐‘กโˆˆ[0,1]. Then, ๐‘ฅโˆ—=๐‘ฆโˆ—.

We denote by ฮ“ the set of strictly increasing, continuous convex functions ๐›พโˆถโ„+โ†’โ„+ with ๐›พ(0)=0. Let ๐ถ be a convex subset of the Banach space ๐‘‹. A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is said to be type (๐›พ) [32] if ๐›พโˆˆฮ“ and 0โฉฝ๐›ผโฉฝ1, ๐›พ(โ€–๐›ผ๐‘‡๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘ฆโˆ’๐‘‡(๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘ฆ)โ€–)โฉฝโ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–(1.22) for all ๐‘ฅ, ๐‘ฆ in ๐ถ. Obviously, every typeโ€‰โ€‰(๐›พ) mapping is nonexpansive. For more information about mappings of type (๐›พ), see [33โ€“35].

Lemma 1.14 (see [36, 37]). Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a convex subset of ๐‘‹. Then, there exists ๐›พโˆˆฮ“ such that for each mapping ๐‘†โˆถ๐ถโ†’๐ถ with Lipschitz constant ๐ฟ, โ€–๐›ผ๐‘†๐‘ฅ+(1โˆ’๐›ผ)๐‘†๐‘ฆโˆ’๐‘†(๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘ฆ)โ€–โฉฝ๐ฟ๐›พโˆ’1๎‚€1โ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ’๐ฟ๎‚โ€–๐‘†๐‘ฅโˆ’๐‘†๐‘ฆโ€–(1.23) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and 0<๐›ผ<1.

2. Main Results

In this section, we prove weak convergence theorems of the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Firstly, we deal with the weak convergence of the sequence {๐‘ฅ๐‘›} defined by (1.14) in a real uniformly convex Banach space ๐‘‹ whose dual ๐‘‹โˆ— has the Kadec-Klee property. In order to prove our main results, the following lemma is needed.

Lemma 2.1. Let ๐‘‹ be a real uniformly convex Banach space and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž, โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be the sequences defined by (1.14). Then, for all ๐‘ข,๐‘ฃโˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2), the limit lim๐‘›โ†’โˆžโ€–๐‘ก๐‘ฅ๐‘›โˆ’(1โˆ’๐‘ก)๐‘ขโˆ’๐‘ฃโ€– exists for all ๐‘กโˆˆ[0,1].

