Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces

Abstract

We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).

1. Introduction

Let 𝐸 be a real normed space and 𝐾 be a nonempty subset of 𝐸. A mapping 𝑇∶𝐾→𝐾 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾. A mapping 𝑇∶𝐾→𝐾 is called asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾 and 𝑛≥1. 𝑇 is called uniformly 𝐿-Lipschitzian if there exists a real number 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾 and 𝑛≥1. 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}.

Iterative techniques for nonexpansive and asymptotically nonexpansive mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors; see [1–8]. However, if the domain of 𝑇, 𝐷(𝑇), is a proper subset of 𝐸 (and this is the case in several applications), and 𝑇 maps 𝐷(𝑇) into 𝐸, then the iteration processes of Mann type and Ishikawa type studied by the authors mentioned above, and their modifications introduced may fail to be well defined.

A subset 𝐾 of 𝐸 is said to be a retract of 𝐸 if there exists a continuous map 𝑃∶𝐸→𝐾 such that 𝑃𝑥=𝑥, for all 𝑥∈𝐾. Every closed convex subset of a uniformly convex Banach space is a retract. A map 𝑃∶𝐸→𝐾 is said to be a retraction if 𝑃²=𝑃. It follows that if a map 𝑃 is a retraction, then 𝑃𝑦=𝑦 for all 𝑦∈𝑅(𝑃), the range of 𝑃.

The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [4] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition (see [4]). Let 𝐾 be a nonempty subset of real normed linear space 𝐸. Let 𝑃∶𝐸→𝐾 be the nonexpansive retraction of 𝐸 onto 𝐾. A nonself mapping 𝑇∶𝐾→E is called asymptotically nonexpansive if there exists sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1(𝑛→∞) such that ‖𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ (1.1) for all 𝑥,𝑦∈𝐾 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‖ (1.2) for all 𝑥,𝑦∈𝐾 and 𝑛≥1.

In [4], they study the following iterative sequence:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑇(𝑃𝑇)^𝑛−1𝑥_𝑛),𝑥₁∈𝐾,𝑛≥1 (1.3) to approximate some fixed point of 𝑇 under suitable conditions. In [9], Wang generalized the iteration process (1.3) as follows:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛𝑦),_𝑛=𝑃((1−𝛼^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛),𝑥₁∈𝐾,𝑛≥1, (1.4) where 𝑇₁,𝑇₂∶𝐾→𝐸 are asymptotically nonexpansive nonself-mappings and {𝛼_𝑛},{𝛼^′_𝑛} are sequences in [0,1]. He studied the strong and weak convergence of the iterative scheme (1.4) under proper conditions. Meanwhile, the results of [9] generalized the results of [4].

In [10], Shahzad studied the following iterative sequence:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑇𝑃[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛]),𝑥₁∈𝐾,𝑛≥1, (1.5) where 𝑇∶𝐾→𝐸 is a nonexpansive nonself-mapping and 𝐾 is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space 𝐸 with 𝑃, nonexpansive retraction.

Recently, Thianwan [11] generalized the iteration process (1.5) as follows:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇𝑃((1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇𝑦_𝑛)+𝛾_𝑛𝑢_𝑛𝑦),_𝑛=𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇𝑃((1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇𝑥_𝑛)+𝛾^′_𝑛𝑣_𝑛),𝑥₁∈𝐾,𝑛≥1, (1.6) where {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛},{𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾. He proved weak and strong convergence theorems for nonexpansive nonself-mappings in uniformly convex Banach spaces.

The purpose of this paper, motivated by the Wang [9], Thianwan [11] and some others, is to construct an iterative scheme for approximating a fixed point of asymptotically nonexpansive nonself-mappings (provided that such a fixed point exists) and to prove some strong and weak convergence theorems for such maps.

Let 𝐸 be a normed space, 𝐾 a nonempty convex subset of 𝐸, 𝑃∶𝐸→𝐾 the nonexpansive retraction of 𝐸 onto 𝐾, and 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself-mappings. Then, for given 𝑥₁∈𝐾 and 𝑛≥1, we define the sequence {𝑥_𝑛} by the iterative scheme:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑃((1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛)+𝛾_𝑛𝑢_𝑛𝑦),_𝑛=𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃((1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛)+𝛾^′_𝑛𝑣_𝑛), (1.7) where {𝛼_𝑛}, {𝛽_𝑛},{𝛾_𝑛},{𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛 and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾. Clearly, the iterative scheme (1.7) is generalized by the iterative schemes (1.4) and (1.6).

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

Let 𝐸 be a Banach space with dimension 𝐸≥2. The modulus of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by

𝛿_𝐸1(𝜀)=inf{1−‖2(𝑥+𝑦)‖∶‖𝑥‖=‖𝑦‖=1,𝜀=‖𝑥−𝑦‖}. (1.8) A Banach space 𝐸 is uniformly convex if and only if 𝛿_𝐸(𝜀)>0 for all 𝜀∈(0,2].

A Banach space 𝐸 is said to satisfy Opial's condition [12] if for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥 implies that

limsup_𝑛→∞‖𝑥_𝑛−𝑥‖<limsup_𝑛→∞‖𝑥_𝑛−𝑦‖ (1.9) for all 𝑦∈𝐸 with 𝑦≠𝑥, where 𝑥_𝑛⇀𝑥 denotes that {𝑥_𝑛}converges weakly to 𝑥.

