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

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 𝐸 which has a Fréchet differentiable norm or its dual 𝐸^∗ 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 Fréchet 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 𝐶 be a nonempty closed convex subset of real normed linear space 𝑋. Let 𝑇∶𝐶→𝐶 be a mapping. A point 𝑥∈𝐶 is called a fixed point of 𝑇 if and only if 𝑇𝑥=𝑥. The set of all fixed points of a mapping 𝑇 is denoted by 𝐹(𝑇). A self-mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇(𝑥)−𝑇(𝑦)‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A self-mapping 𝑇∶𝐶→𝐶 is called asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1 as 𝑛→∞ such that ‖𝑇^𝑛(𝑥)−𝑇^𝑛(𝑦)‖≤𝑘_𝑛‖𝑥−𝑦‖(1.1) for all 𝑥,𝑦∈𝐶 and 𝑛≥1. A mapping 𝑇∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitzian if there exists a constant 𝐿>0 such that ‖𝑇^𝑛(𝑥)−𝑇^𝑛(𝑦)‖≤𝐿‖𝑥−𝑦‖(1.2) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.  𝑇 is uniformly Hölder continuous if there exist positive constants 𝐿 and 𝛼 such that ‖𝑇^𝑛(𝑥)−𝑇^𝑛(𝑦)‖<𝐿‖𝑥−𝑦‖^𝛼(1.3) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.  𝑇 is termed as uniformly equicontinuous if, for any 𝜀>0, there exists 𝛿>0 such that ‖𝑇^𝑛(𝑥)−𝑇^𝑛(𝑦)‖≤𝜀(1.4) whenever ‖𝑥−𝑦‖≤𝛿 for all 𝑥,𝑦∈𝐶 and 𝑛≥1 or, equivalently, 𝑇 is uniformly equicontinuous if and only if ‖‖𝑇^𝑛𝑥_𝑛−𝑇^𝑛𝑦_𝑛‖‖⟶0(1.5) whenever ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞.

It is easy to see that if 𝑇 is an asymptotically nonexpansive, then it is uniformly 𝐿-Lipschitzian with the uniform Lipschitz constant 𝐿=sup{𝑘_𝑛∶𝑛≥1}.

Remark 1.1. It is clear that asymptotically nonexpansiveness ⇒ uniformly L-Lipschitz ⇒ uniformly Hölder continuous ⇒ uniformly equicontinuous.

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

Example 1.2 (see [1]). Define 𝑇∶[0,1]→[0,1] by 𝑇𝑥=(1−𝑥^3/2)^2/3 for all 𝑥∈[0,1].

Fixed-point iteration process for nonexpansive self-mappings including Mann and Ishikawa iteration processes has been studied extensively by various authors [2–8]. For nonexpansive nonself-mappings, some authors (see [9–13]) 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: 𝑥_𝑛+1=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑛≥1,(1.6) to approximate fixed points of nonexpansive self-mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space 𝑋. The mapping 𝑇 remains self-mapping of a nonempty closed convex subset 𝐶 of a uniformly convex Banach space. If, however, the domain 𝐶 of 𝑇 is a proper subset of 𝑋 (and this is the case in several applications) and 𝑇 maps 𝐶 into 𝑋 then, the sequence {𝑥_𝑛} 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 𝑇 in a uniformly convex Banach space which satisfies Opial's condition or has a Fréchet differentiable norm.

Note that each 𝑙^𝑝(1≤𝑝<∞) satisfies Opial's condition, while all 𝐿^𝑝 do not have the property unless 𝑝=2 and the dual of reflexive Banach spaces with a Fréchet differentiable norm has the Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Fréchet 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: 𝑥_𝑛+1=𝑃1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑃1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑛≥1,(1.7) in a uniformly convex Banach space whose dual has the Kadec-Klee property. The result applies not only to 𝐿^𝑝 spaces with (1≤𝑝<∞) but also to other spaces which do not satisfy Opial's condition or have a Fréchet 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 𝐶 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 {𝑥_𝑛} in the following manner: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1,(1.8) where {𝛼_𝑛} is a sequence in the interval (0,1) and 𝑇∶𝐶→𝐶 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 𝑋 be a uniformly convex Banach space satisfying Opial’s condition, ∅≠𝐶⊂𝑋 closed bounded and convex, and 𝑇∶𝐶→𝐶 asymptotically nonexpansive with sequence {𝑘_𝑛}⊂[1,∞) for which ∑^∞𝑛=1(𝑘_𝑛−1)<∞ and {𝛼_𝑛}∈[0,1] is bounded away. Let {𝑥_𝑛} be a sequence generated in (1.8). Then, the sequence {𝑥_𝑛} converges weakly to some fixed point of  𝑇.

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 𝑋 be a uniformly convex Banach space whose norm is Fréchet differentiable, 𝐶 a nonempty closed and convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞ such that 𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be sequence generated in (1.8), where {𝛼_𝑛} is a real sequence bounded away from 0 and 1. Then, the sequence {𝑥_𝑛} converges weakly to some point in 𝐹(𝑇).

In 2001, Khan and Takahashi [22] constructed and studied the following Ishikawa iteration process:𝑦_𝑛𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛1𝑦_𝑛,=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛2𝑥_𝑛,𝑛≥1,(1.9) where  𝑇₁, 𝑇₂  are asymptotically nonexpansive self-mappings on 𝐶 with  ∑^∞𝑛=1(𝑘_𝑛−1)<∞ (rate of convergence) and 0≤𝛼_𝑛, 𝛽_𝑛≤1.