Proof. It follows from Lemma 1.11 that the sequence {๐‘ฅ๐‘›} is bounded. Then, there exists ๐‘…>0 such that {๐‘ฅ๐‘›}โŠ‚๐ต๐‘…(0)โˆฉ๐ถ. Let ๐‘Ž๐‘›(๐‘ก)โˆถ=โ€–๐‘ก๐‘ฅ๐‘›+(1โˆ’๐‘ก)๐‘ขโˆ’๐‘ฃโ€– where ๐‘กโˆˆ[0,1]. Then, lim๐‘›โ†’โˆž๐‘Ž๐‘›(0)=โ€–๐‘ขโˆ’๐‘ฃโ€– and, by Lemma 1.11, lim๐‘›โ†’โˆž๐‘Ž๐‘›(1)=lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฃโ€– exists. Without loss of the generality, we may assume that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฃโ€–=๐‘Ÿ for some positive number ๐‘Ÿ. Let ๐‘ฅโˆˆ๐ถ and ๐‘กโˆˆ(0,1). โ€‰For each ๐‘›โ‰ฅ1, define ๐ด๐‘›โˆถ๐ถโ†’๐ถ by ๐ด๐‘›๎‚€๎€ท๐‘ฅ=๐‘ƒ1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›(๐‘ฅ)+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚(๐‘ฅ),(2.1) where ๐‘ฆ๐‘›๎‚€๎€ท(๐‘ฅ)=๐‘ƒ1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๎‚.(2.2)
Setting ๐‘˜๐‘›=1+๐‘ ๐‘› and ๐‘™๐‘›=1+๐‘ก๐‘›. For ๐‘ฅ,๐‘งโˆˆ๐ถ, we have โ€–โ€–๐ด๐‘›๐‘ฅโˆ’๐ด๐‘›๐‘งโ€–โ€–=โ€–โ€–๐‘ƒ๎‚€๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›(๐‘ฅ)+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚๎‚€๎€ท(๐‘ฅ)โˆ’๐‘ƒ1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›(๐‘ง)+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚โ€–โ€–โ‰คโ€–โ€–๎€ท(๐‘ง)1โˆ’๐›ผ๐‘›๐‘ฆ๎€ธ๎€ท๐‘›(๐‘ฅ)โˆ’๐‘ฆ๐‘›๎€ธ(๐‘ง)+๐›ผ๐‘›๎‚€๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›(๐‘ฅ)โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚โ€–โ€–โ‰ค๎€ท(๐‘ง)1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›(๐‘ฅ)โˆ’๐‘ฆ๐‘›โ€–โ€–(๐‘ง)+๐›ผ๐‘›๐‘˜๐‘›โ€–โ€–๐‘ฆ๐‘›(๐‘ฅ)โˆ’๐‘ฆ๐‘›โ€–โ€–โ‰ค๎€ท(๐‘ง)1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๎€ท1โˆ’๐›ฝ๐‘›๎€ธ(๐‘ฅโˆ’๐‘ง)+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1โ€–โ€–(๐‘ฅโˆ’๐‘ง)+๐›ผ๐‘›๐‘˜๐‘›โ€–โ€–๎€ท1โˆ’๐›ฝ๐‘›๎€ธ(๐‘ฅโˆ’๐‘ง)+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1โ€–โ€–โ‰ค๎€ท(๐‘ฅโˆ’๐‘ง)1โˆ’๐›ผ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–๎€ท๐‘ฅโˆ’๐‘งโ€–+1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐‘™๐‘›โ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๐‘˜๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๐›ฝ๐‘›๐‘˜๐‘›๐‘™๐‘›=๎€ทโ€–๐‘ฅโˆ’๐‘งโ€–1โˆ’๐›ผ๐‘›โˆ’๐›ฝ๐‘›+๐›ผ๐‘›๐›ฝ๐‘›๎€ธ๎€ทโ€–๐‘ฅโˆ’๐‘งโ€–+1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎€ท1+๐‘ก๐‘›๎€ธโ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๎€ท1+๐‘ ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๐›ฝ๐‘›๎€ท1+๐‘ ๐‘›๎€ธ๎€ท1+๐‘ก๐‘›๎€ธโ€–๐‘ฅโˆ’๐‘งโ€–=โ€–๐‘ฅโˆ’๐‘งโ€–+๐›ฝ๐‘›๐‘ก๐‘›โ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๐‘ ๐‘›โ€–๐‘ฅโˆ’๐‘งโ€–+๐›ผ๐‘›๐›ฝ๐‘›๐‘ก๐‘›๐‘ ๐‘›โ‰ค๎€ทโ€–๐‘ฅโˆ’๐‘งโ€–1+๐‘ก๐‘›+๐‘ ๐‘›+๐‘ก๐‘›๐‘ ๐‘›๎€ธโ€–๐‘ฅโˆ’๐‘งโ€–.(2.3)
Set ๐‘†๐‘›,๐‘šโˆถ=๐ด๐‘›+๐‘šโˆ’1๐ด๐‘›+๐‘šโˆ’2โ‹ฏ๐ด๐‘›, ๐‘›,๐‘šโ‰ฅ1 and ๐‘๐‘›,๐‘š=โ€–๐‘†๐‘›,๐‘š(๐‘ก๐‘ฅ๐‘›+(1โˆ’๐‘ก)๐‘ข)โˆ’(๐‘ก๐‘†๐‘›,๐‘š๐‘ฅ๐‘›+(1โˆ’๐‘ก)๐‘ข)โ€–, where 0โ‰ค๐‘กโ‰ค1. Also, โ€–โ€–๐‘†๐‘›,๐‘š๐‘ฅโˆ’๐‘†๐‘›,๐‘š๐‘ฆโ€–โ€–โ‰คโ€–โ€–๐ด๐‘›+๐‘šโˆ’1๎€ท๐ด๐‘›+๐‘šโˆ’2โ‹ฏ๐ด๐‘›๐‘ฅ๎€ธโˆ’๐ด๐‘›+๐‘šโˆ’1๎€ท๐ด๐‘›+๐‘šโˆ’2โ‹ฏ๐ด๐‘›๐‘ฆ๎€ธโ€–โ€–โ‰ค๎€ท1+๐‘ก๐‘›+๐‘šโˆ’1+๐‘ ๐‘›+๐‘šโˆ’1+๐‘ก๐‘›+๐‘šโˆ’1๐‘ ๐‘›+๐‘šโˆ’1๎€ธโ€–โ€–๐ด๐‘›+๐‘šโˆ’2๎€ท๐ด๐‘›+๐‘šโˆ’3โ‹ฏ๐ด๐‘›๐‘ฅ๎€ธโˆ’๐ด๐‘›+๐‘šโˆ’2๎€ท๐ด๐‘›+๐‘šโˆ’3โ‹ฏ๐ด๐‘›๐‘ฆ๎€ธโ€–โ€–โ‹ฎโ‰ค๐‘›+๐‘šโˆ’1๎‘๐‘—=๐‘›๎€ท1+๐‘ก๐‘—+๐‘ ๐‘—+๐‘ก๐‘—๐‘ ๐‘—๎€ธโ€–๐‘ฅโˆ’๐‘ฆโ€–(2.4) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ and ๐‘†๐‘›,๐‘š๐‘ฅ๐‘›=๐‘ฅ๐‘›+๐‘š, ๐‘†๐‘›,๐‘š๐‘ฅโˆ—=๐‘ฅโˆ— for all ๐‘ฅโˆ—โˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2).
Applying Lemma 1.14 with ๐‘ฅ=๐‘ฅ๐‘›, ๐‘ฆ=๐‘ข, ๐‘†=๐‘†๐‘›,๐‘š and using the facts that lim๐‘›โ†’โˆž๐‘ก๐‘›=lim๐‘›โ†’โˆž(๐‘™๐‘›โˆ’1)=0, lim๐‘›โ†’โˆž๐‘ ๐‘›=lim๐‘›โ†’โˆž(๐‘˜๐‘›โˆ’1)=0, and lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€– exist for all ๐‘ฅโˆ—โˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2), we obtain lim๐‘›โ†’โˆž๐‘๐‘›,๐‘š=0. Observe that ๐‘Ž๐‘›+๐‘šโ€–โ€–(๐‘ก)=๐‘ก๐‘ฅ๐‘›+๐‘šโ€–โ€–=โ€–โ€–+(1โˆ’๐‘ก)๐‘ขโˆ’๐‘ฃ๐‘ก๐‘†๐‘›,๐‘š๐‘ฅ๐‘›+(1โˆ’๐‘ก)๐‘ขโˆ’๐‘†๐‘›,๐‘š๐‘ฃโ€–โ€–=โ€–โ€–๐‘†๐‘›,๐‘š๎€ท๐‘ฃโˆ’๐‘ก๐‘†๐‘›,๐‘š๐‘ฅ๐‘›๎€ธโ€–โ€–=โ€–โ€–๐‘†+(1โˆ’๐‘ก)๐‘ข๐‘›,๐‘š๐‘ฃโˆ’๐‘†๐‘›,๐‘š๎€ท๐‘ก๐‘ฅ๐‘›๎€ธ+(1โˆ’๐‘ก)๐‘ข+๐‘†๐‘›,๐‘š๎€ท๐‘ก๐‘ฅ๐‘›๎€ธโˆ’๎€ท+(1โˆ’๐‘ก)๐‘ข๐‘ก๐‘†๐‘›,๐‘š๐‘ฅ๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘†+(1โˆ’๐‘ก)๐‘ข๐‘›,๐‘š๐‘ฃโˆ’๐‘†๐‘›,๐‘š๎€ท๐‘ก๐‘ฅ๐‘›๎€ธโ€–โ€–+(1โˆ’๐‘ก)๐‘ข+๐‘๐‘›,๐‘š=โ€–โ€–๐‘†๐‘›,๐‘š๎€ท๐‘ก๐‘ฅ๐‘›๎€ธ+(1โˆ’๐‘ก)๐‘ขโˆ’๐‘†๐‘›,๐‘š๐‘ฃโ€–โ€–+๐‘๐‘›,๐‘šโ‰ค๐‘›+๐‘šโˆ’1๎‘๐‘—=๐‘›๎€ท1+๐‘ก๐‘—+๐‘ ๐‘—+๐‘ก๐‘—๐‘ ๐‘—๎€ธโ€–โ€–๐‘ก๐‘ฅ๐‘›โ€–โ€–+(1โˆ’๐‘ก)๐‘ขโˆ’๐‘ฃ+๐‘๐‘›,๐‘šโ‰คโˆž๎‘๐‘—=๐‘›๎€ท1+๐‘ก๐‘—+๐‘ ๐‘—+๐‘ก๐‘—๐‘ ๐‘—๎€ธ๐‘Ž๐‘›(๐‘ก)+๐‘๐‘›,๐‘š.(2.5) Consequently, limsup๐‘šโ†’โˆž๐‘Ž๐‘š(๐‘ก)=limsup๐‘šโ†’โˆž๐‘Ž๐‘›+๐‘š(๐‘ก)โ‰คlimsup๐‘šโ†’โˆž๎ƒฉ๐‘๐‘›,๐‘š+โˆžโˆ๐‘—=๐‘›๎€ท1+๐‘ก๐‘—+๐‘ ๐‘—+๐‘ก๐‘—๐‘ ๐‘—๎€ธ๐‘Ž๐‘›๎ƒช(๐‘ก),(2.6)limsup๐‘›โ†’โˆž๐‘Ž๐‘›(๐‘ก)โ‰คliminf๐‘›โ†’โˆž๐‘Ž๐‘›(๐‘ก).(2.7) This implies that lim๐‘›โ†’โˆž๐‘Ž๐‘›(๐‘ก) exists for all ๐‘กโˆˆ[0,1]. This completes the proof.