The mapping 𝑇∶𝐾→𝐸 with 𝐹(𝑇)≠∅ is said to satisfy condition (𝐴) [13] if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that

‖𝑥−𝑇𝑥‖≥𝑓(𝑑(𝑥,𝐹(𝑇))) (1.10) for all 𝑥∈𝐾, where 𝑑(𝑥,𝐹(𝑇))=inf{‖𝑥−𝑝‖∶𝑝∈𝐹(𝑇)}; (see [13, page 337]) for an example of nonexpansive mappings satisfying condition (𝐴).

Two mappings 𝑇₁,𝑇₂∶𝐾→𝐸 are said to satisfy condition (𝐴^′) [14] if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that

12(‖𝑥−𝑇₁𝑥‖+‖𝑥−𝑇₂𝑥‖)≥𝑓(𝑑(𝑥,𝐹(𝑇))) (1.11) for all 𝑥∈𝐾, where 𝑑(𝑥,𝐹(𝑇))=inf{‖𝑥−𝑝‖∶𝑝∈𝐹(𝑇)=𝐹(𝑇₁)∩𝐹(𝑇₂)}.

Note that condition (𝐴^′) reduces to condition (𝐴) when 𝑇₁=𝑇₂ and hence is more general than the demicompactness of 𝑇₁and𝑇₂ [13]. A mapping 𝑇∶𝐾→𝐾 is called: (1) demicompact if any bounded sequence {𝑥_𝑛} in 𝐾 such that {𝑥_𝑛−𝑇𝑥_𝑛} converges has a convergent subsequence, (2) semicompact (or hemicompact) if any bounded sequence {𝑥_𝑛} in 𝐾 such that {𝑥_𝑛−𝑇𝑥_𝑛}→0 as 𝑛→∞ has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general.

Senter and Dotson [13] have approximated fixed points of a nonexpansive mapping 𝑇 by Mann iterates, whereas Maiti and Ghosh [14] and Tan and Xu [5] have approximated the fixed points using Ishikawa iterates under the condition (𝐴) of Senter and Dotson [13]. Tan and Xu [5] pointed out that condition (𝐴) is weaker than the compactness of 𝐾. Khan and Takahashi [6] have studied the two mappings case for asymptotically nonexpansive mappings under the assumption that the domain of the mappings is compact. We shall use condition (𝐴^′) instead of compactness of 𝐾 to study the strong convergence of {𝑥_𝑛} defined in (1.7).

In the sequel, we need the following usefull known lemmas to prove our main results.

Lemma 1.2 (see [5]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤(1+𝛿_𝑛)𝑎_𝑛+𝑏_𝑛,𝑛≥1. (1.12) If ∑^∞_𝑛=1𝑏_𝑛<∞ and ∑^∞_𝑛=1𝛿_𝑛<∞, then (i)lim_𝑛→∞𝑎_𝑛 exists;(ii)In particular, if {𝑎_𝑛} has a subsequence which converges strongly to zero, then lim_𝑛→∞𝑎_𝑛=0.

Lemma 1.3 (see [2]). Suppose that 𝐸 is a uniformly convex Banach space and 0<𝑝≤𝑡_𝑛≤𝑞<1 for all 𝑛≥1. Suppose further that {𝑥_𝑛} and {𝑦_𝑛} are sequences of 𝐸such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑟,limsup_𝑛→∞‖𝑦_𝑛‖≤𝑟,lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑟 (1.13) hold for some 𝑟≥0. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 1.4 (see [4]). Let 𝐸 be a uniformly convex Banach space, 𝐾 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐾→𝐸 be a nonexpansive mapping. 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), where 𝐹(𝑇) is the set fixed point of 𝑇.

2. Main Results

We shall make use of the following lemmas.

Lemma. Let 𝐸 be a normed space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself-mappings of 𝐸 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)∶={𝑥∈𝐾∶𝑇₁𝑥=𝑇₂𝑥=𝑥}≠∅. Suppose that {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Starting from an arbitrary 𝑥₁∈𝐾, define the sequence {𝑥_𝑛} by the recursion (1.7). Then, lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂).

