Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

Abstract

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

1. Introduction

Let 𝐶 be a subset of real Banach space 𝑋. Let 𝑇 be a self-mapping of 𝐶 and let 𝐹(𝑇) denote the fixed points set of 𝑇, that is, 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}. Recall that a mapping 𝑇 is said to be asymptotically nonexpansive on 𝐶 if there exists a sequence {𝑏_𝑛} in [0,∞) with lim_𝑛→∞𝑏_𝑛=0 such that for each 𝑥,𝑦∈𝐶,

‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝑏_𝑛)‖𝑥−𝑦‖,∀𝑛≥1.(1.1)

If 𝑏_𝑛=0 for all 𝑛≥1, then 𝑇 is known as a nonexpansive mapping. 𝑇 is called generalized asymptotically quasi-nonexpansive [1] if there exist sequences {𝑏_𝑛},{𝑐_𝑛} in [0,∞) with lim_𝑛→∞𝑏_𝑛=0=lim_𝑛→∞𝑐_𝑛 such that

‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑏_𝑛)‖𝑥−𝑝‖+𝑐_𝑛,∀𝑛≥1,(1.2) for all 𝑥∈𝐶 and all 𝑝∈𝐹(𝑇). If 𝑐_𝑛=0 for all 𝑛≥1, then 𝑇 is known as an asymptotically quasi-nonexpansive mapping. 𝑇  is called asymptotically nonexpansive mapping in the intermediate sense [2] provided that 𝑇 is uniformly continuous and

limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)≤0.(1.3)T is said to be (𝐿−𝛾)  uniform Lipschitz [3] if there are constants 𝐿>0 and 𝛾>0 such that

‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖^𝛾,∀𝑛≥1,(1.4) for all 𝑥,𝑦∈𝐶. A mapping 𝑇 is called semicompact if any bounded sequence {𝑥_𝑛} in 𝐶 with lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑥_𝑛𝑖} converges strongly to some 𝑥^∗ in 𝐶.

Remark 1.1. Let 𝑇 be asymptotically nonexpansive mapping in the intermediate sense. Put 𝐺_𝑛=sup_{𝑥,𝑦∈𝐶}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)∨0, ∀𝑛≥1.
If 𝐹(𝑇)≠∅, we obtain that ‖𝑇^𝑛𝑥−𝑝‖≤‖𝑥−𝑝‖+𝐺_𝑛 for all 𝑥∈𝐶 and all 𝑝∈𝐹(𝑇). Since lim_𝑛→∞𝐺_𝑛=0, therefore 𝑇 is a generalized asymptotically quasi-nonexpansive mapping.

Recall that a mapping 𝑇∶𝐶→𝐶 with 𝐹(𝑇)≠∅ is said to satisfy condition (I) [4] if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥−𝑇𝑥‖≥𝑓(𝑑(𝑥,𝐹(𝑇))) for all 𝑥∈𝐶, where 𝑑(𝑥,𝐹(𝑇))=inf{‖𝑥−𝑝‖∶𝑝∈𝐹(𝑇)}.

Fixed-point iteration processes for asymptotically quasi-nonexpansive mapping in Banach spaces including Mann and Ishikawa iterations processes have been studied extensively by many authors; see [3, 5–11]. Many of them are used widely to study the approximate solutions of the certain problems. In 1974, Senter and Dotson [4] studied the convergence of the Mann iteration scheme defined by 𝑥₁∈𝐶,

𝑥_𝑛+1=𝛼_𝑛𝑇𝑥_𝑛+(1−𝛼_𝑛)_𝑥𝑛,∀𝑛≥1,(1.5) in a uniformly convex Banach space, where {𝛼_𝑛} is a sequence satisfying 0<𝑎≤𝛼_𝑛≤𝑏<1forall𝑛≥1 and 𝑇 is a nonexpansive (or a quasi-nonexpansive) mapping. They established a relation between condition (I) and demicompactness.

Recall that a mapping 𝑇∶𝐶→𝐶 is demicompact if for every bounded sequence {𝑥_𝑛} in 𝐶 such that {𝑥_𝑛−𝑇𝑥_𝑛} converges, there exists a subsequence say {𝑥_𝑛𝑗} of {𝑥_𝑛} that converges strongly to some 𝑦 in 𝐶. Every compact and semicompact mapping is demicompact. They actually showed that condition (I) is weaker than demicompactness for a nonexpansive mapping defined on bounded set.

Xu and Noor [12], in 2002, introduced a three-step iterative scheme as follows:

𝑧_𝑛=𝑎_𝑛𝑇^𝑛𝑥_𝑛+(1−𝑎_𝑛)_𝑥𝑛,𝑦_𝑛=𝑏_𝑛𝑇^𝑛𝑧_𝑛+(1−𝑏_𝑛)_𝑥𝑛,𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑦_𝑛+(1−𝛼_𝑛)_𝑥𝑛,𝑛≥1,(1.6) where {𝑎_𝑛},{𝑏_𝑛},{𝛼_𝑛} are appropriate sequences in [0,1]. The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied (e.g., see Khan et al. [6], Plubtieng and Wangkeeree [7], Fukhar-ud-din and Khan [5], Petrot [13], and Suantai [14]). It has been shown in [15] that three-step method performs better than two-step and one-step methods for solving variational inequalities.