Theorem 2.2. Let ๐‘‹ be a real uniformly convex Banach space which has a Frรฉchet differentiable norm and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ‘โˆ’1)<โˆž,โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be the sequences defined by (1.14). Then, {๐‘ฅ๐‘›} converges weakly to a fixed point of ๐‘‡1 and ๐‘‡2.

Proof. Set ๐‘ฅ=๐‘1โˆ’๐‘2 and โ„Ž=๐‘ก(๐‘ฅ๐‘›โˆ’๐‘1) in (1.18). By using Lemmas 1.11, and 2.1 and the same proof of Lemmaโ€‰โ€‰4 of Osilike and Udomene [7], we can show that, for every ๐‘1,๐‘2โˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2), ๎ซ๎€ท๐‘๐‘โˆ’๐‘ž,๐ฝ1โˆ’๐‘2๎€ธ๎ฌ=0,(2.8) for all ๐‘,๐‘žโˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›). Since ๐ธ is reflexive and {๐‘ฅ๐‘›} is bounded, we from Lemma 1.13 conclude that ๐œ”๐‘ค(๐‘ฅ๐‘›)โŠ‚๐น(๐‘‡๐‘–) for each ๐‘–=1,2. Let ๐‘,๐‘žโˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›). It follows that ๐‘,๐‘žโˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2); that is, โ€–๐‘โˆ’๐‘žโ€–2=โŸจ๐‘โˆ’๐‘ž,๐ฝ(๐‘โˆ’๐‘ž)โŸฉ=0.(2.9) Therefore, ๐‘=๐‘ž. This completes the proof.