Proof. Let 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂). Since {𝑢_𝑛} and {𝑣_𝑛} are bounded sequences in 𝐾, we have 𝑟=max{sup_𝑛≥1‖𝑢_𝑛−𝑝‖,sup_𝑛≥1‖𝑣_𝑛−𝑝‖}. (2.1) Set 𝜎_𝑛 = (1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛 and 𝛿_𝑛= (1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛. Firstly, we note that ‖𝜎_𝑛−𝑝‖=‖(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖≤𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖+(1−𝛽_𝑛)‖𝑦_𝑛−𝑝‖≤𝛽_𝑛𝑘_𝑛‖𝑦_𝑛−𝑝‖+(1−𝛽_𝑛)‖𝑦_𝑛−𝑝‖≤𝑘_𝑛‖𝑦_𝑛−𝑝‖,‖𝛿_𝑛−𝑝‖=‖(1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝‖≤𝛽^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝‖+(1−𝛽^′_𝑛)‖𝑥_𝑛−𝑝‖≤𝛽^′_𝑛𝑙_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛽^′_𝑛)‖𝑥_𝑛−𝑝‖≤𝑙_𝑛‖x_𝑛−𝑝‖. (2.2)(2.3) From (1.7) and (2.3), we have ‖𝑦_𝑛−𝑝‖=‖𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛)−𝑝‖≤‖(1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛−𝑝‖≤𝛼^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝‖+(1−𝛼^′_𝑛−𝛾^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛‖𝑣_𝑛−𝑝‖≤𝛼^′_𝑛𝑙_𝑛‖𝛿_𝑛−𝑝‖+(1−𝛼^′_𝑛−𝛾^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛‖𝑣_𝑛−𝑝‖≤𝛼^′_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼^′_𝑛−𝛾^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛𝑟≤𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛𝑟. (2.4) Substituting (2.4) into (2.2), we obtain ‖𝜎_𝑛−𝑝‖≤𝑘_𝑛‖𝑦_𝑛−𝑝‖≤𝑘_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝑘_𝑛𝛾^′_𝑛𝑟. (2.5) It follows from (1.7) and (2.5) that ‖𝑥_𝑛+1−𝑝‖=‖𝑃((1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛+𝛾_𝑛𝑢_𝑛)−𝑝‖≤‖(1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛+𝛾_𝑛𝑢_𝑛−𝑝‖≤𝛼_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝‖+(1−𝛼_𝑛−𝛾_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑝‖≤𝛼_𝑛𝑘_𝑛‖𝜎_𝑛−𝑝‖+(1−𝛼_𝑛−𝛾_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑝‖≤𝛼_𝑛(𝑘²_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝑘²_𝑛𝛾^′_𝑛𝑟)+(1−𝛼_𝑛−𝛾_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛𝑟≤𝑘²_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝑘²_𝑛𝛾^′_𝑛𝑟+𝛾_𝑛𝑟=(1+(𝑙²_𝑛−1)(𝑘²_𝑛−1)+(𝑙²_𝑛−1)+(𝑘²_𝑛−1))‖𝑥_𝑛−𝑝‖+(𝑘²_𝑛𝛾^′_𝑛+𝛾_𝑛)𝑟. (2.6) Note that ∑^∞_𝑛=1𝑘_𝑛−1<∞ and ∑^∞_𝑛=1𝑙_𝑛−1<∞ are equivalent to ∑^∞_𝑛=1𝑘²_𝑛−1<∞ and ∑^∞_𝑛=1𝑙²_𝑛−1<∞, respectively. Since ∑^∞_𝑛=1𝛾_𝑛<∞ and ∑^∞_𝑛=1𝛾^′_𝑛<∞, we have ∑^∞_𝑛=1(𝑘²_𝑛𝛾^′_𝑛+𝛾_𝑛)𝑟<∞. We obtained from (2.6) and Lemma 1.2 that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹(𝑇). This completes the proof.

Lemma. Let 𝐸 be a normed space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be nonself uniformly 𝐿₁-Lipschitzian, 𝐿₂-Lipschitzian, respectively. Suppose that {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Starting from an arbitrary 𝑥₁∈𝐾, define the sequence {𝑥_𝑛} by the recursion (1.7) and set 𝐶_𝑛=‖𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖,𝐶^′_𝑛=‖𝑥_𝑛−𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛‖ for all 𝑛≥1. If lim_𝑛→∞𝐶_𝑛=lim_𝑛→∞𝐶^′_𝑛=0, then lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0. (2.7)