Note that the rate of convergence condition, namely, ∑^∞𝑛=1(𝑘_𝑛−1)<∞ 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 Fréchet 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 Opial’s condition, Kadec-Klee property or Fréchet differentiable norm.

Theorem 1.5 (see [23]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋. Let 𝑇₁,𝑇₂∶𝐶→𝐶 be asymptotically nonexpansive maps with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that lim_𝑛→∞𝑘_𝑛=1, lim_𝑛→∞𝑙_𝑛=1, respectively. Let the sequence {𝑥_𝑛} be as in (1.9) with 𝛿≤𝛼_𝑛,𝛽_𝑛≤1−𝛿. for some 𝛿∈(0,1/2). If 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅, then {𝑥_𝑛} converges weakly to a common fixed point of 𝑇₁ and 𝑇₂.

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 𝐶 be a nonempty subset of a real normed linear space 𝑋. Let 𝑃∶𝑋→𝐶 be a nonexpansive retraction of 𝑋 onto 𝐶. A nonself-mapping 𝑇∶𝐶→𝑋 is called asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1 as 𝑛→∞ such that ‖‖𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦‖‖≤𝑘_𝑛‖𝑥−𝑦‖(1.10) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.  𝑇 is said to be uniformly L-Lipschitzian if there exists a constant 𝐿>0 such that ‖‖𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦‖‖≤𝐿‖𝑥−𝑦‖(1.11) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.

By studying the following iteration process: 𝑥₁∈𝐶,𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇(𝑃𝑇)^𝑛−1𝑥_𝑛,(1.12)Chidume et al. [24] got the following weak convergence theorem for asymptotically nonexpansive nonself-mapping.

Theorem 1.7 (see [24]). Let 𝑋 be a real uniformly convex Banach space which has a Fréchet differentiable norm and 𝐶 a nonempty closed convex subset of 𝑋. Let 𝑇∶𝐶→𝑋 be an asymptotically nonexpansive map with sequence {𝑘_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘^2𝑛−1)<∞ and 𝐹(𝑇)≠∅. Let {𝛼_𝑛}⊂(0,1) be such that 𝜖≤1−𝛼_𝑛≤1−𝜖, for all 𝑛≥1 and some 𝜖>0. From an arbitrary 𝑥₁∈𝐶, define the sequence {𝑥_𝑛} by (1.12). Then, {𝑥_𝑛} converges weakly to some fixed point of 𝑇.

If 𝑇 is a self-mapping, then 𝑃 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: 𝑥₁∈𝐶,𝑦_𝑛𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛,=𝑃1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛,𝑛≥1,(1.13) where 𝑇₁,𝑇₂∶𝐶→𝑋 are asymptotically nonexpansive nonself-mappings and {𝛼_𝑛}, {𝛽_𝑛} 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:𝑦_𝑛𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛,=𝑃1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛,𝑛≥1,(1.14) where {𝛼_𝑛} and {𝛽_𝑛} are appropriate real sequences in [0,1).

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

Theorem 1.8. Let 𝑋 be a uniformly convex Banach space which satisfies Opial's condition and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞, ∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences defined by (1.14). Then, {𝑥_𝑛} and {𝑦_𝑛} converge weakly to a common fixed point of 𝑇₁ and 𝑇₂.

The iterative schemes (1.14) and (1.13) are independent: neither reduces to the other. If 𝑇₁=𝑇₂ and 𝛽_𝑛=0 for all 𝑛≥1, 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 𝑋 be a Banach space with dimension 𝑋≥2. The modulus of 𝑋 is the function 𝛿_𝑋∶(0,2]→[0,1] defined by 𝛿_𝑋(𝜖)=inf1−‖‖‖12(𝑥+𝑦)‖‖‖∶‖𝑥‖=1,‖𝑦‖=1,𝜖=‖𝑥−𝑦‖.(1.15) Banach space 𝑋 is uniformly convex if and only if 𝛿_𝑋(𝜖)>0 for all 𝜖∈(0,2]. It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that a Banach space 𝑋 is said to satisfy Opial's condition [27] if 𝑥_𝑛→𝑥 weakly as 𝑛→∞ and 𝑥≠𝑦 implying that limsup_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<limsup_𝑛→∞‖‖𝑥_𝑛−𝑦‖‖.(1.16) The norm of 𝑋 is said to be Fréchet differentiable if for each 𝑥∈𝑋 with ‖𝑥‖=1 the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(1.17) exists and is attained uniformly for 𝑦, with ‖𝑦‖=1. In the case of Fréchet differentiable norm, it has been obtained in [21] that⟨ℎ,𝐽(𝑥)⟩+12‖𝑥‖²≤12‖𝑥+ℎ‖²≤⟨ℎ,𝐽(𝑥)⟩+12‖𝑥‖²+𝑏(‖ℎ‖)(1.18) for all 𝑥, ℎ in 𝐸, where 𝐽 is the normalized duality map from 𝐸 to 𝐸^∗ defined by𝐽(𝑥)=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑠^∗⟩=‖𝑥‖²=‖𝑥^∗‖²,(1.19)

⟨⋅,⋅⟩ is the duality pairing between 𝐸 and 𝐸^∗ and 𝑏 is an increasing function defined on [0,∞) such that lim_𝑡↓0𝑏(𝑡)/𝑡=0.