Theorem 2.3. Let ๐‘‹ be a real uniformly convex Banach space such that its dual ๐‘‹โˆ— has the Kadec-Klee property and ๐ถ a nonempty closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž, โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be the sequences defined by (1.14). Then, {๐‘ฅ๐‘›} converges weakly to a fixed point of ๐‘‡1 and ๐‘‡2.

Proof. It follows from Lemma 1.11 that the sequence {๐‘ฅ๐‘›} is bounded. Then, there exists a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} converging weakly to a point ๐‘ฅโˆ—โˆˆ๐ถ. By Lemma 1.12, we have lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›๐‘—โˆ’๐‘‡1๐‘ฅ๐‘›๐‘—โ€–โ€–=0=lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›๐‘—โˆ’๐‘‡2๐‘ฅ๐‘›๐‘—โ€–โ€–.(2.10) Now, using Lemma 1.10, we have (๐ผโˆ’๐‘‡)๐‘ฅโˆ—=0; that is, ๐‘‡๐‘ฅโˆ—=๐‘ฅโˆ—. Thus, ๐‘ฅโˆ—โˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2). It remains to show that {๐‘ฅ๐‘›} converges weakly to ๐‘ฅโˆ—. Suppose that {๐‘ฅ๐‘›๐‘–} is another subsequence of {๐‘ฅ๐‘›} converging weakly to some ๐‘ฆโˆ—. Then, ๐‘ฆโˆ—โˆˆ๐ถ and so ๐‘ฅโˆ—,๐‘ฆโˆ—โˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›)โˆฉ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2). By Lemma 2.1, lim๐‘›โ†’โˆžโ€–โ€–๐‘ก๐‘ฅ๐‘›โˆ’(1โˆ’๐‘ก)๐‘ฅโˆ—โˆ’๐‘ฆโˆ—โ€–โ€–(2.11) exists for all ๐‘กโˆˆ[0,1]. It follows from Lemma 1.13 that ๐‘ฅโˆ—=๐‘ฆโˆ—. As a result, ๐œ”๐‘ค(๐‘ฅ๐‘›) is a singleton, and so {๐‘ฅ๐‘›} converges weakly to a fixed point of ๐‘‡.

In the remainder of this section, we deal with the weak convergence of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space without any of the Opial's condition, Kadec-Klee property, or Frรฉchet differentiable norm.

Let ๐‘‡1 and ๐‘‡2 be two asymptotically nonexpansive nonself-mappings of ๐ถ with {๐‘˜๐‘›}โŠ‚[1,โˆž), lim๐‘›โ†’โˆž๐‘˜๐‘›=1, and {๐‘™๐‘›}โŠ‚[1,โˆž), lim๐‘›โ†’โˆž๐‘™๐‘›=1, respectively. In the sequel, we take {๐‘ก๐‘›}โŠ‚[1,โˆž), where ๐‘ก๐‘›=max{๐‘˜๐‘›,๐‘™๐‘›}.

We start with proving the following lemma for later use.

Lemma 2.4. Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a nonempty bounded closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that ๐‘˜๐‘›โ†’1, ๐‘™๐‘›โ†’1 as ๐‘›โ†’โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Then, for the sequence {๐‘ฅ๐‘›} given in (1.14), we have that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡1๐‘ฅ๐‘›โ€–โ€–=0=lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๐‘ฅ๐‘›โ€–โ€–.(2.12)