Proof. Since {𝑢_𝑛}, {𝑣_𝑛} are bounded, it follows from Lemma 2.1 that {𝑢_𝑛−𝑥_𝑛} and {𝑣_𝑛−𝑥_𝑛} are all bounded. We set 𝑟₁=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛≥1},𝑟₂=sup{‖𝑣_𝑛−𝑥_𝑛𝑟‖∶𝑛≥1},₃=sup{‖𝑢_𝑛−1−𝑥_𝑛−1‖∶𝑛≥1},𝑟=max{𝑟_𝑖∶𝑖=1,2,3}. (2.8) Let 𝜎_𝑛=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛 and 𝛿_𝑛=(1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛. Then, we have ‖𝜎_𝑛−𝑥_𝑛‖=‖(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑥_𝑛‖≤𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖+𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑥_𝑛‖+(1−𝛽_𝑛)‖𝑦_𝑛−𝑥_𝑛‖≤(𝐿₁+1)‖𝑦_𝑛−𝑥_𝑛‖+𝐶_𝑛,‖𝛿_𝑛−𝑥_𝑛‖=‖(1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑥_𝑛‖≤𝛽^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑥_𝑛‖≤𝐶^′_𝑛. (2.9)(2.10) We find the following from (1.7) and (2.10): ‖𝑦_𝑛−𝑥_𝑛‖=‖𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛)−𝑥_𝑛‖≤‖(1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛−𝑥_𝑛‖≤𝛼^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛‖+𝛼^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑥_𝑛‖+𝛾^′_𝑛‖𝑣_𝑛−𝑥_𝑛‖≤𝐿₂‖𝛿_𝑛−𝑥_𝑛‖+𝐶^′_𝑛+𝛾^′_𝑛𝑟≤𝐿₂𝐶^′_𝑛+𝐶^′_𝑛+𝛾^′_𝑛𝑟=(𝐿₂+1)𝐶^′_𝑛+𝛾^′_𝑛𝑟. (2.11) Substituting (2.11) into (2.9), we get ‖𝜎_𝑛−𝑥_𝑛‖≤(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛+(𝐿₁+1)𝛾^′_𝑛𝑟+𝐶_𝑛. (2.12) It follows from (1.7) and (2.12) that ‖𝑥_𝑛+1−𝑥_𝑛‖≤‖𝑃((1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛+𝛾_𝑛𝑢_𝑛)−𝑥_𝑛‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑥_𝑛‖+𝛾_𝑛‖𝑢_𝑛−𝑥_𝑛‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖+‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑥_𝑛‖+𝛾_𝑛𝑟≤𝐿₁‖𝜎_𝑛−𝑥_𝑛‖+𝐶_𝑛+𝛾_𝑛𝑟≤𝐿₁((𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛+(𝐿₁+1)𝛾^′_𝑛𝑟+𝐶_𝑛)+𝐶_𝑛+𝛾_𝑛𝑟=(𝐿₁+1)𝐶_𝑛+𝐿₁(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛+𝐿₁(𝐿₁+1)𝛾^′_𝑛𝑟+𝛾_𝑛𝑟. (2.13) Using (2.11) and (2.13), we obtain ‖𝜎_𝑛−1−𝑥_𝑛‖=‖(1−𝛽_𝑛−1)𝑦_𝑛−1+𝛽_𝑛−1𝑇₁(𝑃𝑇₁)^𝑛−2𝑦_𝑛−1−𝑥_𝑛‖≤𝛽_𝑛−1‖𝑇₁(𝑃𝑇₁)^𝑛−2𝑦_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛−1‖+𝛽_𝑛−1‖𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛−1−𝑥_𝑛−1‖+𝛽_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+(1−𝛽_𝑛−1)‖𝑦_𝑛−1−𝑥_𝑛‖≤𝐿₁‖𝑦_𝑛−1−𝑥_𝑛−1‖+𝐶_𝑛−1+‖𝑥_𝑛−𝑥_𝑛−1‖+‖𝑦_𝑛−1−𝑥_𝑛−1‖+‖𝑥_𝑛−𝑥_𝑛−1‖≤(𝐿₁+1)[(𝐿₂+1)𝐶^′_𝑛−1+𝛾^′_𝑛−1𝑟]+2(𝐿₁+1)𝐶_𝑛−1+𝐿₁(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛−1+𝐿₁(𝐿₁+1)𝛾^′_𝑛−1𝑟+𝛾_𝑛−1𝑟+𝐶_𝑛−1=(2𝐿₁+3)𝐶_𝑛−1+(2𝐿₁+1)(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛−1+(2𝐿₁+1)(𝐿₁+1)𝛾^′_𝑛−1𝑟+2𝛾_𝑛−1𝑟. (2.14) Combine (2.13) with (2.14) yields that ‖𝑥_𝑛−(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖=‖𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖≤‖(1−𝛼_𝑛−1−𝛾_𝑛−1)𝑥_𝑛−1+𝛼_𝑛−1𝑇₁(𝑃𝑇₁)^𝑛−2𝑃𝜎_𝑛−1+𝛾_𝑛−1𝑢_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖≤𝛼_𝑛−1‖𝑇₁(𝑃𝑇₁)^𝑛−2𝑃𝜎_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖+(1−𝛼_𝑛−1)‖𝑥_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖+𝛾_𝑛−1‖𝑢_𝑛−1−𝑥_𝑛−1‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−2𝑃𝜎_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖+‖𝑥_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛‖+𝛾_𝑛−1𝑟≤𝐿₁‖𝜎_𝑛−1−𝑥_𝑛‖+‖𝑥_𝑛−1−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛−1‖+‖𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−2𝑥_𝑛−1‖+𝛾_𝑛−1𝑟≤𝐿₁(2𝐿₁+3)𝐶_𝑛−1+(2𝐿₁+1)(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛−1+(2𝐿₁+1)(𝐿₁+1)𝛾^′_𝑛−1𝑟+2𝛾_𝑛−1𝑟+𝐶_𝑛−1+(𝐿₁+1)𝐶_𝑛−1+𝐿₁(𝐿₁+1)(𝐿₂+1)𝐶^′_𝑛−1+𝐿₁(𝐿₁+1)𝛾^′_𝑛−1𝑟+2𝛾_𝑛−1𝑟=2(𝐿₁+1)²𝐶_𝑛−1+2𝐿₁(𝐿₁+1)²(𝐿₂+1)𝐶^′_𝑛−1+2𝐿₁(𝐿₁+1)²𝛾^′_𝑛−1𝑟+2(𝐿₁+1)𝛾_𝑛−1𝑟, (2.15) from which it follows that ‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=‖𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛+𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑇₁𝑥_𝑛‖≤‖𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛‖+‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑇₁𝑥_𝑛‖≤𝐶_𝑛+𝐿₁‖(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑥_𝑛‖≤𝐶_𝑛+2𝐿₁(𝐿₁+1)²𝐶_𝑛−1+2𝐿²₁(𝐿₁+1)²(𝐿₂+1)𝐶^′_𝑛−1+2𝐿²₁(𝐿₁+1)²𝛾^′_𝑛−1𝑟+2𝐿₁(𝐿₁+1)𝛾_𝑛−1𝑟. (2.16) It follows from lim_𝑛→∞𝐶_𝑛=lim_𝑛→∞𝐶^′_𝑛=0 that lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=0. Similarly, we can show that lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0. This completes the proof.

