An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

Let 𝐾 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸. Let {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ be two infinite families of asymptotically nonexpansive mappings from 𝐾 to itself with 𝐹∶={𝑥∈𝐾∶𝑇_𝜆𝑥=𝑥=𝑆_𝜆𝑥,𝜆∈Λ}≠∅. For an arbitrary initial point 𝑥₀∈𝐾, {𝑥_𝑛} is defined as follows: 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛(𝑇^∗_𝑛−1)^𝑚𝑛−1𝑥_𝑛−1+𝛾_𝑛(𝑇^∗_𝑛)^𝑚𝑛𝑦_𝑛, 𝑦_𝑛=𝛼^′_𝑛𝑥_𝑛+𝛽^′_𝑛(𝑆^∗_𝑛−1)^𝑚𝑛−1𝑥_𝑛−1+𝛾^′_𝑛(𝑆^∗_𝑛)^𝑚𝑛𝑥_𝑛, 𝑛=1,2,3,…, where 𝑇^∗_𝑛=𝑇_𝜆𝑖𝑛 and 𝑆^∗_𝑛=𝑆_𝜆𝑖𝑛 with 𝑖_𝑛 and 𝑚_𝑛 satisfying the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2, 𝑚≥𝑖; {𝑇_𝜆𝑖}^∞_𝑖=1 and {𝑆_𝜆𝑖}^∞_𝑖=1 are two countable subsets of {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ, respectively; {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, and {𝛾^′_𝑛} are sequences in [𝛿,1−𝛿] for some 𝛿∈(0,1), satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 and 𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛=1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

1. Introduction

Let 𝐾 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸. A mapping 𝑇∶𝐾→𝐾 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾. 𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 (𝑛→∞) such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐾,𝑛=1,2,3,…. (1) It is obvious that a nonexpansive mapping is an asymptotically nonexpansive one, but the converse is not true. Denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}. Throughout this paper, we always assume that 𝐹(𝑇)≠∅. As an important generalization of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972, who proved that if 𝐾 is a nonempty, closed, and convex subset of a real uniformly convex Banach space and 𝑇∶𝐾→𝐾 is an asymptotically nonexpansive mapping, then 𝑇 has a fixed point.

Since then, iterative techniques for approximating fixed points of asymptotically nonexpansive mappings have been studied by various authors (see, e.g., [2–9]). However, these researches are all restricted to the situation of at most finite families of asymptotically nonexpansive mappings. For the extension of finite families to infinite ones, we develop an original method, namely, a specific way of choosing the indexes, for the iterative approximation of common fixed points of the involved mappings.

We now cite an announced result as the object of our extension. In 2010, Wang et al. [10] constructed the following iteration process for two asymptotically nonexpansive mappings and obtained some strong convergence theorems for common fixed points of the given mappings in Banach spaces. For an arbitrary initial point 𝑥₀∈𝐾, {𝑥_𝑛},{𝑦_𝑛} are defined as follows: 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇^𝑛−1𝑥_𝑛−1+𝛾_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=𝛼^′_𝑛𝑥_𝑛+𝛽^′_𝑛𝑆^𝑛−1𝑥_𝑛−1+𝛾^′_𝑛𝑆^𝑛𝑥_𝑛,𝑛=1,2,3,…, (2) where 𝑇,𝑆∶𝐾→𝐾 are two asymptotically nonexpansive mappings; {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, and {𝛾^′_𝑛} are real sequences in [0,1) satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 and 𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛=1.

In this paper, a modified iteration scheme of (2) is used for approximating common fixed points of two infinite families of asymptotically nonexpansive mappings; a strong convergence theorem is established in the framework of uniformly convex Banach spaces. The results show the feasibility of the newly developed technique and extend those of the authors whose related researches are restricted to the situation of finite families of such mappings.

2. Preliminaries

Throughout this paper, we use 𝐹 to denote the set of common fixed points of two infinite families of asymptotically nonexpansive mappings {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ, that is, 𝐹∶={𝑥∈𝐾∶𝑇_𝜆𝑥=𝑥=𝑆_𝜆,𝜆∈Λ}.

Let 𝐾 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let {𝑇_𝜆𝑖}^∞_𝑖=1 and {𝑆_𝜆𝑖}^∞_𝑖=1 be two countable subsets of {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ, respectively. In order to approximate some member of 𝐹, we define, from an arbitrary 𝑥₀∈𝐾, the following implicit iteration scheme: 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1+𝛾_𝑛𝑇^∗_𝑛^𝑚𝑛𝑦_𝑛,𝑦_𝑛=𝛼^′_𝑛𝑥_𝑛+𝛽^′_𝑛𝑆^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1+𝛾^′_𝑛𝑆^∗_𝑛^𝑚𝑛𝑥_𝑛,𝑛=1,2,3,…, (3) where {𝛼_𝑛}, {𝛽_𝑛},  {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, and {𝛾^′_𝑛} are sequences in [𝛿,1−𝛿] for some 𝛿∈(0,1), satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 and 𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛=1; 𝑇^∗_𝑛=𝑇_𝜆𝑖𝑛 and 𝑆^∗_𝑛=𝑆_𝜆𝑖𝑛 with 𝑖_𝑛 and 𝑚_𝑛 being the solutions to the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑛=1,2,3,…), that is, for each 𝑛≥1, there exist unique 𝑖_𝑛 and 𝑚_𝑛 such that 𝑖₁=1,𝑖₂=1,𝑖₃𝑖=2,₄=1,𝑖₅=2,𝑖₆𝑖=3,₇=1,𝑖₈𝑚=2,…,₁=1,𝑚₂=2,𝑚₃𝑚=2,₄=3,𝑚₅=3,𝑚₆𝑚=3,₇=4,𝑚₈=4,…. (4) For convenience, we restate the following concepts and results.