Proof. By setting ๐‘ก๐‘›=max{๐‘˜๐‘›,๐‘™๐‘›}, then lim๐‘›โ†’โˆž๐‘ก๐‘›=1 if lim๐‘›โ†’โˆž๐‘˜๐‘›=1=lim๐‘›โ†’โˆž๐‘™๐‘›. Let ๐‘โˆˆ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2). Since ๐ถ is bounded, there exists ๐ต๐‘Ÿ(0) such that ๐ถโŠ‚๐ต๐‘Ÿ(0) for some ๐‘Ÿ>0. Applying Lemma 1.9 for scheme (1.14), we have โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘2=โ€–โ€–๐‘ƒ๎‚€๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ฝ๐‘›๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›๎‚โ€–โ€–โˆ’๐‘2โ‰คโ€–โ€–๎€ท1โˆ’๐›ฝ๐‘›๐‘ฅ๎€ธ๎€ท๐‘›๎€ธโˆ’๐‘+๐›ฝ๐‘›๎‚€๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›๎‚โ€–โ€–โˆ’๐‘2=๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2+๐›ฝ๐‘›๐‘™2๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚=๎€ท1โˆ’๐›ฝ๐‘›+๐›ฝ๐‘›๐‘™2๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚(2.13) and so, โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘2=โ€–โ€–๐‘ƒ๎‚€๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฆ๐‘›+๐›ผ๐‘›๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›๎‚โ€–โ€–โˆ’๐‘2โ‰คโ€–โ€–๎€ท1โˆ’๐›ผ๐‘›๐‘ฆ๎€ธ๎€ท๐‘›๎€ธโˆ’๐‘+๐›ผ๐‘›(๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘)2=๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘2+๐›ผ๐‘›๐‘˜2๐‘›โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚=๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚โ‰ค๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธ๎‚€๎€ท1โˆ’๐›ฝ๐‘›+๐›ฝ๐‘›๐‘™2๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚=๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›+๐›ฝ๐‘›๐‘™2๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธ๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚=๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐‘™2๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐›ผ๐‘›๐‘˜2๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๐›ฝ๐‘›๐‘™2๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธ๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚โ‰ค๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐‘ก2๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐›ผ๐‘›๐‘ก2๐‘›+๐›ผ๐‘›๐›ฝ๐‘›๐‘ก4๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๎€ท1โˆ’๐›ผ๐‘›+๐›ผ๐‘›๐‘˜2๐‘›๎€ธ๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚โ‰ค๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ก4๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐‘ก4๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐›ผ๐‘›๐‘ก4๐‘›+๐›ผ๐‘›๐›ฝ๐‘›๐‘ก4๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๎€ท1+๐›ผ๐‘›๎€ท๐‘˜2๐‘›๐›ฝโˆ’1๎€ธ๎€ธ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚โ‰ค๎€ท๎€ท1โˆ’๐›ผ๐‘›๎€ธ๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ก4๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐‘ก4๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐›ผ๐‘›๐‘ก4๐‘›+๐›ผ๐‘›๐›ฝ๐‘›๐‘ก4๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐›ฝ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2๎€ท๐‘ก+๐‘Ÿ4๐‘›๎€ธโˆ’1โˆ’๐œ€2๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐œ€2๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚.(2.14) From (2.14), we obtain the following two important inequalities: โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2๎€ท๐‘ก+๐‘Ÿ4๐‘›๎€ธโˆ’1โˆ’๐œ€2๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โ€–โ€–๐‘ฅ,(2.15)๐‘›+1โ€–โ€–โˆ’๐‘2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2๎€ท๐‘ก+๐‘Ÿ4๐‘›๎€ธโˆ’1โˆ’๐œ€2๐‘”๎‚€โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–๎‚.(2.16) Now, we prove that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–=0=lim๐‘›โ†’โˆžโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–.(2.17) Assume that limsup๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡2(๐‘ƒ๐‘‡2)๐‘›โˆ’1๐‘ฅ๐‘›โ€–>0. Then, there exists a subsequence (use the same notation for subsequence as for the sequence) of {๐‘ฅ๐‘›} and ๐œ‡>0 such that โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–โ‰ฅ๐œ‡>0.(2.18) By definition of ๐‘”, we have ๐‘”๎‚€โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–๎‚โ‰ฅ๐‘”(๐œ‡)>0.(2.19) From (2.15), we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2๎€ท๐‘ก+๐‘Ÿ4๐‘›๎€ธโˆ’1โˆ’๐œ€2=โ€–โ€–๐‘ฅ๐‘”(๐œ‡)๐‘›โ€–โ€–โˆ’๐‘2๎‚ต๎€ท๐‘ก+๐‘Ÿ4๐‘›๎€ธโˆ’๐œ€โˆ’12๎‚ถโˆ’๐œ€2๐‘Ÿ๐‘”(๐œ‡)22๐‘”(๐œ‡).(2.20)
In addition, ๐‘ก4๐‘›โ†’1 and (๐œ€2/2๐‘Ÿ)๐‘”(๐œ‡)>0; there exists ๐‘›0โ‰ฅ1 such that (๐‘ก4๐‘›โˆ’1)<(๐œ€2/2๐‘Ÿ)๐‘”(๐œ‡) for all ๐‘›โ‰ฅ๐‘›0. From (2.20), we obtain ๐œ€22โ€–โ€–๐‘ฅ๐‘”(๐œ‡)โ‰ค๐‘›โ€–โ€–โˆ’๐‘2โˆ’โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘2(2.21) for all ๐‘›โ‰ฅ๐‘›0.
Let ๐‘šโ‰ฅ๐‘›0. It follows from (2.21) that ๐œ€22๐‘š๎“๐‘›=๐‘›0๐‘”(๐œ‡)โ‰ค๐‘š๎“๐‘›=๐‘›0๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘2๎‚=โ€–โ€–๐‘ฅ๐‘›0โ€–โ€–โˆ’๐‘2.(2.22)
By letting ๐‘šโ†’โˆž in (2.22), we obtain โ€–โ€–๐‘ฅโˆž=๐‘›0โ€–โ€–โˆ’๐‘2<โˆž(2.23) which contradicts the reality. This proves that ๐œ‡=0. Thus, limsup๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡2(๐‘ƒ๐‘‡2)๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ‰ค0. Consequently, we have lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–=0.(2.24)
Similarly, using (2.16), we may show that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–=0.(2.25)
Using (2.24), we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(2.26)
From (2.25), (2.26), and the uniform equicontinuous of ๐‘‡1 (see Remark 1.1), we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1(๐‘ƒ๐‘‡1)๐‘›โˆ’1๐‘ฆ๐‘›โˆ’๐‘‡1(๐‘ƒ๐‘‡1)๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(2.27)
Since โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–โ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–=๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฆ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–,(2.28) it follows from (2.26), (2.27), and the uniform equi-continuity of ๐‘‡1 (see Remark 1.1) that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–=0.(2.29)
Since lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡1(๐‘ƒ๐‘‡1)๐‘›โˆ’1๐‘ฅ๐‘›โ€–=0 and again from the fact that ๐‘‡1 is uniformly equicontinuous mapping, by Using (2.29), we have โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+๐‘ฅ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โŸถ0(as๐‘›โŸถโˆž).(2.30) In addition, โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+๐‘ฅ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ+๐ฟ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–,(2.31) where ๐ฟ=sup{๐‘˜๐‘›โˆถ๐‘›โ‰ฅ1}. It follows from (2.29) and (2.30) that limโ†’โˆžโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1โ€–โ€–=0.(2.32) We denote (๐‘ƒ๐‘‡1)1โˆ’1 to be the identity maps from ๐ถ onto itself. Thus, by the inequality (2.30) and (2.32), we have โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1+๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โˆ’๐‘‡1๐‘ฅ๐‘›+1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โˆ’๐‘‡1๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ1โˆ’1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ1โˆ’1๐‘ฅ๐‘›+1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–โ€–โ€–๎€ท+๐ฟ๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–โ€–โ€–๎€ท+๐ฟ๐‘ƒ๐‘‡1๎€ธ๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1๎€ท๐‘ฅโˆ’๐‘ƒ๐‘›+1๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’1๐‘ฅ๐‘›+1โ€–โ€–โ€–โ€–๐‘‡+๐ฟ1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›โˆ’2๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+1โ€–โ€–โŸถ0(as๐‘›โŸถโˆž),(2.33) which implies that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡1๐‘ฅ๐‘›โ€–=0. Similarly, we may show that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡2๐‘ฅ๐‘›โ€–=0. The proof is completed.