Lemma. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself-mappings of 𝐸 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}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛, and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_𝑛≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). Starting from an arbitrary 𝑥₁∈𝐾, define the sequence {𝑥_𝑛} by the recursion (1.7). Then, lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0. (2.17)

Proof. Let 𝜎_𝑛=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛 and 𝛿_𝑛=(1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛. By Lemma 2.1, we see that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Assume that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑐. If 𝑐=0, then by the continuity of 𝑇₁ and 𝑇₂ the conclusion follows. Now, suppose 𝑐>0. Taking limsup on both sides in the inequalities (2.2), (2.3), and (2.4), we have limsup_𝑛→∞‖𝜎_𝑛−𝑝‖≤𝑐,limsup_𝑛→∞‖𝛿_𝑛−𝑝‖≤𝑐,limsup_𝑛→∞‖𝑦_𝑛−𝑝‖≤𝑐, (2.18) respectively. Next, we consider ‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛)‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑥_𝑛‖≤𝑘_𝑛‖𝜎_𝑛−𝑝‖+𝛾_𝑛𝑟. (2.19) Taking limsup on both sides in the above inequality and using (2.18), we get limsup_𝑛→∞‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛)‖≤𝑐. (2.20) Observe that ‖𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛)‖≤‖𝑥_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛−𝑝‖+𝛾_𝑛𝑟, (2.21) which implies that limsup_𝑛→∞‖𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛)‖≤𝑐. (2.22)limsup_𝑛→∞‖𝑥_𝑛+1−𝑝‖=𝑐 means that liminf_𝑛→∞‖𝛼_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))+(1−𝛼_𝑛)(𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))‖≥𝑐. (2.23) On the other hand, by using (2.23) and (2.5), we have ‖𝛼_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))+(1−𝛼_𝑛)(𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))‖≤𝛼_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑥_𝑛‖≤𝛼_𝑛𝑘_𝑛‖𝜎_𝑛−𝑝‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛‖𝑢_𝑛−𝑥_𝑛‖≤𝛼_𝑛𝑘_𝑛(𝑘_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝑘_𝑛𝛾^′_𝑛𝑟)+(1−𝛼_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑛𝑟≤𝑘²_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝑘²_𝑛𝛾^′_𝑛𝑟+𝛾_𝑛𝑟. (2.24) Therefore, we have limsup_𝑛→∞‖𝛼_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))+(1−𝛼_𝑛)(𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))‖≤𝑐. (2.25) Combining (2.23) with (2.25), we obtain lim_𝑛→∞‖𝛼_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))+(1−𝛼_𝑛)(𝑥_𝑛−𝑝+𝛾_𝑛(𝑢_𝑛−𝑥_𝑛))‖=𝑐. (2.26) Hence, applying Lemma 1.3, we find lim_𝑛→∞‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑥_𝑛‖=0. (2.27) Note that ‖𝑥_𝑛−𝑝‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑝‖+‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑃𝜎_𝑛−𝑥_𝑛‖≤𝑘_𝑛‖𝜎_𝑛−𝑝‖ (2.28) which yields that 𝑐≤liminf_𝑛→∞‖𝜎_𝑛−𝑝‖≤limsup_𝑛→∞‖𝜎_𝑛−𝑝‖≤𝑐. (2.29) That is, lim_𝑛→∞‖𝜎_𝑛−𝑝‖=𝑐. This implies that liminf_𝑛→∞‖𝛽_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝)+(1−𝛽_𝑛)(𝑦_𝑛−𝑝)‖≥𝑐. (2.30) Similarly, we have ‖𝛽_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝)+(1−𝛽_𝑛)(𝑦_𝑛−𝑝)‖≤𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖+(1−𝛽_𝑛)‖(𝑦_𝑛−𝑝)‖≤𝑘_𝑛‖𝑦_𝑛−𝑝‖,limsup_𝑛→∞‖𝛽_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝)+(1−𝛽_𝑛)(𝑦_𝑛−𝑝)‖≤𝑐. (2.31)(2.32) Combining (2.30) with (2.32), we obtain lim_𝑛→∞‖𝛽_𝑛(𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝)+(1−𝛽_𝑛)(𝑦_𝑛−𝑝)‖=𝑐. (2.33) On the other hand, we have ‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−p‖≤𝑘_𝑛‖𝑦_𝑛−𝑝‖,limsup_𝑛→∞‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖≤𝑐. (2.34)(2.35) Hence, using (2.32), (2.33), (2.35), and Lemma 1.3, we find lim_𝑛→∞‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑦_𝑛‖=0. (2.36) Note that from (2.36), we have ‖𝜎_𝑛−𝑝‖=‖(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖≤(1−𝛽_𝑛)‖𝑦_𝑛−𝑝‖+𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑝‖≤(1−𝛽_𝑛)‖𝑦_𝑛−𝑝‖+𝛽_𝑛‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑦_𝑛‖+𝛽_𝑛‖𝑦_𝑛−𝑝‖=‖𝑦_𝑛−𝑝‖ (2.37) which yields that 𝑐≤liminf_𝑛→∞‖𝑦_𝑛−𝑝‖≤limsup_𝑛→∞‖𝑦_𝑛−𝑝‖≤𝑐. (2.38) That is, lim_𝑛→∞‖𝑦_𝑛−𝑝‖=𝑐.