In 2001, Khan and Takahashi [16] have approximated common fixed points of two asymptotically nonexpansive mappings by the modified Ishikawa iteration. Jeong and Kim [17] have approximated common fixed points of two asymptotically nonexpansive mappings. Plubtieng et al. [18], in 2006, modified Noor iterations with errors and have approximated common fixed points of three asymptotically nonexpansive mappings. Shahzad and Udomene [10] established convergence theorems for the modified Ishikawa iteration process of to asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. Plubtieng and Wangkeeree [7], in 2006, established strong convergence theorems of the modified multistep Noor iterations with errors for an asymptotically quasi-nonexpansive mapping and asymptotically nonexpansive mapping in the intermediate sense.

Very recently, Khan et al. [6], in 2008, established convergence theorems for the modified multistep Noor iterations process of finite family of asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. For rerated results with errors terms, we refer to [5–7, 17–21]. Inspired and motivated by these facts, we introduce a new iteration process for a finite family of {𝑇_𝑖∶𝑖=1,2,…,𝑘} of generalized asymptotically quasi-nonexpansive mappings as follows.

Let 𝑇_𝑖∶𝐶→𝐶(𝑖=1,2,…,𝑘) be mappings and 𝐹∶=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖). For a given 𝑥₁∈𝐶, and a fixed 𝑘∈ℕ (ℕ denote the set of all positive integers), compute the iterative sequences {𝑥_𝑛} and {𝑦_𝑖𝑛} by

𝑥_𝑛+1=𝑦_𝑘𝑛=𝛼_𝑘𝑛𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}+𝛽_𝑘𝑛𝑥_𝑛+𝛾_𝑘𝑛𝑢_𝑘𝑛,𝑦_{(𝑘−1)𝑛}=𝛼_{(𝑘−1)𝑛}𝑇^𝑛_𝑘−1𝑦_{(𝑘−2)𝑛}+𝛽_{(𝑘−1)𝑛}𝑥_𝑛+𝛾_{(𝑘−1)𝑛}𝑢_{(𝑘−1)𝑛},⋮𝑦_3𝑛=𝛼_3𝑛𝑇^𝑛₃𝑦_2𝑛+𝛽_3𝑛𝑥_𝑛+𝛾_3𝑛𝑢_3𝑛,𝑦_2𝑛=𝛼_2𝑛𝑇^𝑛₂𝑦_1𝑛+𝛽_2𝑛𝑥_𝑛+𝛾_2𝑛𝑢_2𝑛,𝑦_1𝑛=𝛼_1𝑛𝑇^𝑛₁𝑦_0𝑛+𝛽_1𝑛𝑥_𝑛+𝛾_1𝑛𝑢_1𝑛,(1.7) where 𝑦_0𝑛=𝑥_𝑛 and {𝑢_1𝑛},{𝑢_2𝑛},…,{𝑢_𝑘𝑛} are bounded sequences in 𝐶 with {𝛼_𝑖𝑛},{𝛽_𝑖𝑛}, and {𝛾_𝑖𝑛} are appropriate real sequences in [0,1] such that 𝛼_𝑖𝑛+𝛽_𝑖𝑛+𝛾_𝑖𝑛=1 for all 𝑖=1,2,…,𝑘 and all 𝑛. Our iteration includes and extends the Mann iteration (1.5), three-step iteration by Xu and Noor (1.6), the multistep Noor iterations with errors by Plubtieng and Wangkeeree [7], and the iteration defined by Khan et al. [6] simultaneously.

The purpose of this paper is to establish several strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings when one mapping 𝑇_𝑖 satisfies a condition which is weaker than demicompactness and we also weak convergence theorem for a finite family of generalized asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space satisfying Opial's property. Our results generalize and improve the corresponding ones announced by Khan et al. [6], Fukhar-ud-din and Khan [5], and many others.

2. Preliminaries

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

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in X is said to be 𝑑𝑒𝑚𝑖𝑐𝑙𝑜𝑠𝑒𝑑 at 0 if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that {𝑥_𝑛} converges weakly to 𝑥∈𝐷(𝑇) and {𝑇𝑥_𝑛} converging strongly to 0, we have 𝑇𝑥=0.

A Banach space 𝑋 is said to satisfy Opial's property if for each 𝑥 in 𝑋 and each sequence {𝑥_𝑛} weakly convergent to 𝑥, the following condition holds for 𝑥≠𝑦:

liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖.(2.1) It is well known that all Hilbert spaces and 𝑙_𝑝(1<𝑝<∞) spaces have Opial's property while 𝐿_𝑝 spaces (𝑝≠2) have not. A family {𝑇_𝑖∶𝑖=1,2,…,𝑘} of self-mappings of 𝐶 with 𝐹∶=⋂^𝑘_𝑖=1𝐹(𝑇_𝑖)≠∅ is said to satisfy the following conditions.

(1)Condition (𝐴) [22]. If there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that 1/𝑘∑^𝑘_𝑖=1‖𝑥−𝑇_𝑖𝑥‖≥𝑓(𝑑(𝑥,𝐹)) for all 𝑥∈𝐶, where 𝑑(𝑥,𝐹)=inf{‖𝑥−𝑝‖∶𝑝∈𝐹}.(2)Condition (𝐵) [22]. If there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that max_{1≤𝑖≤𝑘}{‖𝑥−𝑇_𝑖𝑥‖}≥𝑓(𝑑(𝑥,𝐹)) for all 𝑥∈𝐶.(3)Condition (𝐶) [22]. If there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥−𝑇_𝑙𝑥‖≥𝑓(𝑑(𝑥,𝐹)) for all 𝑥∈𝐶 and for at least one 𝑇_𝑙,𝑙=1,2,…,𝑘.

Note that (𝐵) and (𝐶) are equivalent, condition (𝐵) reduces to condition (I) when all but one of 𝑇_𝑖's are identities, and in addition, it also condition (𝐴).