Our weak convergence theorem is as follows. We do not use the rate of convergence conditions, namely, โˆ‘โˆž๐‘›=1(๐‘˜๐‘›โˆ’1)<โˆž and โˆ‘โˆž๐‘›=1(๐‘™๐‘›โˆ’1)<โˆž in its proof.

Theorem 2.5. Let ๐‘‹ be a uniformly convex Banach space and ๐ถ a nonempty bounded closed convex nonexpansive retract of ๐‘‹ with ๐‘ƒ as a nonexpansive retraction. Let ๐‘‡1,๐‘‡2โˆถ๐ถโ†’๐‘‹ be two asymptotically nonexpansive nonself-mappings of ๐ถ with sequences {๐‘˜๐‘›},{๐‘™๐‘›}โŠ‚[1,โˆž) such that ๐‘˜๐‘›โ†’1, ๐‘™๐‘›โ†’1 as ๐‘›โ†’โˆž, respectively, and ๐น(๐‘‡1)โˆฉ๐น(๐‘‡2)โ‰ โˆ…. Suppose that {๐›ผ๐‘›} and {๐›ฝ๐‘›} are real sequences in [๐œ–,1โˆ’๐œ–] for some ๐œ–โˆˆ(0,1). Then, the sequence {๐‘ฅ๐‘›} given in (1.14) converges weakly to a common fixed point of ๐‘‡1 and ๐‘‡2.