Again, lim_𝑛→∞‖𝑦_𝑛−𝑝‖=𝑐 means that
liminf_𝑛→∞‖𝛼^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))+(1−𝛼^′_𝑛)(𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))‖≥𝑐. (2.39) By using (2.39) and (2.3), we obtain ‖𝛼^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))+(1−𝛼^′_𝑛)(𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))‖≤𝛼^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝‖+(1−𝛼^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛‖(𝑣_𝑛−𝑥_𝑛)‖≤𝛼^′_𝑛𝑙_𝑛‖𝛿_𝑛−𝑝‖+(1−𝛼^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛‖(𝑣_𝑛−𝑥_𝑛)‖≤𝛼^′_𝑛𝑙²_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼^′_𝑛)‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛𝑟≤𝑙²_𝑛‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛𝑟. (2.40) Therefore, we have limsup_𝑛→∞‖𝛼^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))+(1−𝛼^′_𝑛)(𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))‖≤𝑐. (2.41) Combining (2.39) with (2.41), we obtain lim_𝑛→∞‖𝛼^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))+(1−𝛼^′_𝑛)(𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛))‖=𝑐. (2.42) On the other hand, we have ‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛)‖≤‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝‖+𝛾^′_𝑛‖𝑣_𝑛−𝑥_𝑛‖≤𝑙_𝑛‖𝛿_𝑛−𝑝‖+𝛾^′_𝑛𝑟 (2.43) which implies that limsup_𝑛→∞‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛)‖≤𝑐. (2.44) Notice that ‖𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛)‖≤‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛‖𝑣_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛−𝑝‖+𝛾^′_𝑛𝑟, (2.45) which implies that limsup_𝑛→∞‖𝑥_𝑛−𝑝+𝛾^′_𝑛(𝑣_𝑛−𝑥_𝑛)‖≤𝑐. (2.46) Using (2.42), (2.44), (2.46), and Lemma 1.3, we find lim_𝑛→∞‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑥_𝑛‖=0. (2.47) Observe that ‖𝑥_𝑛−𝑝‖≤‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑥_𝑛‖+‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑝‖≤𝑙_𝑛‖𝛿_𝑛−𝑝‖ (2.48) which yields that 𝑐≤liminf_𝑛→∞‖𝛿_𝑛−𝑝‖≤limsup_𝑛→∞‖𝛿_𝑛−𝑝‖≤𝑐. (2.49) That is, lim_𝑛→∞‖𝛿_𝑛−𝑝‖=𝑐. This implies that liminf_𝑛→∞‖𝛽^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝)+(1−𝛽^′_𝑛)(𝑥_𝑛−𝑝)‖≥𝑐. (2.50) Similarly, we have ‖𝛽^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝)+(1−𝛽^′_𝑛)(𝑥_𝑛−𝑝)‖≤𝛽^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝‖+(1−𝛽^′_𝑛)‖𝑥_𝑛−𝑝‖≤𝑙_𝑛‖𝑥_𝑛−𝑝‖,limsup_𝑛→∞‖𝛽^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝)+(1−𝛽^′_𝑛)(𝑥_𝑛−𝑝)‖≤𝑐. (2.51)(2.52) Combining (2.50) with (2.52), we obtain lim_𝑛→∞‖𝛽^′_𝑛(𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝)+(1−𝛽^′_𝑛)(𝑥_𝑛−𝑝)‖=𝑐. (2.53) On the other hand, we have ‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝‖≤𝑙_𝑛‖𝑥_𝑛−𝑝‖,limsup_𝑛→∞‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑝‖≤𝑐,limsup_𝑛→∞‖𝑥_𝑛−𝑝‖≤𝑐. (2.54)(2.55) Hence, using (2.53), (2.54), (2.55), and Lemma 1.3, we find lim_𝑛→∞‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑥_𝑛−𝑥_𝑛‖=0. (2.56) In addition, from 𝑦_𝑛=𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛) and (2.47), we have ‖𝑦_𝑛−𝑥_𝑛‖=‖𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛+𝛾^′_𝑛𝑣_𝑛)−𝑥_𝑛‖≤𝛼^′_𝑛‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑥_𝑛‖+𝛾^′_𝑛‖𝑣_𝑛−𝑥_𝑛‖≤‖𝑇₂(𝑃𝑇₂)^𝑛−1𝑃𝛿_𝑛−𝑥_𝑛‖+𝛾^′_𝑛𝑟.→0,(as𝑛→∞). (2.57) Hence, from (2.36) and (2.57), we find ‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑥_𝑛‖≤‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛‖+‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝑥_𝑛‖≤𝑘_𝑛‖𝑦_𝑛−𝑥_𝑛‖+‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑦_𝑛−𝑦_𝑛‖+‖𝑦_𝑛−𝑥_𝑛‖→0,(as𝑛→∞). (2.58) That is, lim_𝑛→∞‖𝑇₁(𝑃𝑇₁)^𝑛−1𝑥_𝑛−𝑥_𝑛‖=0. (2.59) Since 𝑇₁ and 𝑇₂ are uniformly 𝐿₁-Lipschitzian and uniformly 𝐿₂-Lipschitzian, respectively, for some 𝐿₁,𝐿₂≥0, it follows from (2.56), (2.59), and Lemma 2.2 that lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0. (2.60) This completes the proof.