It is well known that every continuous and demicompact mapping must satisfy condition (I) (see [4]). Since every completely continuous 𝑇∶𝐶→𝐶 is continuous and demicompact so that it satisfies condition (I). Thus we will use condition (𝐶) instead of the demicompactness and complete continuity of a family {𝑇_𝑖∶𝑖=1,2,…,𝑘}.

Lemma 2.1 (see [8, Lemma 1]). Let {𝑎_𝑛},{𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤(1+𝛿_𝑛)_𝑎𝑛+𝑏_𝑛,∀𝑛=1,2,….(2.2) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then(i)lim_𝑛→∞𝑎_𝑛 exists;(ii)lim_𝑛→∞𝑎_𝑛=0 whenever liminf_𝑛→∞𝑎_𝑛=0.

Lemma 2.2 (see [7, Lemma 3.1]). Let 𝑋 be a uniformly convex Banach space, {𝑥_𝑛}, {𝑦_𝑛}⊂𝑋, real numbers 𝑎≥0,𝛼,𝛽∈(0,1), and let {𝛼_𝑛} be a real sequence number which satisfies(i)0<𝛼≤𝛼_𝑛≤𝛽<1,𝑓𝑜𝑟𝑎𝑙𝑙𝑛≥𝑛₀ and for some 𝑛₀∈ℕ;(ii)limsup_𝑛→∞‖𝑥_𝑛‖≤𝑎 and limsup_𝑛→∞‖𝑦_𝑛‖≤𝑎;(iii)lim_𝑛→∞‖𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑦_𝑛‖=𝑎. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.3 (see [14, Lemma 2.7]). Let 𝑋 be a Banach space which satisfies Opial's property and let {𝑥_𝑛} be a sequence in 𝑋. Let 𝑢,𝑣∈𝑋 be such that lim_𝑛→∞‖𝑥_𝑛−𝑢‖ and lim_𝑛→∞‖𝑥_𝑛−𝑣‖ exist. If {𝑥_𝑛𝑘} and {𝑥_𝑚𝑘} are subsequences of {𝑥_𝑛} which converge weakly to 𝑢 and 𝑣, respectively, then 𝑢=𝑣.

3. Convergence Theorems in Banach Spaces

Our first result is the strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings in a Banach space. In order to prove our main results, the following lemma is needed.

Lemma 3.1. Let 𝑋 be a Banach space and 𝐶 a nonempty closed and convex subset of 𝑋, and {𝑇_𝑖∶𝑖=1,2,…,𝑘} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for each 𝑖=1,2,…,𝑘. For a given 𝑥₁∈𝐶, let the sequences {𝑥_𝑛} and {𝑦_𝑖𝑛} be defined by (1.7). Then(a)there exist sequences {𝑣_𝑛} and {𝑒_𝑖𝑛} in [0,∞) such that ∑^∞_𝑛=1𝑣_𝑛<∞, ∑^∞_𝑛=1𝑒_𝑖𝑛<∞, and ‖𝑦_𝑖𝑛−𝑝‖≤(1+𝑣_𝑛)^𝑖‖𝑥_𝑛−𝑝‖+𝑒_𝑖𝑛, for all 𝑖=1,2,…,𝑘 and all 𝑝∈𝐹;(b)lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹;(c)there exist constant 𝑀>0 and {𝑠_𝑖} in [0,∞) such that ∑^∞_𝑖=1𝑠_𝑖<∞ and ‖𝑥_𝑛+𝑚−𝑝‖≤𝑀‖𝑥_𝑛−𝑝‖+∑^∞_𝑖=𝑛𝑠_𝑖 for all 𝑝∈𝐹 and 𝑛,𝑚∈ℕ.