A subset 𝐶 of 𝑋 is said to be retract if there exists continuous mapping 𝑃∶𝑋→𝐶 such that 𝑃𝑥=𝑥 for all 𝑥∈𝐶. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping 𝑃∶𝑋→𝑋 is said to be a retraction if 𝑃²=𝑃. If a mapping 𝑃 is a retraction, then 𝑃𝑧=𝑧 for every 𝑧∈𝑅(𝑃), range of 𝑃. A set 𝐶 is optimal if each point outside 𝐶 can be moved to be closer to all points of 𝐶. It is well known (see [28]) that(1)if 𝑋 is a separable, strictly convex, smooth, reflexive Banach space, and if 𝐶⊂𝑋 is an optimal set with interior, then 𝐶 is a nonexpansive retract of 𝑋;(2)a subset of 𝑙^𝑝, with 1<𝑝<∞, 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 𝑋 as follows.

A sequence {𝑥_𝑛} in a normed space 𝑋 is said to be weakly convergent if there is an 𝑥∈𝑋 such that lim_𝑛→∞𝑓(𝑥_𝑛)=𝑓(𝑥) for every bounded linear functional 𝑓 on 𝑋. The element 𝑥 is called the weak limit of {𝑥_𝑛}, and we say that {𝑥_𝑛} converges weakly to 𝑥. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.

A Banach space 𝑋 is said to have the Kadec-Klee property if, for every sequence {𝑥_𝑛} in 𝑋, 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖ together imply ‖𝑥_𝑛−𝑥‖→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 𝑝>1, 𝑟>0 be two fixed numbers. Then, a Banach space 𝑋 is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞)→[0,∞), 𝑔(0)=0 such that ‖𝜆𝑥+(1−𝜆)𝑦‖^𝑝≤𝜆‖𝑥‖^𝑝+(1−𝜆)‖𝑦‖^𝑝−𝑤_𝑝(𝜆)𝑔(‖𝑥−𝑦‖)(1.20) for all 𝑥, 𝑦 in 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}, 𝜆∈[0,1], where 𝑤_𝑝(𝜆)=𝜆(1−𝜆)^𝑝+𝜆^𝑝(1−𝜆).(1.21)

Lemma 1.10 (see [24]). Let 𝑋 be a uniformly convex Banach space and 𝐶 a nonempty closed convex subset of 𝑋, and let 𝑇∶𝐶→𝑋 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1 as 𝑛→∞. Then, 𝐼−𝑇 is demiclosed at zero; that is, if 𝑥_𝑛→𝑥 weakly and 𝑥_𝑛−𝑇𝑥_𝑛→0 strongly, then 𝑥∈𝐹(𝑇).

Lemma 1.11 (see [26]). Let 𝑋 be a uniformly convex Banach space and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞, ∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [0,1). From an arbitrary 𝑥₁∈𝐶, define the sequence {𝑥_𝑛} by (1.14). If 𝑞∈𝐹(𝑇₁)∩𝐹(𝑇₂), then lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists.

Lemma 1.12 (see [26]). Let 𝑋 be a uniformly convex Banach space and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞, ∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). From an arbitrary 𝑥₁∈𝐶, define the sequence {𝑥_𝑛} by (1.14). Then, lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0.

Lemma 1.13 (see [16]). Let 𝑋 be a real reflexive Banach space such that its dual 𝑋^∗ has the Kadec-Klee property. Let {𝑥_𝑛} be a bounded sequence in 𝑋 and 𝑥^∗,𝑦^∗∈𝜔_𝑤(𝑥_𝑛), where 𝜔_𝑤(𝑥_𝑛) denotes the set of all weak subsequential limits of {𝑥_𝑛}. Suppose that lim_𝑛→∞‖𝑡𝑥_𝑛+(1−𝑡)𝑥^∗−𝑦^∗‖ exists for all 𝑡∈[0,1]. Then, 𝑥^∗=𝑦^∗.

We denote by Γ the set of strictly increasing, continuous convex functions 𝛾∶ℝ⁺→ℝ⁺ with 𝛾(0)=0. Let 𝐶 be a convex subset of the Banach space 𝑋. A mapping 𝑇∶𝐶→𝐶 is said to be type (𝛾) [32] if 𝛾∈Γ and 0⩽𝛼⩽1, 𝛾(‖𝛼𝑇𝑥+(1−𝛼)𝑇𝑦−𝑇(𝛼𝑥+(1−𝛼)𝑦)‖)⩽‖𝑥−𝑦‖−‖𝑇𝑥−𝑇𝑦‖(1.22) for all 𝑥, 𝑦 in 𝐶. Obviously, every type  (𝛾) mapping is nonexpansive. For more information about mappings of type (𝛾), see [33–35].

Lemma 1.14 (see [36, 37]). Let 𝑋 be a uniformly convex Banach space and 𝐶 a convex subset of 𝑋. Then, there exists 𝛾∈Γ such that for each mapping 𝑆∶𝐶→𝐶 with Lipschitz constant 𝐿, ‖𝛼𝑆𝑥+(1−𝛼)𝑆𝑦−𝑆(𝛼𝑥+(1−𝛼)𝑦)‖⩽𝐿𝛾⁻¹‖𝑥−𝑦‖−1𝐿‖𝑆𝑥−𝑆𝑦‖(1.23) for all 𝑥,𝑦∈𝐶 and 0<𝛼<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 {𝑥_𝑛} defined by (1.14) in a real uniformly convex Banach space 𝑋 whose dual 𝑋^∗ has the Kadec-Klee property. In order to prove our main results, the following lemma is needed.