Theorem. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself -mappings of 𝐸 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}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛, and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_𝑛≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). If one of 𝑇₁𝑎𝑛𝑑𝑇₂ is completely continuous, then the sequence {𝑥_𝑛} defined by the recursion (1.7) converges strongly to some common fixed point of 𝑇₁and𝑇₂.

Proof. By Lemma 2.1, {𝑥_𝑛} is bounded. In addition, by Lemma 2.3;lim_𝑛→∞‖𝑥_𝑛−𝑇₁𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑇₂𝑥_𝑛‖=0; then {𝑇₁𝑥_𝑛} and {𝑇₂𝑥_𝑛} are also bounded. If 𝑇₁ is completely continuous, there exists subsequence {𝑇₁𝑥_𝑛𝑗} of {𝑇₁𝑥_𝑛} such that 𝑇₁𝑥_𝑛𝑗→𝑝 as 𝑗→∞. It follows from Lemma 2.3 that lim_𝑗→∞‖𝑥_𝑛𝑗−𝑇₁𝑥_𝑛𝑗‖=lim_𝑗→∞‖𝑥_𝑛𝑗−𝑇₂𝑥_𝑛𝑗‖=0. So by the continuity of 𝑇₁ and Lemma 1.4, we have lim_𝑗→∞‖𝑥_𝑛𝑗−𝑝‖=0 and 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂). Furthermore, by Lemma 2.1, we get that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Thus lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0. The proof is completed.

The following result gives a strong convergence theorem for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying condition (𝐴^′).

Theorem. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself-mappings of 𝐸 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}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛, and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_𝑛≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). Suppose that 𝑇₁and𝑇₂ satisfy condition (𝐴^′). Then, the sequence {𝑥_𝑛} defined by the recursion (1.7) converges strongly to some common fixed point of 𝑇₁𝑎𝑛𝑑𝑇₂.

Proof. By Lemma 2.1, we readily see that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ and so, lim_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇₁)∩𝐹(𝑇₂)) exists for all 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂). Also, by Lemma 2.3, lim_𝑛→∞‖𝑇₁𝑥_𝑛−𝑥_𝑛‖=lim_𝑛→∞‖𝑇₂𝑥_𝑛−𝑥_𝑛‖=0. It follows from condition (𝐴^′) that lim_𝑛→∞𝑓(𝑑(𝑥_𝑛,𝐹(𝑇₁)∩𝐹(𝑇₂)))≤lim_𝑛→∞(12(‖𝑥_𝑛−𝑇₁𝑥_𝑛‖+‖𝑥_𝑛−𝑇₂𝑥_𝑛‖))=0. (2.61) That is, lim_𝑛→∞𝑓(𝑑(𝑥_𝑛,𝐹(𝑇₁)∩𝐹(𝑇₂)))=0. (2.62) Since 𝑓∶[0,∞)→[0,∞) is a nondecreasing function satisfying 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞), therefore, we have lim_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇₁)∩𝐹(𝑇₂))=0. (2.63) Now we can take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} and sequence {𝑦_𝑗}⊂𝐹 such that ‖𝑥_𝑛𝑗−𝑦_𝑗‖<2^−𝑗 for all integers 𝑗≥1. Using the proof method of Tan and Xu [5], we have ‖𝑥_𝑛𝑗+1−𝑦_𝑗‖≤‖𝑥_𝑛𝑗−𝑦_𝑗‖<2^−𝑗, (2.64) and hence ‖𝑦_𝑗+1−𝑦_𝑗‖≤‖𝑦_𝑗+1−𝑥_𝑛𝑗+1‖+‖𝑥_𝑛𝑗+1−𝑦_𝑗‖≤2^−(𝑗+1)+2^−𝑗<2^−𝑗+1. (2.65) We get that {𝑦_𝑗} is a Cauchy sequence in 𝐹 and so it converges. Let 𝑦_𝑗→𝑦. Since 𝐹 is closed, therefore, 𝑦∈𝐹 and then 𝑥_𝑛𝑗→𝑦. As lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, 𝑥_𝑛→𝑦∈𝐹(𝑇₁)∩𝐹(𝑇₂). Thereby completing the proof.