Proof. (a) Let 𝑝∈𝐹, 𝑣_𝑛=max_{1≤𝑖≤𝑘}{𝑏_𝑖𝑛} and 𝑑_𝑛=max_{1≤𝑖≤𝑘}{𝑐_𝑖𝑛} for all 𝑛.
Since ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞, for all 𝑖=1,2,…,𝑘, therefore ∑^∞_𝑛=1𝑣_𝑛<∞ and ∑^∞_𝑛=1𝑑_𝑛<∞. For each 𝑛≥1, we note that ‖𝑦_1𝑛−𝑝‖=‖𝛼_1𝑛𝑇^𝑛₁𝑦_0𝑛+𝛽_1𝑛𝑥_𝑛+𝛾_1𝑛𝑢_1𝑛−𝑝‖≤𝛼_1𝑛‖𝑇^𝑛₁𝑥_𝑛−𝑝‖+𝛽_1𝑛‖𝑥_𝑛−𝑝‖+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤𝛼_1𝑛(1+𝑏_1𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑐_1𝑛+𝛽_1𝑛‖𝑥_𝑛−𝑝‖+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤𝛼_1𝑛(1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑑_𝑛+𝛽_1𝑛(1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤(1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝑒_1𝑛,(3.1) where 𝑒_1𝑛=𝛼_1𝑛𝑑_𝑛+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖. Since {𝑢_1𝑛} is bounded, ∑^∞_𝑛=1𝛾_1𝑛<∞ and ∑^∞_𝑛=1𝑑_𝑛<∞, we obtain that ∑^∞_𝑛=1𝑒_1𝑛<∞. It follows from (3.1) that ‖𝑦_2𝑛−𝑝‖≤𝛼_2𝑛‖𝑇^𝑛₂𝑦_1𝑛−𝑝‖+𝛽_2𝑛‖𝑥_𝑛−𝑝‖+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤𝛼_2𝑛(1+𝑣_𝑛)‖𝑦_1𝑛−𝑝‖+𝛼_2𝑛𝑑_𝑛+𝛽_2𝑛‖𝑥_𝑛−𝑝‖+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤𝛼_2𝑛(1+𝑣_𝑛)((1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝑒_1𝑛)+𝛼_2𝑛𝑑_𝑛+𝛽_2𝑛(1+𝑣_𝑛)²‖𝑥_𝑛−𝑝‖+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖=(𝛼_2𝑛+𝛽_2𝑛)(1+𝑣_𝑛)²‖𝑥_𝑛−𝑝‖+𝛼_2𝑛(1+𝑣_𝑛)𝑒_1𝑛+𝛼_2𝑛𝑑_𝑛+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤(1+𝑣_𝑛)²‖𝑥_𝑛−𝑝‖+𝑒_2𝑛,(3.2) where 𝑒_2𝑛=𝛼_2𝑛(1+𝑣_𝑛)𝑒_1𝑛+𝛼_2𝑛𝑑_𝑛+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖. Since {𝑢_2𝑛}, {𝑣_𝑛} are bounded, ∑^∞_𝑛=1𝑒_1𝑛<∞, ∑^∞_𝑛=1𝑑_𝑛<∞, and ∑^∞_𝑛=1𝛾_2𝑛<∞, it follows that ∑^∞_𝑛=1𝑒_2𝑛<∞. Moreover, we see that ‖𝑦_3𝑛−𝑝‖≤𝛼_3𝑛‖𝑇^𝑛₃𝑦_2𝑛−𝑝‖+𝛽_3𝑛‖𝑥_𝑛−𝑝‖+𝛾_3𝑛‖𝑢_3𝑛−𝑝‖≤𝛼_3𝑛(1+𝑣_𝑛)‖𝑦_2𝑛−𝑝‖+𝛼_3𝑛𝑑_𝑛+𝛽_3𝑛‖𝑥_𝑛−𝑝‖+𝛾_3𝑛‖𝑢_3𝑛−𝑝‖≤𝛼_3𝑛(1+𝑣_𝑛)((1+𝑣_𝑛)²‖𝑥_𝑛−𝑝‖+𝑒_2𝑛)+𝛼_3𝑛𝑑_𝑛+𝛽_3𝑛(1+𝑣_𝑛)³‖𝑥_𝑛−𝑝‖+𝛾_3𝑛‖𝑢_3𝑛−𝑝‖=(𝛼_3𝑛+𝛽_3𝑛)(1+𝑣_𝑛)³‖𝑥_𝑛−𝑝‖+𝛼_3𝑛(1+𝑣_𝑛)𝑒_2𝑛+𝛼_3𝑛𝑑_𝑛+𝛾_3𝑛‖𝑢_3𝑛−𝑝‖≤(1+𝑣_𝑛)³‖𝑥_𝑛−𝑝‖+𝑒_3𝑛,(3.3) where 𝑒_3𝑛=𝛼_3𝑛(1+𝑣_𝑛)𝑒_2𝑛+𝛼_3𝑛𝑑_𝑛+𝛾_3𝑛‖𝑢_3𝑛−𝑝‖. Since {𝑢_3𝑛}, {𝑣_𝑛} are bounded, ∑^∞_𝑛=1𝑒_2𝑛<∞, ∑^∞_𝑛=1𝑑_𝑛<∞, and ∑^∞_𝑛=1𝛾_3𝑛<∞, it follows that ∑^∞_𝑛=1𝑒_3𝑛<∞. By continuing the above method, there are nonnegative real sequences {𝑒_𝑖𝑛} in [0,∞) such that ∑^∞_𝑛=1𝑒_𝑖𝑛<∞ and ‖𝑦_𝑖𝑛−𝑝‖≤(1+𝑣_𝑛)^𝑖‖𝑥_𝑛−𝑝‖+𝑒_𝑖𝑛∀𝑖=1,2,…,𝑘.(3.4) This completes the proof of (a).
(b) From part (a), for the case 𝑖=𝑘, we have ‖𝑥_𝑛+1−𝑝‖≤(1+𝑣_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑒_𝑘𝑛,∀𝑛,𝑝∈𝐹.(3.5) It follows from Lemma 2.1(i) that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, for all 𝑝∈𝐹.
(c) If 𝑡≥0, then 1+𝑡≤𝑒^𝑡 and so, (1+𝑡)^𝑘≤𝑒^𝑘𝑡, for 𝑘=1,2,…. Thus, from (3.5), it follows that ‖𝑥_𝑛+𝑚−𝑝‖≤(1+𝑣_{𝑛+𝑚−1})^𝑘‖𝑥_{𝑛+𝑚−1}−𝑝‖+𝑒_{𝑘(𝑛+𝑚−1)}≤exp{𝑘𝑣_{𝑛+𝑚−1}}‖𝑥_{𝑛+𝑚−1}−𝑝‖+𝑒_{𝑘(𝑛+𝑚−1)}≤⋯≤exp{𝑘^{𝑛+𝑚−1}_𝑖=𝑛𝑣_𝑖}‖𝑥_𝑛−𝑝‖+^{𝑛+𝑚−1}_𝑖=𝑛𝑒_𝑘𝑖≤exp{𝑘^∞_𝑖=1𝑣_𝑖}‖𝑥_𝑛−𝑝‖+^∞_𝑖=𝑛𝑒_𝑘𝑖≤𝑀‖𝑥_𝑛−𝑝‖+^∞_𝑖=𝑛𝑠_𝑖,(3.6) where 𝑀=exp{𝑘∑^∞_𝑖=1𝑣_𝑖} and 𝑠_𝑖=𝑒_𝑘𝑖.