Lemma 2.1. Let 𝑋 be a real uniformly convex Banach space and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞, ∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences defined by (1.14). Then, for all 𝑢,𝑣∈𝐹(𝑇₁)∩𝐹(𝑇₂), the limit lim_𝑛→∞‖𝑡𝑥_𝑛−(1−𝑡)𝑢−𝑣‖ exists for all 𝑡∈[0,1].

Proof. It follows from Lemma 1.11 that the sequence {𝑥_𝑛} is bounded. Then, there exists 𝑅>0 such that {𝑥_𝑛}⊂𝐵_𝑅(0)∩𝐶. Let 𝑎_𝑛(𝑡)∶=‖𝑡𝑥_𝑛+(1−𝑡)𝑢−𝑣‖ where 𝑡∈[0,1]. Then, lim_𝑛→∞𝑎_𝑛(0)=‖𝑢−𝑣‖ and, by Lemma 1.11, lim_𝑛→∞𝑎_𝑛(1)=lim_𝑛→∞‖𝑥_𝑛−𝑣‖ exists. Without loss of the generality, we may assume that lim_𝑛→∞‖𝑥_𝑛−𝑣‖=𝑟 for some positive number 𝑟. Let 𝑥∈𝐶 and 𝑡∈(0,1).  For each 𝑛≥1, define 𝐴_𝑛∶𝐶→𝐶 by 𝐴_𝑛𝑥=𝑃1−𝛼_𝑛𝑦_𝑛(𝑥)+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛(𝑥),(2.1) where 𝑦_𝑛(𝑥)=𝑃1−𝛽_𝑛𝑥+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥.(2.2)
Setting 𝑘_𝑛=1+𝑠_𝑛 and 𝑙_𝑛=1+𝑡_𝑛. For 𝑥,𝑧∈𝐶, we have ‖‖𝐴_𝑛𝑥−𝐴_𝑛𝑧‖‖=‖‖𝑃−𝑃1−1−𝛼_𝑛𝛼_𝑛𝑦_𝑛(𝑥)+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛(𝑥)≤‖‖1−𝛼_𝑛𝑦_𝑛(𝑧)+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛(𝑧)‖‖+𝛼_𝑛𝑦_𝑛(𝑥)−𝑦_𝑛(𝑧)≤1−𝛼_𝑛𝑇¹𝑃𝑇₁^𝑛−1𝑦_𝑛(𝑥)−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛(𝑧)‖‖≤1−𝛼_𝑛‖‖𝑦_𝑛(𝑥)−𝑦_𝑛(𝑧)‖‖+𝛼_𝑛𝑘_𝑛‖‖𝑦_𝑛(𝑥)−𝑦_𝑛(𝑧)‖‖+𝛼_𝑛‖‖1−𝛽_𝑛(𝑥−𝑧)+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1(𝑥−𝑧)‖‖≤1−𝑘_𝑛𝛼‖‖_𝑛1−𝛽_𝑛(𝑥−𝑧)+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1(𝑥−𝑧)‖‖+𝛼_𝑛1−𝛽_𝑛‖𝑥−𝑧‖+1−𝛼_𝑛𝛽_𝑛𝑙_𝑛‖𝑥−𝑧‖=1−𝑘𝛼_𝑛𝑛1−𝛽_𝑛‖𝑥−𝑧‖+𝛼_𝑛𝛽_𝑛𝑘_𝑛𝑙_𝑛‖𝑥−𝑧‖+𝛼_𝑛−𝛽_𝑛+𝛼_𝑛𝛽_𝑛‖𝑥−𝑧‖+1−𝛼_𝑛𝛽_𝑛1+𝑡_𝑛‖𝑥−𝑧‖=‖𝑥−𝑧1‖++𝑠_𝑛𝛽_𝑛1−𝛽_𝑛‖𝑥−𝑧‖+𝛼_𝑛𝛽_𝑛1+𝑠_𝑛1+𝑡_𝑛‖𝑥−𝑧‖≤1+𝑡_𝑛𝑡_𝑛‖𝑥−𝑧‖+𝛼_𝑛𝑠_𝑛‖𝑥−𝑧‖+𝛼_𝑛𝛽_𝑛𝑡_𝑛𝑠_𝑛‖𝑥−𝑧‖+𝑠_𝑛+𝑡_𝑛𝑠_𝑛‖𝑥−𝑧‖.(2.3)
Set 𝑆_𝑛,𝑚∶=𝐴_{𝑛+𝑚−1}𝐴_{𝑛+𝑚−2}⋯𝐴_𝑛, 𝑛,𝑚≥1 and 𝑏_𝑛,𝑚=‖𝑆_𝑛,𝑚(𝑡𝑥_𝑛+(1−𝑡)𝑢)−(𝑡𝑆_𝑛,𝑚𝑥_𝑛+(1−𝑡)𝑢)‖, where 0≤𝑡≤1. Also, ‖‖𝑆_𝑛,𝑚𝑥−𝑆_𝑛,𝑚𝑦‖‖≤≤‖‖1𝐴+𝑡_{𝑛+𝑚−1}𝐴_{𝑛+𝑚−2𝑛+𝑚−1}⋯𝐴_𝑛𝑥−𝐴_{𝑛+𝑚−1}𝐴_{𝑛+𝑚−2}⋯𝐴_𝑛𝑦‖‖‖‖𝐴_{𝑛+𝑚−2}+𝑠_{𝑛+𝑚−1}+𝑡_{𝑛+𝑚−1}𝑠_{𝑛+𝑚−1}≤⋮_{𝑛+𝑚−1}𝐴_{𝑛+𝑚−3}⋯𝐴_𝑛𝑥−𝐴_{𝑛+𝑚−2}𝐴_{𝑛+𝑚−3}⋯𝐴_𝑛𝑦‖‖_𝑗=𝑛1+𝑡_𝑗+𝑠_𝑗+𝑡_𝑗𝑠_𝑗‖𝑥−𝑦‖(2.4) for all 𝑥,𝑦∈𝐶 and 𝑆_𝑛,𝑚𝑥_𝑛=𝑥_𝑛+𝑚, 𝑆_𝑛,𝑚𝑥^∗=𝑥^∗ for all 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂).
Applying Lemma 1.14 with 𝑥=𝑥_𝑛, 𝑦=𝑢, 𝑆=𝑆_𝑛,𝑚 and using the facts that lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝑙_𝑛−1)=0, lim_𝑛→∞𝑠_𝑛=lim_𝑛→∞(𝑘_𝑛−1)=0, and lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exist for all 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂), we obtain lim_𝑛→∞𝑏_𝑛,𝑚=0. Observe that 𝑎_𝑛+𝑚(𝑡)==‖‖‖‖𝑡𝑡𝑥𝑆_{𝑛+𝑚𝑛,𝑚}+(1−𝑡)𝑢−𝑣‖‖=‖‖𝑆_𝑛,𝑚𝑥_𝑛+(1−𝑡)𝑢−𝑆_𝑛,𝑚𝑣‖‖=‖‖𝑆𝑣−𝑡𝑆_𝑛,𝑚𝑥_𝑛+(1−𝑡)𝑢_𝑛,𝑚‖‖−𝑡𝑣𝑆−_𝑛,𝑚𝑆_𝑛,𝑚𝑡𝑥_𝑛+(1−𝑡)𝑢+𝑆_𝑛,𝑚𝑡𝑥_𝑛+(1−𝑡)𝑢≤‖‖𝑆𝑥_𝑛+(1−𝑡)𝑢_𝑛,𝑚‖‖𝑣−𝑆_𝑛,𝑚𝑡𝑥_𝑛+(1−𝑡)𝑢‖‖+𝑏_𝑛,𝑚=‖‖𝑆_𝑛,𝑚𝑡𝑥_𝑛+(1−𝑡)𝑢−𝑆_𝑛,𝑚𝑣‖‖+𝑏_𝑛,𝑚≤^{𝑛+𝑚−1}_𝑗=𝑛1+𝑡_𝑗+𝑠_𝑗+𝑡_𝑗𝑠_𝑗‖‖𝑡𝑥_𝑛+(1−𝑡)𝑢−𝑣‖‖+𝑏_𝑛,𝑚≤^∞_𝑗=𝑛1+𝑡_𝑗+𝑠_𝑗+𝑡_𝑗𝑠_𝑗𝑎_𝑛(𝑡)+𝑏_𝑛,𝑚.(2.5) Consequently, limsup_𝑚→∞𝑎_𝑚(𝑡)=limsup_𝑚→∞𝑎_𝑛+𝑚(𝑡)≤limsup_𝑚→∞𝑏^𝑛,𝑚limsup_𝑛→∞+∏^∞_𝑗=𝑛1+𝑡_𝑗+𝑠_𝑗+𝑡_𝑗𝑠_𝑗𝑎_𝑛(𝑡),(2.6)𝑎_𝑛(𝑡)≤liminf_𝑛→∞𝑎_𝑛(𝑡).(2.7) This implies that lim_𝑛→∞𝑎_𝑛(𝑡) exists for all 𝑡∈[0,1]. This completes the proof.