Proof. Since ๐ถ is a nonempty bounded closed convex subset of a uniformly convex Banach space ๐‘‹, there exists a subsequence {๐‘ฅ๐‘›๐‘—} of {๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘— converges weakly to ๐‘žโˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›), where ๐œ”๐‘ค(๐‘ฅ๐‘›) denotes the set of all weak subsequential limits of {๐‘ฅ๐‘›}. This show that ๐œ”๐‘ค(๐‘ฅ๐‘›)โ‰ โˆ… and, by Lemma 2.4, lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›๐‘—โˆ’๐‘‡1๐‘ฅ๐‘›๐‘—โ€–=lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›๐‘—โˆ’๐‘‡2๐‘ฅ๐‘›๐‘—โ€–=0. Since ๐ผโˆ’๐‘‡1 and ๐ผโˆ’๐‘‡2 are demiclosed at zero, using Lemma 1.10, we have ๐‘‡1๐‘ž=๐‘ž=๐‘‡2๐‘ž. Therefore, ๐œ”๐‘ค(๐‘ฅ๐‘›)โŠ‚๐น(๐‘‡1)โˆฉ๐น(๐‘‡2).For any ๐‘žโˆˆ๐œ”๐‘ค(๐‘ฅ๐‘›), there exists a subsequence {๐‘ฅ๐‘›๐‘–} of {๐‘ฅ๐‘›} such that ๐‘ฅ๐‘›๐‘–โ‡€๐‘ž(as๐‘–โŸถโˆž).(2.34) It follows from (2.24) and (2.34) that ๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›๐‘—=๎‚€๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›๐‘—โˆ’๐‘ฅ๐‘›๐‘—๎‚+๐‘ฅ๐‘›๐‘—โ‡€๐‘ž.(2.35) Now, from (1.14), (2.34), and (2.35), ๐‘ฆ๐‘›๐‘—=๐‘ƒ๎‚€๎‚€1โˆ’๐›ฝ๐‘›๐‘—๎‚๐‘ฅ๐‘›๐‘—+๐›ฝ๐‘›๐‘—๐‘‡2๎€ท๐‘ƒ๐‘‡2๎€ธ๐‘›๐‘—โˆ’1๐‘ฅ๐‘›๐‘—๎‚โ‡€๐‘ž.(2.36) Also, from (2.25) and (2.36), we have ๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›๐‘—โˆ’1๐‘ฆ๐‘›๐‘—=๎‚€๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›๐‘—โˆ’1๐‘ฆ๐‘›๐‘—โˆ’๐‘ฆ๐‘›๐‘—๎‚+๐‘ฆ๐‘›๐‘—โ‡€๐‘ž.(2.37) It follows from (2.36) and (2.37) that ๐‘ฅ๐‘›๐‘—+1=๐‘ƒ๎‚€๎‚€1โˆ’๐›ผ๐‘›๐‘—๎‚๐‘ฆ๐‘›๐‘—+๐›ผ๐‘›๐‘—๐‘‡1๎€ท๐‘ƒ๐‘‡1๎€ธ๐‘›๐‘—โˆ’1๐‘ฆ๐‘›๐‘—๎‚โ‡€๐‘ž.(2.38) Continuing in this way, by induction, we can prove that, for any ๐‘šโ‰ฅ0, ๐‘ฅ๐‘›๐‘—+๐‘šโ‡€๐‘ž.(2.39) By induction, one can prove that โˆชโˆž๐‘š=0{๐‘ฅ๐‘›๐‘—+๐‘š} converges weakly to ๐‘ž as ๐‘—โ†’โˆž; in fact, {๐‘ฅ๐‘›}โˆž๐‘›=๐‘›1=โˆชโˆž๐‘š=0{๐‘ฅ๐‘›๐‘—+๐‘š}โˆž๐‘—=1 gives that ๐‘ฅ๐‘›โ‡€๐‘ž as ๐‘›โ†’โˆž. This completes the prove.

Acknowledgments

The author would like to thank the Thailand Research Fund, The Commission on Higher Education(MRG5380226), and University of Phayao, Phayao, Thailand, for financial support during the preparation of this paper. Thanks are also extended to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper.