Theorem 3.2. Let 𝑋 be a Banach space and 𝐶 a nonempty closed and convex subset of 𝑋 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ is closed and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for each 𝑖=1,2,…,𝑘. Then the iterative sequence {𝑥_𝑛} defined by (1.7) converges strongly to a common fixed point of the family of mappings if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Proof. We prove only the sufficiency because the necessity is obvious. From (3.5), we have ‖𝑥_𝑛+1−𝑝‖≤(1+𝑣_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑒_𝑘𝑛,forall𝑛andall𝑝∈𝐹.
Hence, we have 𝑑(𝑥_𝑛+1,𝐹)≤(1+𝑣_𝑛)^𝑘𝑑(𝑥_𝑛,𝐹)+𝑒_𝑘𝑛=(1+^𝑘_𝑟=1𝑘(𝑘−1)…(𝑘−𝑟+1)𝑣𝑟!^𝑟𝑛)𝑑(𝑥_𝑛,𝐹)+𝑒_𝑘𝑛.(3.7) Since ∑^∞_𝑛=1𝑣_𝑛<∞, it follows that ∑^∞_𝑛=1∑^𝑘_𝑟=1(𝑘(𝑘−1)⋯(𝑘−𝑟+1)/𝑟!)𝑣^𝑟_𝑛<∞. Since ∑^∞_𝑛=1𝑒_𝑘𝑛<∞ and liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, it follows from Lemma 2.1(ii) that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. Next, we prove that {𝑥_𝑛} is a Cauchy sequence. From Lemma 3.1(c), we have ‖𝑥_𝑛+𝑚−𝑝‖≤𝑀‖𝑥_𝑛−𝑝‖+^∞_𝑖=𝑛𝑠_𝑖,∀𝑝∈𝐹,𝑛,𝑚∈ℕ.(3.8) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 and ∑^∞_𝑖=1𝑠_𝑖<∞, therefore for 𝜖>0, there exists 𝑛₀∈ℕ such that 𝑑(𝑥_𝑛𝜖,𝐹)<,4𝑀^∞_𝑖=𝑛0𝑠_𝑖<𝜖4,∀𝑛≥𝑛₀.(3.9) Therefore, there exists 𝑧₁ in 𝐹 such that ‖𝑥_𝑛0−𝑧₁𝜖‖<.4𝑀(3.10) From (3.8) to (3.10), for all 𝑛≥𝑛₀ and 𝑚≥1, we have ‖𝑥_𝑛+𝑚−𝑥_𝑛‖≤‖𝑥_𝑛+𝑚−𝑧₁‖+‖𝑥_𝑛−𝑧₁‖≤𝑀‖𝑥_𝑛0−𝑧₁‖+^∞_𝑖=𝑛0𝑠_𝑖+𝑀‖𝑥_𝑛0−𝑧₁‖+^∞_𝑖=𝑛0𝑠_𝑖𝜖<𝑀+𝜖4𝑀4𝜖+𝑀+𝜖4𝑀4=𝜖.(3.11) This shows that {𝑥_𝑛} is a Cauchy sequence, hence 𝑥_𝑛→𝑧∈𝐶. It remains to show that 𝑧∈𝐹. Notice that |𝑑(𝑧,𝐹)−𝑑(𝑥_𝑛,𝐹)|≤‖𝑧−𝑥_𝑛‖,∀𝑛.(3.12) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, we obtain that 𝑧∈𝐹.

The following corollary follows from Theorem 3.2.

Corollary 3.3. Let 𝑋 be a Banach space and 𝐶 a nonempty closed and convex subset of 𝑋 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ is closed and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for each 𝑖=1,2,…,𝑘. Then the iterative sequence {𝑥_𝑛}, defined by (1.7), converges strongly to a point 𝑝∈𝐹 if and only if there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} converging to 𝑝.

Since an asymptotically quasi-nonexpansive mapping is generalized asymptotically quasi-nonexpansive mapping, so we have the following result.

Corollary 3.4. Let 𝑋 be a Banach space and 𝐶 a nonempty closed and convex subset of 𝑋 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} a finite family of asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for each 𝑖=1,2,…,𝑘. Then the iterative sequence {𝑥_𝑛}, defined by (1.7), converges strongly to a common fixed point of the family of mappings if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Remark 3.5. Theorem 3.2 generalizes and extends Theorem 2.2 of Khan et al. [6], for a finite family of asymptotically quasi-nonexpansive mappings, Theorem 1 of Fukhar-ud-din and Khan [5], and Theorem 3.2 of Shahzad and Udomene [10] for two asymptotically quasi-nonexpanaive mappings to the more general class of generalized asymptotically quasi-nonexpansive mappings.

Theorem 3.6. Let 𝑋 be a Banach space and 𝐶 a nonempty closed and convex subset of 𝑋 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Suppose that 𝐹≠∅ is closed. Let 𝑥₁∈𝐶 and {𝑥_𝑛} be the sequence defined by (1.7). If ∑^∞_𝑛=1𝛾_𝑖𝑛<∞, lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for all 𝑖=1,2,…,𝑘 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} satisfies condition (𝐶), then {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings.

Proof. From lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for all 𝑖=1,2,…,𝑘 and {𝑇_𝑖∶𝑖=1,2,…,𝑘} satisfying condition (𝐶), there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥_𝑛−𝑇_𝑖0𝑥_𝑛‖≥𝑓(𝑑(𝑥_𝑛,𝐹)) for some 𝑖₀∈{1,2,…,𝑘}, it follows that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. From Theorem 3.2, we obtain that {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings.

4. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we establish weak and strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive and (𝐿−𝛾) uniform Lipschitz mappings in a uniformly convex Banach space. In order to prove our main results, we need the following lemma.