Theorem 2.2. Let 𝑋 be a real uniformly convex Banach space which has a Fréchet differentiable norm and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞,∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences defined by (1.14). Then, {𝑥_𝑛} converges weakly to a fixed point of 𝑇₁ and 𝑇₂.

Proof. Set 𝑥=𝑝₁−𝑝₂ and ℎ=𝑡(𝑥_𝑛−𝑝₁) in (1.18). By using Lemmas 1.11, and 2.1 and the same proof of Lemma  4 of Osilike and Udomene [7], we can show that, for every 𝑝₁,𝑝₂∈𝐹(𝑇₁)∩𝐹(𝑇₂), 𝑝−𝑞,𝐽𝑝₁−𝑝₂=0,(2.8) for all 𝑝,𝑞∈𝜔_𝑤(𝑥_𝑛). Since 𝐸 is reflexive and {𝑥_𝑛} is bounded, we from Lemma 1.13 conclude that 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑇_𝑖) for each 𝑖=1,2. Let 𝑝,𝑞∈𝜔_𝑤(𝑥_𝑛). It follows that 𝑝,𝑞∈𝐹(𝑇₁)∩𝐹(𝑇₂); that is, ‖𝑝−𝑞‖²=⟨𝑝−𝑞,𝐽(𝑝−𝑞)⟩=0.(2.9) Therefore, 𝑝=𝑞. This completes the proof.

Theorem 2.3. Let 𝑋 be a real uniformly convex Banach space such that its dual 𝑋^∗ has the Kadec-Klee property and 𝐶 a nonempty closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that ∑^∞𝑛=1(𝑘_𝑛−1)<∞, ∑^∞𝑛=1(𝑙_𝑛−1)<∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences defined by (1.14). Then, {𝑥_𝑛} converges weakly to a fixed point of 𝑇₁ and 𝑇₂.