Remark. If 𝛾_𝑛=𝛾^′_𝑛=𝛽_𝑛=𝛽^′_𝑛=0, then the iterative scheme (1.7) reduces to the iterative scheme (1.4) of [9]. Moreover, the condition (𝐴^′) is weaker than both the compactness of 𝐾 and the semicompactness of the asymptotically nonexpansive nonself-mappings 𝑇₁,𝑇₂∶𝐾→𝐸. Also, the condition 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛≤𝑏<1 for all 𝑛≥1 is weaker than the condition 0<𝜀≤𝛼_n,𝛼^′_𝑛,≤1−𝜀, for all 𝑛≥1 and some 𝜀∈[0,1). Hence, Theorems 2.4 and 2.5 generalize Theorems 3.3 and 3.4 in [9], respectively.

In the next result, we prove the weak convergence of the iterative scheme (1.7) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive nonself-mappings of 𝐸 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}, {𝛽_𝑛},{𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛,and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_𝑛≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). Suppose that 𝑇₁and𝑇₂ satisfy Opial's condition. Then, the sequence {𝑥_𝑛} defined by the recursion (1.7) converges weakly to some common fixed point of 𝑇and𝑇₂.

Proof. Let 𝑝∈F(𝑇₁)∩𝐹(𝑇₂). By Lemma 2.1, we see that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists and {𝑥_𝑛} bounded. Now we prove that {𝑥_𝑛} has a unique weak subsequential limit in 𝐹(𝑇₁)∩𝐹(𝑇₂). Firstly, suppose that subsequences {𝑥_𝑛𝑘} and {𝑥_𝑛𝑗} of {𝑥_𝑛} converge weakly to 𝑝₁ and 𝑝₂, respectively. By Lemma 2.3, we have lim_𝑛→∞‖𝑥_𝑛𝑘−𝑇₁𝑥_𝑛𝑘‖=0. And Lemma 1.4 guarantees that (𝐼−𝑇₁)𝑝₁=0, that is., 𝑇₁𝑝₁=𝑝₁. Similarly, 𝑇₂𝑝₁=𝑝₁. Again in the same way, we can prove that 𝑝₂∈𝐹(𝑇₁)∩𝐹(𝑇₂).
Secondly, assume 𝑝₁≠𝑝₂, then by Opial’s condition, we have
lim_𝑛→∞‖𝑥_𝑛−𝑝₁‖=lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝₁‖<lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝₂‖=lim_𝑗→∞‖𝑥_𝑛𝑗−𝑝₂‖<lim_𝑘→∞‖𝑥_𝑛𝑘−𝑝₁‖=lim_𝑛→∞‖𝑥_𝑛−𝑝₁‖, (2.66) which is a contradiction, hence, 𝑝₁=𝑝₂. Then, {𝑥_𝑛} converges weakly to a common fixed point of 𝑇₁ and 𝑇₂. This completes the proof.

Remark. The above Theorem generalizes Theorem 3.5 of Wang [9].

3. Case of Two Nonself-Nonexpansive Mappings

Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonexpansive nonself-mappings. Then, the iterative scheme (1.7) is written as follows:

𝑥_𝑛+1=𝑃((1−𝛼_𝑛−𝛾_𝑛)𝑥_𝑛+𝛼_𝑛𝑇₁𝑃(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑇₁𝑦_𝑛+𝛾_𝑛𝑢_𝑛𝑦),_𝑛=𝑃((1−𝛼^′_𝑛−𝛾^′_𝑛)𝑥_𝑛+𝛼^′_𝑛𝑇₂𝑃((1−𝛽^′_𝑛)𝑥_𝑛+𝛽^′_𝑛𝑇₂𝑥_𝑛)+𝛾^′_𝑛𝑣_𝑛),𝑥₁∈𝐾,𝑛≥1. (3.1)

Nothing prevents one from proving the results of the previous section for nonexpansive nonself-mappings case. Thus, one can easily prove the following.

Theorem. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonexpansive nonself-mappings of 𝐸 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}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛, and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_n≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). Suppose that 𝑇₁and𝑇₂ satisfy condition (𝐴^′). Then, the sequence {𝑥_𝑛} defined by the recursion (3.1) converges strongly to some common fixed point of 𝑇₁and𝑇₂.

Theorem. Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonexpansive nonself-mappings of 𝐸 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}, {𝛽_𝑛},{𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are appropriate sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛, and {𝑢_𝑛}, {𝑣_n} are bounded sequences in 𝐾 such that ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞. Moreover, 0<𝑎≤𝛼_𝑛,𝛼^′_𝑛,𝛽_𝑛,𝛽^′_𝑛≤𝑏<1 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1). Suppose that 𝑇₁and𝑇₂ satisfy Opial's condition. Then, the sequence {𝑥_𝑛} defined by the recursion (3.1) converges weakly to some common fixed point of 𝑇and𝑇₂.

Remark. If 𝑇₁=𝑇₂=𝑇 and 𝑇 is a nonexpansive nonself-mapping, then the iterative scheme (3.1) reduces to the iterative scheme (1.6) of Thianwan [11]. Then, Theorems 3.1-3.2 generalize Theorems 2.4 and 2.6 in [11], respectively.

Acknowledgment

The authors would like to thank the referees for their helpful comments.