Lemma 4.1. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex Banach space 𝑋 and {𝑇_𝑖,𝑖=1,2,…,𝑘} a finite family of (𝐿−𝛾) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. For a given 𝑥₁∈𝐶 let {𝑥_𝑛} and {𝑦_𝑖𝑛} be the sequences defined by (1.7) with 0<𝜂≤𝛼_𝑖𝑛≤𝜌<1, for all 𝑖=1,2,…,𝑘 and all 𝑛≥𝑛₀ and for some 𝑛₀∈ℕ. Then (i)lim_𝑛→∞‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖=0 for all 𝑗=1,2,…,𝑘,(ii)lim_𝑛→∞‖𝑇_𝑗𝑥_𝑛−𝑥_𝑛‖=0 for all 𝑗=1,2,…,𝑘,(iii)lim_𝑛→∞‖𝑦_𝑗𝑛−𝑥_𝑛‖=0 for all 𝑗=1,2,…,𝑘.

Proof. Let 𝑝∈𝐹, 𝑣_𝑛=max_{1≤𝑖≤𝑘}{𝑏_𝑖𝑛} and 𝑑_𝑛=max_{1≤𝑖≤𝑘}{𝑐_𝑖𝑛} for all 𝑛.
(i) From Lemma 3.1(b), we have that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹. Suppose that
lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑎.(4.1) From (3.4) and (4.1), we get that limsup_𝑛→∞‖𝑦_𝑗𝑛−𝑝‖≤𝑎,for1≤𝑗≤𝑘−1.(4.2) For each 𝑗∈{1,2,…,𝑘−1} and 𝑛∈ℕ, we have ‖𝑦_𝑗𝑛−𝑝‖≤𝛼_𝑗𝑛‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑝‖+𝛽_𝑗𝑛‖𝑥_𝑛−𝑝‖+𝛾_𝑗𝑛‖𝑢_𝑗𝑛−𝑝‖≤𝛼_𝑗𝑛(1+𝑣_𝑛)‖𝑦_{(𝑗−1)𝑛}−𝑝‖+𝛼_𝑗𝑛(1+𝑣_𝑛)𝑑_𝑛+(1−𝛼_𝑗𝑛)(1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝛾_𝑗𝑛(1+𝑣_𝑛)‖𝑢_𝑗𝑛−𝑝‖.(4.3) By using (4.3) and (1.7), for each 𝑗=1,2,…,𝑘−1, we have ‖𝑥_𝑛+1−𝑝‖=‖𝛼_𝑘𝑛(𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑝)+𝛽_𝑘𝑛(𝑥_𝑛−𝑝)+𝛾_𝑘𝑛(𝑢_𝑘𝑛−𝑝)‖≤𝛼_𝑘𝑛(1+𝑣_𝑛)‖𝑦_{(𝑘−1)𝑛}−𝑝‖+(1−𝛼_𝑘𝑛)(1+𝑣_𝑛)‖𝑥_𝑛−𝑝‖+𝛼_𝑘𝑛(1+𝑣_𝑛)𝑑_𝑛+𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑝‖(1+𝑣_𝑛)≤(𝛼_𝑘𝑛𝛼_{(𝑘−1)𝑛})(1+𝑣_𝑛)²‖𝑦_{(𝑘−2)𝑛}−𝑝‖+(1−𝛼_𝑘𝑛𝛼_{(𝑘−1)𝑛})(1+𝑣_𝑛)²‖𝑥_𝑛−𝑝‖+(𝛼_𝑘𝑛+𝛼_{(𝑘−1)𝑛})(1+𝑣_𝑛)²𝑑_𝑛+(𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑝‖+𝛾_{(𝑘−1)𝑛}‖𝑢_{(𝑘−1)𝑛}−𝑝‖)(1+𝑣_𝑛)²⋮≤(𝛼_𝑘𝑛𝛼_{(𝑘−1)𝑛}⋯𝛼_(𝑗+1)𝑛)(1+𝑣_𝑛)^{(𝑘−𝑗)}‖𝑦_𝑗𝑛−𝑝‖+(1−𝛼_𝑘𝑛𝛼_{(𝑘−1)𝑛}⋯𝛼_(𝑗+1)𝑛)(1+𝑣_𝑛)^{(𝑘−𝑗)}‖𝑥_𝑛−𝑝‖+(𝛼_𝑘𝑛+𝛼_{(𝑘−1)𝑛}+⋯+𝛼_(𝑗+1)𝑛)(1+𝑣_𝑛)^{(𝑘−𝑗)}𝑑_𝑛+(𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑝‖+𝛾_{(𝑘−1)𝑛}‖𝑢_{(𝑘−1)𝑛}−𝑝‖+⋯+𝛾_(𝑗+1)𝑛‖𝑢_(𝑗+1)𝑛−𝑝‖)(1+𝑣_𝑛)^𝑘−𝑗.(4.4) Since 0<𝜂≤𝛼_𝑖𝑛≤𝜌<1, for all 𝑖=1,2,…,𝑘 and all 𝑛≥𝑛₀, we have that for all 𝑛≥𝑛₀ and all 𝑗=1,2,…,𝑘−1, ‖𝑥_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖𝜂^𝑘−𝑗−‖𝑥_𝑛+1−𝑝‖𝜂^𝑘−𝑗(1+𝑣_𝑛)^𝑘−𝑗+‖𝑦_𝑗𝑛𝜉−𝑝‖+^𝑗𝑛𝜂^𝑘−𝑗𝑑_𝑛+𝜗_𝑗𝑛𝜂^𝑘−𝑗,(4.5) where 𝜉_𝑗𝑛=𝛼_𝑘𝑛+𝛼_{(𝑘−1)𝑛}+⋯+𝛼_(𝑗+1)𝑛 and 𝜗_𝑗𝑛=𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑝‖+𝛾_{(𝑘−1)𝑛}‖𝑢_{(𝑘−1)𝑛}−𝑝‖+⋯+𝛾_(𝑗+1)𝑛‖𝑢_(𝑗+1)𝑛−𝑝‖. Since lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑎 and lim_𝑛→∞𝜗_𝑗𝑛=lim_𝑛→∞𝑑_𝑛=lim_𝑛→∞𝑣_𝑛=0, it follows that 𝑎≤liminf_𝑛→∞‖𝑦_𝑗𝑛−𝑝‖,∀𝑗=1,2,…,𝑘−1.