Proof. It follows from Lemma 1.11 that the sequence {𝑥_𝑛} is bounded. Then, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} converging weakly to a point 𝑥^∗∈𝐶. By Lemma 1.12, we have lim_𝑛→∞‖‖𝑥_𝑛𝑗−𝑇₁𝑥_𝑛𝑗‖‖=0=lim_𝑛→∞‖‖𝑥_𝑛𝑗−𝑇₂𝑥_𝑛𝑗‖‖.(2.10) Now, using Lemma 1.10, we have (𝐼−𝑇)𝑥^∗=0; that is, 𝑇𝑥^∗=𝑥^∗. Thus, 𝑥^∗∈𝐹(𝑇₁)∩𝐹(𝑇₂). It remains to show that {𝑥_𝑛} converges weakly to 𝑥^∗. Suppose that {𝑥_𝑛𝑖} is another subsequence of {𝑥_𝑛} converging weakly to some 𝑦^∗. Then, 𝑦^∗∈𝐶 and so 𝑥^∗,𝑦^∗∈𝜔_𝑤(𝑥_𝑛)∩𝐹(𝑇₁)∩𝐹(𝑇₂). By Lemma 2.1, lim_𝑛→∞‖‖𝑡𝑥_𝑛−(1−𝑡)𝑥^∗−𝑦^∗‖‖(2.11) exists for all 𝑡∈[0,1]. It follows from Lemma 1.13 that 𝑥^∗=𝑦^∗. As a result, 𝜔_𝑤(𝑥_𝑛) is a singleton, and so {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

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 Fréchet differentiable norm.

Let 𝑇₁ and 𝑇₂ be two asymptotically nonexpansive nonself-mappings of 𝐶 with {𝑘_𝑛}⊂[1,∞), lim_𝑛→∞𝑘_𝑛=1, and {𝑙_𝑛}⊂[1,∞), lim_𝑛→∞𝑙_𝑛=1, respectively. In the sequel, we take {𝑡_𝑛}⊂[1,∞), where 𝑡_𝑛=max{𝑘_𝑛,𝑙_𝑛}.

We start with proving the following lemma for later use.

Lemma 2.4. Let 𝑋 be a uniformly convex Banach space and 𝐶 a nonempty bounded closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that 𝑘_𝑛→1, 𝑙_𝑛→1 as 𝑛→∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Then, for the sequence {𝑥_𝑛} given in (1.14), we have that lim_𝑛→∞‖‖𝑥_𝑛−𝑇₁𝑥_𝑛‖‖=0=lim_𝑛→∞‖‖𝑥_𝑛−𝑇₂𝑥_𝑛‖‖.(2.12)