References

  1. H. Y. Zhou, G. T. Guo, H. J. Hwang, and Y. J. Cho, โ€œOn the iterative methods for nonlinear operator equations in Banach spaces,โ€ Panamerican Mathematical Journal, vol. 14, no. 4, pp. 61โ€“68, 2004. View at Google Scholar ยท View at Zentralblatt MATH
  2. S. S. Chang, Y. J. Cho, and H. Zhou, โ€œDemi-closed principle and weak convergence problems for asymptotically nonexpansive mappings,โ€ Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1245โ€“1260, 2001. View at Google Scholar ยท View at Zentralblatt MATH
  3. S. Ishikawa, โ€œFixed points and iteration of nonexpansive mappings of in a Banach spaces,โ€ Proceedings of the American Mathematical Society, vol. 73, pp. 61โ€“71, 1976. View at Google Scholar
  4. S. H. Khan and H. Fukhar-ud-din, โ€œWeak and strong convergence of a scheme with errors for two nonexpansive mappings,โ€ Nonlinear Analysis, vol. 61, no. 8, pp. 1295โ€“1301, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  5. M. Maiti and M. K. Ghosh, โ€œApproximating fixed points by Ishikawa iterates,โ€ Bulletin of the Australian Mathematical Society, vol. 40, no. 1, pp. 113โ€“117, 1989. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  6. M. O. Osilike and S. C. Aniagbosor, โ€œWeak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,โ€ Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181โ€“1191, 2000. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  7. M. O. Osilike and A. Udomene, โ€œDemiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,โ€ Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 431โ€“445, 2001. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  8. J. Schu, โ€œIterative construction of fixed points of asymptotically nonexpansive mappings,โ€ Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407โ€“413, 1991. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  9. J. S. Jung and S. S. Kim, โ€œStrong convergence theorems for nonexpansive nonself-mappings in Banach spaces,โ€ Nonlinear Analysis, vol. 33, no. 3, pp. 321โ€“329, 1998. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  10. S. Y. Matsushita and D. Kuroiwa, โ€œStrong convergence of averaging iterations of nonexpansive nonself-mappings,โ€ Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 206โ€“214, 2004. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  11. N. Shahzad, โ€œApproximating fixed points of non-self nonexpansive mappings in Banach spaces,โ€ Nonlinear Analysis, vol. 61, no. 6, pp. 1031โ€“1039, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  12. W. Takahashi and G.-E. Kim, โ€œStrong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces,โ€ Nonlinear Analysis, vol. 32, no. 3, pp. 447โ€“454, 1998. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  13. H. K. Xu and X. M. Yin, โ€œStrong convergence theorems for nonexpansive nonself-mappings,โ€ Nonlinear Analysis, vol. 242, pp. 23โ€“228, 1995. View at Google Scholar ยท View at Zentralblatt MATH
  14. K. K. Tan and H. K. Xu, โ€œApproximating fixed points of nonexpansive mappings by the Ishikawa iteration process,โ€ Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301โ€“308, 1993. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  15. J. G. Falset, W. Kaczor, T. Kuczumow, and S. Reich, โ€œWeak convergence theorems for asymptotically nonexpansive mappings and semigroups,โ€ Nonlinear Analysis, vol. 43, no. 3, pp. 377โ€“401, 2001. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  16. W. Kaczor, โ€œWeak convergence of almost orbits of asymptotically nonexpansive commutative semigroups,โ€ Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 565โ€“574, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  17. K. Goebel and W. A. Kirk, โ€œA fixed point theorem for asymptotically nonexpansive mappings,โ€ Proceedings of the American Mathematical Society, vol. 35, pp. 171โ€“174, 1972. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  18. J. Schu, โ€œWeak and strong convergence to fixed points of asymptotically nonexpansive mappings,โ€ Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153โ€“159, 1991. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  19. S. Reich, โ€œWeak convergence theorems for nonexpansive mappings in Banach spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274โ€“276, 1979. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  20. B. E. Rhoades, โ€œFixed point iterations for certain nonlinear mappings,โ€ Journal of Mathematical Analysis and Applications, vol. 183, no. 1, pp. 118โ€“120, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  21. K. K. Tan and H. K. Xu, โ€œFixed point iteration processes for asymptotically nonexpansive mappings,โ€ Proceedings of the American Mathematical Society, vol. 122, no. 3, pp. 733โ€“739, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  22. S. H. Khan and W. Takahashi, โ€œApproximating common fixed points of two asymptotically nonexpansive mappings,โ€ Scientiae Mathematicae Japonicae, vol. 53, no. 1, pp. 143โ€“148, 2001. View at Google Scholar ยท View at Zentralblatt MATH
  23. S. H. Khan and H. Fukhar-Ud-Din, โ€œWeak and strong convergence theorems without some widely used conditions,โ€ International Journal of Pure and Applied Mathematics, vol. 63, no. 2, pp. 137โ€“148, 2010. View at Google Scholar ยท View at Zentralblatt MATH
  24. C. E. Chidume, E. U. Ofoedu, and H. Zegeye, โ€œStrong and weak convergence theorems for asymptotically nonexpansive mappings,โ€ Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 364โ€“374, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  25. L. Wang, โ€œStrong and weak convergence theorems for common fixed point of nonself asymptotically nonexpansive mappings,โ€ Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 550โ€“557, 2006. View at Publisher ยท View at Google Scholar
  26. S. Thianwan, โ€œCommon fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space,โ€ Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 688โ€“695, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  27. Z. Opial, โ€œWeak convergence of the sequence of successive approximations for nonexpansive mappings,โ€ Bulletin of the American Mathematical Society, vol. 73, pp. 591โ€“597, 1967. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  28. W. J. Davis and P. Enflo, โ€œContractive projections on lp spaces,โ€ in Analysis at Urbana, Vol. I (Urbana, IL, 1986โ€“1987), vol. 137 of London Mathematical Society Lecture Note Series, pp. 151โ€“161, Cambridge University Press, Cambridge, UK, 1989. View at Google Scholar
  29. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1990.
  30. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000.
  31. H. K. Xu, โ€œInequalities in Banach spaces with applications,โ€ Nonlinear Analysis, vol. 16, no. 12, pp. 1127โ€“1138, 1991. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  32. J.-B. Baillon, Comportement asymptotique des contractions et semi-groupes de contractions. Equations de Schrรถdinger nonlinรฉaires et divers,, thesis, prรฉsentรฉes ร  l'a Universitรฉ, Paris VI, 1978.
  33. A. G. Aksoy and M. A. Khamsi, Nonstandard methods in fixed point theory, Universitext, Springer, New York, NY, USA, 1990.
  34. R. E. Bruck, โ€œOn the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces,โ€ Israel Journal of Mathematics, vol. 38, no. 4, pp. 304โ€“314, 1981. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  35. M. A. Khamsi, โ€œOn normal structure, fixed-point property and contractions of type (ฮณ),โ€ Proceedings of the American Mathematical Society, vol. 106, no. 4, pp. 995โ€“1001, 1989. View at Publisher ยท View at Google Scholar
  36. R. Bruck, T. Kuczumow, and S. Reich, โ€œConvergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,โ€ Colloquium Mathematicum, vol. 65, no. 2, pp. 169โ€“179, 1993. View at Google Scholar ยท View at Zentralblatt MATH
  37. H. Oka, โ€œA nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings,โ€ Nonlinear Analysis, vol. 18, no. 7, pp. 619โ€“635, 1992. View at Publisher ยท View at Google Scholar