(4.6) From (4.2) and (4.6), we have lim_𝑛→∞‖𝑦_𝑗𝑛−𝑝‖=𝑎=lim_𝑛→∞‖𝑥_𝑛−𝑝‖,∀𝑗=1,2,…,𝑘−1.(4.7) That is, for each 𝑗=1,2,…,𝑘, we have lim_𝑛→∞‖𝛼_𝑗𝑛(𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛))+(1−𝛼_𝑗𝑛)(𝑥_𝑛−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛))‖=𝑎.(4.8) Since ‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛)‖≤‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑝‖+𝛾_𝑗𝑛‖𝑢_𝑗𝑛−𝑥_𝑛‖≤(1+𝑣_𝑛)‖𝑦_{(𝑗−1)𝑛}−𝑝‖+𝑑_𝑛+𝛾_𝑗𝑛‖𝑢_𝑗𝑛−𝑥_𝑛‖,‖𝑥_𝑛−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛)‖≤‖𝑥_𝑛−𝑝‖+𝛾_𝑗𝑛‖𝑢_𝑗𝑛−𝑥_𝑛‖,(4.9) it follows that limsup_𝑛→∞‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛)‖≤𝑎,(4.10)limsup_𝑛→∞‖𝑥_𝑛−𝑝+𝛾_𝑗𝑛(𝑢_𝑗𝑛−𝑥_𝑛)‖≤𝑎,∀𝑗=1,2,…,𝑘.(4.11) From (4.8) to (4.11), we can conclude from Lemma 2.2 that lim_𝑛→∞‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖=0,∀𝑗=1,2,…,𝑘.(4.12)
(ii) It follows from part (i) in the case 𝑗=1 that lim_𝑛→∞‖𝑇^𝑛₁𝑥_𝑛−𝑥_𝑛‖=0. For 𝑗=2,3,…,𝑘, we obtain from part (i) that ‖𝑇^𝑛_𝑗𝑥_𝑛−𝑥_𝑛‖≤‖𝑇^𝑛_𝑗𝑥_𝑛−𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}‖+‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖≤𝐿‖𝑥_𝑛−𝑦_{(𝑗−1)𝑛}‖^𝛾+‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖≤𝐿(𝛼_{(𝑗−1)𝑛}‖𝑇^𝑛_𝑗−1𝑦_{(𝑗−2)𝑛}−𝑥_𝑛‖+𝛾_{(𝑗−1)𝑛}‖𝑢_{(𝑗−1)𝑛}−𝑥_𝑛‖)^𝛾+‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖→0as𝑛→∞.(4.13) Therefore, lim_𝑛→∞‖𝑇^𝑛_𝑗𝑥_𝑛−𝑥_𝑛‖=0,∀𝑗=1,2,…,𝑘.(4.14) Since ‖𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛+1−𝑇_𝑗^𝑛+1𝑥_𝑛+1‖+‖𝑇_𝑗^𝑛+1𝑥_𝑛+1−𝑇_𝑗^𝑛+1𝑥_𝑛‖+‖𝑇_𝑗^𝑛+1𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖≤𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖+𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑥_𝑛‖+‖𝑥_𝑛+1−𝑇_𝑗^𝑛+1𝑥_𝑛+1‖+𝐿(𝛼_𝑘𝑛‖𝑇^𝑛_𝑘𝑦_{(𝑘−1)𝑛}−𝑥_𝑛‖+𝛾_𝑘𝑛‖𝑢_𝑘𝑛−𝑥_𝑛‖)^𝛾+𝐿‖𝑇^𝑛_𝑗𝑥_𝑛−𝑥_𝑛‖^𝛾,(4.15) it follows from (i) and (4.14) that lim_𝑛→∞‖𝑇_𝑗𝑥_𝑛−𝑥_𝑛‖=0,∀𝑗=1,2,…,𝑘.(4.16)
(iii) Since lim_𝑛→∞𝛾_𝑗𝑛=0 and ‖𝑦_𝑗𝑛−𝑥_𝑛‖≤𝛼_𝑗𝑛‖𝑇^𝑛_𝑗𝑦_{(𝑗−1)𝑛}−𝑥_𝑛‖+𝛾_𝑗𝑛‖𝑢_𝑗𝑛−𝑥_𝑛‖(4.17) for all 𝑗∈{1,2,…,𝑘}, (iii) is directly obtained by (i).