Proof. By setting 𝑡_𝑛=max{𝑘_𝑛,𝑙_𝑛}, then lim_𝑛→∞𝑡_𝑛=1 if lim_𝑛→∞𝑘_𝑛=1=lim_𝑛→∞𝑙_𝑛. Let 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂). Since 𝐶 is bounded, there exists 𝐵_𝑟(0) such that 𝐶⊂𝐵_𝑟(0) for some 𝑟>0. Applying Lemma 1.9 for scheme (1.14), we have ‖‖𝑦_𝑛−𝑝‖‖²=‖‖𝑃1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛−𝑝‖‖²≤‖‖1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑇²𝑃𝑇₂^𝑛−1𝑥_𝑛−𝑝‖‖²=1−𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖²+𝛽_𝑛𝑙^2𝑛‖‖𝑥_𝑛−𝑝‖‖²=−1𝛽−_𝑛𝛽1_𝑛−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖+𝛽_𝑛𝑙^2𝑛‖‖𝑥_𝑛−𝑝‖‖²−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖(2.13) and so, ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖𝑃1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛−𝑝‖‖²≤‖‖1−𝛼_𝑛𝑦_𝑛−𝑝+𝛼_𝑛(𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛−𝑝)‖‖²=1−𝛼_𝑛‖‖𝑦_𝑛−𝑝‖‖²+𝛼_𝑛𝑘^2𝑛‖‖𝑦_𝑛−𝑝‖‖²=−1𝛼−_𝑛𝛼1_𝑛−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖+𝛼_𝑛𝑘^2𝑛‖‖𝑦_𝑛−𝑝‖‖²≤−1𝛼_𝑛−𝛼1_𝑛−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖−𝛼_𝑛+𝛼_𝑛𝑘^2𝑛1−𝛽_𝑛+𝛽_𝑛𝑙^2𝑛‖‖𝑥_𝑛−𝑝‖‖²−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛=1−𝛼_𝑛‖‖+𝛼_𝑛𝑘^2𝑛1−𝛽_𝑛+𝛽_𝑛𝑙^2𝑛‖‖𝑥_𝑛−𝑝‖‖²−−𝛼1_𝑛−𝛼_𝑛+𝛼_𝑛𝑘^2𝑛𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛=1−𝛼_𝑛‖‖1−𝛽_𝑛+1−𝛼_𝑛𝛽_𝑛𝑙^2𝑛+1−𝛽_𝑛𝛼_𝑛𝑘^2𝑛+𝛼_𝑛𝑘^2𝑛𝛽_𝑛𝑙^2𝑛‖‖𝑥_𝑛−𝑝‖‖²−−𝛼1_𝑛−𝛼_𝑛+𝛼_𝑛𝑘^2𝑛𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛≤1−𝛼_𝑛‖‖1−𝛽_𝑛+1−𝛼_𝑛𝛽_𝑛𝑡^2𝑛+1−𝛽_𝑛𝛼_𝑛𝑡^2𝑛+𝛼_𝑛𝛽_𝑛𝑡^4𝑛‖‖𝑥_𝑛−𝑝‖‖²−−𝛼1_𝑛−𝛼_𝑛+𝛼_𝑛𝑘^2𝑛𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛≤1−𝛼_𝑛‖‖1−𝛽_𝑛𝑡^4𝑛+1−𝛼_𝑛𝛽_𝑛𝑡^4𝑛+1−𝛽_𝑛𝛼_𝑛𝑡^4𝑛+𝛼_𝑛𝛽_𝑛𝑡^4𝑛‖‖𝑥_𝑛−𝑝‖‖²−−𝛼1_𝑛+𝛼_𝑛𝑘^2𝑛−1𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛≤1−𝛼_𝑛‖‖1−𝛽_𝑛𝑡^4𝑛+1−𝛼_𝑛𝛽_𝑛𝑡^4𝑛+1−𝛽_𝑛𝛼_𝑛𝑡^4𝑛+𝛼_𝑛𝛽_𝑛𝑡^4𝑛‖‖𝑥_𝑛−𝑝‖‖²−−𝛼𝛽_𝑛𝑛1−𝛽_𝑛𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖1−𝛼_𝑛𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛≤‖‖𝑥_𝑛‖‖−𝜀₂−𝑝‖‖²+𝑟𝑡^4𝑛−1−𝜀²𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖.(2.14) From (2.14), we obtain the following two important inequalities: ‖‖‖‖𝑥𝑥_𝑛+1−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²+𝑟𝑡^4𝑛−1−𝜀²𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖,(2.15)_𝑛+1−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²+𝑟𝑡^4𝑛−1−𝜀²𝑔‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖.(2.16) Now, we prove that lim_𝑛→∞‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖=0=lim_𝑛→∞‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖.(2.17) Assume that limsup_𝑛→∞‖𝑥_𝑛−𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛‖>0. Then, there exists a subsequence (use the same notation for subsequence as for the sequence) of {𝑥_𝑛} and 𝜇>0 such that ‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖≥𝜇>0.(2.18) By definition of 𝑔, we have 𝑔‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖≥𝑔(𝜇)>0.(2.19) From (2.15), we have ‖‖𝑥_𝑛+1−𝑝‖‖²≤=‖‖‖‖𝑥𝑥_𝑛−𝑝‖‖²+𝑟𝑡^4𝑛−1−𝜀²𝑔(𝜇)_𝑛−𝑝‖‖²+𝑟𝑡^4𝑛−1−𝜀²2𝑟𝑔(𝜇)−𝜀2²𝑔(𝜇).(2.20)
In addition, 𝑡^4𝑛→1 and (𝜀²/2𝑟)𝑔(𝜇)>0; there exists 𝑛₀≥1 such that (𝑡^4𝑛−1)<(𝜀²/2𝑟)𝑔(𝜇) for all 𝑛≥𝑛₀. From (2.20), we obtain 𝜀²2𝑔(𝜇)≤‖‖𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²(2.21) for all 𝑛≥𝑛₀.
Let 𝑚≥𝑛₀. It follows from (2.21) that 𝜀²2^𝑚_𝑛=𝑛0𝑔(𝜇)≤^𝑚_𝑛_=𝑛0‖‖𝑥^𝑛=‖‖𝑥_𝑛0−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²−𝑝‖‖².(2.22)
By letting 𝑚→∞ in (2.22), we obtain ∞=‖‖𝑥_𝑛0−𝑝‖‖²<∞(2.23) which contradicts the reality. This proves that 𝜇=0. Thus, limsup_𝑛→∞‖𝑥_𝑛−𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛‖≤0. Consequently, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖=0.(2.24)
Similarly, using (2.16), we may show that lim_𝑛→∞‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖=0.(2.25)
Using (2.24), we have ‖‖𝑥_𝑛−𝑦_𝑛‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(2.26)
From (2.25), (2.26), and the uniform equicontinuous of 𝑇₁ (see Remark 1.1), we have ‖‖𝑥_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛≤‖‖𝑥_𝑛‖‖+‖‖𝑇−₁𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛‖‖(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(2.27)
Since ‖‖𝑥_𝑛−𝑥_𝑛+1‖‖≤=11−−𝛼𝛼_𝑛𝑛‖‖𝑦_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛−𝑥_𝑛‖‖‖‖𝑦_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛−𝑥_𝑛≤‖‖𝑦_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛−𝑥_𝑛‖‖,(2.28) it follows from (2.26), (2.27), and the uniform equi-continuity of 𝑇₁ (see Remark 1.1) that lim_𝑛→∞‖‖𝑥_𝑛−𝑥_𝑛+1‖‖=0.(2.29)
Since lim_𝑛→∞‖𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖=0 and again from the fact that 𝑇₁ is uniformly equicontinuous mapping, by Using (2.29), we have ‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖=‖‖𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1≤‖‖𝑥_𝑛+1‖‖+‖‖𝑇₁−𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛‖‖𝑃𝑇₁^𝑛−1𝑥_𝑛−𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(2.30) In addition, ‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+1=‖‖𝑥_𝑛+1‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛−𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+1‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛−𝑥_𝑛‖‖+𝐿‖‖𝑥_𝑛+1−𝑥_𝑛‖‖,(2.31) where 𝐿=sup{𝑘_𝑛∶𝑛≥1}. It follows from (2.29) and (2.30) that lim_→∞‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+1‖‖=0.(2.32) We denote (𝑃𝑇₁)¹⁻¹ to be the identity maps from 𝐶 onto itself. Thus, by the inequality (2.30) and (2.32), we have ‖‖𝑥_𝑛+1−𝑇₁𝑥_𝑛+1‖‖=≤‖‖‖‖𝑥𝑥_𝑛+1𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1+𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1−𝑇₁𝑥_𝑛+1‖‖=‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖+‖‖𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1−𝑇₁𝑥_𝑛+1‖‖≤‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖+‖‖𝑇₁𝑃𝑇₁¹⁻¹𝑃𝑇₁^𝑛−1𝑥_𝑛+1−𝑇₁𝑃𝑇₁¹⁻¹𝑥_𝑛+1‖‖=‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖+𝐿‖‖𝑃𝑇₁^𝑛−1𝑥_𝑛+1−𝑥_𝑛+1‖‖≤‖‖𝑥_𝑛+1−𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖+𝐿‖‖𝑃𝑇₁𝑃𝑇₁^𝑛−2𝑥_𝑛+1−𝑃𝑥_𝑛+1‖‖+𝐿‖‖𝑇−₁𝑇₁𝑃𝑇₁^𝑛−1𝑥_𝑛+1‖‖𝑃𝑇₁^𝑛−2𝑥_𝑛+1−𝑥_𝑛+1‖‖⟶0(as𝑛⟶∞),(2.33) which implies that lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=0. Similarly, we may show that lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0. The proof is completed.