Theorem 4.2. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex Banach space 𝑋 satisfying the Opial's property, and {𝑇_𝑖,𝑖=1,2,…,𝑘} a finite family of (𝐿−𝛾) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequences {𝑏_𝑖𝑛},{𝑐_𝑖𝑛}⊂[0,∞) such that ∑^∞_𝑛=1𝑏_𝑖𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. Assume that 𝐹≠∅ and ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for all 𝑖=1,2,…,𝑘. For a given 𝑥₁∈𝐶 let {𝑥_𝑛} be the sequence defined by (1.7) with 0<𝜂≤𝛼_𝑖𝑛≤𝜌<1, for all 𝑖=1,2,…,𝑘 and all 𝑛≥𝑛₀ and for some 𝑛₀∈ℕ. If 𝐼−𝑇_𝑖, 𝑖=1,2,…,𝑘, is demiclosed at 0, then {𝑥_𝑛} converges weakly to a common fixed point of the family {𝑇_𝑖∶𝑖=1,2,…,𝑘}.

Proof. By Lemma 4.1(ii), we have lim_𝑛→∞‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖=0, for all 𝑖=1,2,…,𝑘. Since 𝑋 is uniformly convex and {𝑥_𝑛} is bounded, without loss of generality we may assume that 𝑥_𝑛→𝑢 weakly as 𝑛→∞ for some 𝑢∈𝐶. Since 𝐼−𝑇_𝑖, 𝑖=1,2,…,𝑘, is demiclosed at 0, we have 𝑢∈𝐹. Suppose that there are subsequences {𝑥_𝑛𝑘} and {𝑥_𝑚𝑘} of {𝑥_𝑛} that converge weakly to 𝑢 and 𝑣, respectively. Again, as above, we can prove that 𝑢,𝑣∈𝐹. By Lemma 3.1(b), lim_𝑛→∞‖𝑥_𝑛−𝑢‖ and lim_𝑛→∞‖𝑥_𝑛−𝑣‖ exist. It follows from Lemma 2.3 that 𝑢=𝑣. Therefore {𝑥_𝑛} converges weakly to a common fixed point of {𝑇_𝑖∶𝑖=1,2,…,𝑘}.

Theorem 4.3. Under the hypotheses of Lemma 4.1, assume that the family {𝑇_𝑖∶𝑖=1,2,…,𝑘} satisfies condition (𝐶). Then {𝑥_𝑛} and {𝑦_𝑗𝑛} converge strongly to a common fixed point of the family of mappings for all 𝑗=1,2,…,𝑘.

Proof. From (3.5), we have ‖𝑥_𝑛+1−𝑝‖≤(1+𝑣_𝑛)^𝑘‖𝑥_𝑛−𝑝‖+𝑒_𝑘𝑛,∀𝑛,𝑝∈𝐹.(4.18) Therefore, 𝑑(𝑥_𝑛+1,𝐹)≤(1+𝑣_𝑛)^𝑘𝑑(𝑥_𝑛,𝐹)+𝑒_𝑘𝑛=(1+^𝑘_𝑟=1𝑘(𝑘−1)⋯(𝑘−𝑟+1)𝑣𝑟!^𝑟𝑛)𝑑(𝑥_𝑛,𝐹)+𝑒_𝑘𝑛.(4.19) Since ∑^∞_𝑛=1𝑣_𝑛<∞, it follows that ∑^∞_𝑛=1∑^𝑘_𝑟=1(𝑘(𝑘−1)⋯(𝑘−𝑟+1)/𝑟!)𝑣^𝑟_𝑛<∞. Since ∑^∞_𝑛=1𝑒_𝑘𝑛<∞, we obtain from Lemma 2.1(i) that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists. By Lemma 4.1(ii), we have lim_𝑛→∞‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖=0 for all 𝑖=1,2,…,𝑘. Since {𝑇_𝑖∶𝑖=1,2,…,𝑘} satisfies condition (𝐶), there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥_𝑛−𝑇_𝑖0𝑥_𝑛‖≥𝑓(𝑑(𝑥_𝑛,𝐹)) for some 𝑖₀∈{1,2,…,𝑘}, it follows that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. By Theorem 3.2, we can conclude that {𝑥_𝑛} converges strongly to a common fixed point 𝑞 of the family {𝑇_𝑖∶𝑖=1,2,…,𝑘}. From Lemma 4.1(iii), we have lim_𝑛→∞‖𝑦_𝑗𝑛−𝑥_𝑛‖=0 for all 𝑗=1,2,…,𝑘, and we obtain that lim_𝑛→∞𝑦_𝑗𝑛=𝑞 for all 𝑗=1,2,…,𝑘.

Remark 4.4. The family of generalized asymptotically quasi-nonexpansive mappings in Theorem 4.2 and 4.3 can be replaced by a family of asymptotically quasi-nonexpansive mappings. Lemma 3.1 and 4.1 generalize and improve [6, Lemma 3.1], [19, Lemmas 3.4 and 3.5], [18, Lemma 2.3], [7, Lemma 4.2], and [17, Lemma 3.3] to a finite family of (𝐿−𝛾) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.2 generalizes and improves [6, Theorems 3.2 and 4.2], [18, Theorem 2.9], [17, Theorem 3.1], and [21, Theorem 1] to the more general class of a finite family of (𝐿−𝛾) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.3 generalizes and improves [6, Theorem 3.3], [7, Theorem 4.3], [5, Theorem 2], [18, Theorem 2.4], [19, Theorem 4.2], [17, Theorem 3.2], [10, Theorem 3.4], [20, Theorem 2] and [21, Theorem 2] by using condition (𝐶) instead of condition (𝐴) or semicompactness or completely continuous or compactness to the more general class of a finite family of (𝐿−𝛾) uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings.

Remark 4.5. From Remark 1.1, Theorems 3.2 to 4.3 hold true for a finite family {𝑇_𝑖}^𝑘_𝑖=1 of asymptotically nonexpansive mappings in the intermediate sense.

Acknowledgments

The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin University, and the Graduate School of Chiang Mai University, Thailand for their financial support.