Our weak convergence theorem is as follows. We do not use the rate of convergence conditions, namely, ∑^∞𝑛=1(𝑘_𝑛−1)<∞ and ∑^∞𝑛=1(𝑙_𝑛−1)<∞ in its proof.

Theorem 2.5. Let 𝑋 be a uniformly convex Banach space and 𝐶 a nonempty bounded closed convex nonexpansive retract of 𝑋 with 𝑃 as a nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐶→𝑋 be two asymptotically nonexpansive nonself-mappings of 𝐶 with sequences {𝑘_𝑛},{𝑙_𝑛}⊂[1,∞) such that 𝑘_𝑛→1, 𝑙_𝑛→1 as 𝑛→∞, respectively, and 𝐹(𝑇₁)∩𝐹(𝑇₂)≠∅. Suppose that {𝛼_𝑛} and {𝛽_𝑛} are real sequences in [𝜖,1−𝜖] for some 𝜖∈(0,1). Then, the sequence {𝑥_𝑛} given in (1.14) converges weakly to a common fixed point of 𝑇₁ and 𝑇₂.

Proof. Since 𝐶 is a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗 converges weakly to 𝑞∈𝜔_𝑤(𝑥_𝑛), where 𝜔_𝑤(𝑥_𝑛) denotes the set of all weak subsequential limits of {𝑥_𝑛}. This show that 𝜔_𝑤(𝑥_𝑛)≠∅ and, by Lemma 2.4, lim_𝑛→∞‖𝑥_𝑛𝑗−𝑇₁𝑥_𝑛𝑗‖=lim_𝑛→∞‖𝑥_𝑛𝑗−𝑇₂𝑥_𝑛𝑗‖=0. Since 𝐼−𝑇₁ and 𝐼−𝑇₂ are demiclosed at zero, using Lemma 1.10, we have 𝑇₁𝑞=𝑞=𝑇₂𝑞. Therefore, 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑇₁)∩𝐹(𝑇₂).For any 𝑞∈𝜔_𝑤(𝑥_𝑛), there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑞(as𝑖⟶∞).(2.34) It follows from (2.24) and (2.34) that 𝑇₂𝑃𝑇₂^𝑛𝑗−1𝑥_𝑛𝑗=𝑇₂𝑃𝑇₂^𝑛𝑗−1𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥^𝑛_𝑗⇀𝑞.(2.35) Now, from (1.14), (2.34), and (2.35), 𝑦_𝑛𝑗=𝑃1−𝛽_𝑛𝑗𝑥^𝑛_𝑗+𝛽_𝑛𝑗𝑇₂𝑃𝑇₂^𝑛𝑗−1𝑥_𝑛𝑗⇀𝑞.(2.36) Also, from (2.25) and (2.36), we have 𝑇₁𝑃𝑇₁^𝑛𝑗−1𝑦_𝑛𝑗=𝑇₁𝑃𝑇₁^𝑛𝑗−1𝑦_𝑛𝑗−𝑦_𝑛𝑗+𝑦^𝑛_𝑗⇀𝑞.(2.37) It follows from (2.36) and (2.37) that 𝑥_𝑛𝑗+1=𝑃1−𝛼_𝑛𝑗𝑦^𝑛_𝑗+𝛼_𝑛𝑗𝑇₁𝑃𝑇₁^𝑛𝑗−1𝑦_𝑛𝑗⇀𝑞.(2.38) Continuing in this way, by induction, we can prove that, for any 𝑚≥0, 𝑥_{𝑛𝑗+𝑚}⇀𝑞.(2.39) By induction, one can prove that ∪^∞𝑚=0{𝑥_{𝑛𝑗+𝑚}} converges weakly to 𝑞 as 𝑗→∞; in fact, {𝑥_𝑛}^∞𝑛=𝑛₁=∪^∞𝑚=0{𝑥_{𝑛𝑗+𝑚}}^∞𝑗=1 gives that 𝑥_𝑛⇀𝑞 as 𝑛→∞. 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.