A Banach space 𝐸 is said to satisfy Opial’s condition if, for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥 implies that limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦, (5) for all 𝑦∈𝐸 with 𝑦≠𝑥, where 𝑥_𝑛⇀𝑥 denotes that {𝑥_𝑛} converges weakly to 𝑥.

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at 𝑝 if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that {𝑥_𝑛} converges weakly to 𝑥^∗∈𝐷(𝑇) and {𝑇𝑥_𝑛} converges strongly to 𝑝, then 𝑇𝑥^∗=𝑝.

We now need the following lemmas for our main results.

Lemma 1 (see [11]). Let {𝑎_𝑛},{𝛿_𝑛}, and {𝑏_𝑛} be sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,𝑛=1,2,3,…. (6) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists.

Lemma 2 (see [6]). Let 𝐸 be a real uniformly convex Banach space, and let 𝑎,and𝑏 be two constants with 0<𝑎<𝑏<1. Suppose that {𝑡_𝑛}⊂[𝑎,𝑏] is a real sequence and {𝑥_𝑛}and{𝑦_𝑛} are two sequences in 𝐸. Then, the conditions lim_𝑛→∞𝑡_𝑛𝑥_𝑛+1−𝑡_𝑛𝑦_𝑛=𝑑,limsup_𝑛→∞𝑥_𝑛≤𝑑,limsup_𝑛→∞𝑦_𝑛≤𝑑 (7) imply that lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, where 𝑑≥0 is a constant.

Lemma 3 (see [2]). Let 𝐸 be a real uniformly convex Banach space, 𝐾 a nonempty, closed, convex subset of 𝐸, and let 𝑇∶𝐾→𝐸 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1 as 𝑛→∞. Then, 𝐼−𝑇 is demiclosed at zero.

Lemma 4. The unique solutions to the positive integer equation ()𝑚𝑛=𝑖+𝑚−12,𝑚≥𝑖,𝑛=1,2,3,… (8) are ()𝑚𝑖=𝑛−𝑚−121,𝑚=−2−12𝑛+4,𝑛=1,2,3,…, (9) where [𝑥] denotes the maximal integer that is not larger than x.

Proof. It follows from (8) that ()𝑚𝑖=𝑛−𝑚−12,𝑖≤𝑚,𝑛=1,2,3,…, (10) and hence ()𝑚1≤𝑖=𝑛−𝑚−12≤𝑚,𝑛=1,2,3,…, (11) that is, (𝑚−1)𝑚2()𝑚+1≤𝑛≤𝑚+12,𝑛=1,2,3,…, (12) which implies that 1𝑚−2²7≤2𝑛−4,1𝑚+2²1≥2𝑛+4,𝑛=1,2,3,…. (13) Thus, 12𝑛+4−121≤𝑚≤2+72𝑛−4,𝑛=1,2,3…, (14) that is, −72𝑛−4−121≤−𝑚≤2−12𝑛+4,𝑛=1,2,3,…, (15) while the difference of the two sides of the inequality above is 1−12𝑛+4−72𝑛−42=1−2𝑛+1/4+∈[),2𝑛−7/40,1𝑛=1,2,3,…. (16) Then, it follows from (15) that (9) holds obviously.

3. Main Results

Lemma 5. Let 𝐾 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸, and let {𝑇_𝜆𝑖}^∞_𝑖=1 and {𝑆_𝜆𝑖}^∞_𝑖=1 be two countable subsets of the asymptotically nonexpansive mappings {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ from 𝐾 to itself, respectively, with corresponding sequences {𝑘_𝑛^(𝑖)}⊂[1,∞) and {𝑟_𝑛^(𝑖)}⊂[1,∞) such that ∑^∞_𝑖=1∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞ and ∑^∞_𝑖=1∑^∞_𝑛=1(𝑟_𝑛^(𝑖)−1)<∞. Suppose that {𝑥_𝑛} is generated by (3), where 𝛾_𝑛𝑘_𝑛^(𝑖)(𝛼^′_𝑛+𝛾^′_𝑛𝑟_𝑛^(𝑖))<1 for all 𝑖≥1. If 𝐹≠∅, then lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists for each 𝑞∈𝐹.

Proof. Set 𝑢^(𝑖𝑛)_𝑚𝑛=𝑘^(𝑖𝑛)_𝑚𝑛−1 and 𝑣^(𝑖𝑛)_𝑚𝑛=𝑟^(𝑖𝑛)_𝑚𝑛−1 for each positive integer 𝑛≥1, where 𝑖_𝑛 and 𝑚_𝑛 satisfy the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑚∈ℕ). For any 𝑞∈𝐹, it follows from (3) that 𝑦_𝑛−𝑞≤𝛼^′_𝑛𝑥_𝑛−𝑞+𝛽^′_𝑛𝑟^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞+𝛾^′_𝑛𝑟^(𝑖𝑛)_𝑚𝑛𝑥_𝑛−1−𝑞=𝛼^′_𝑛+𝛾^′_𝑛𝑟^(𝑖𝑛)_𝑚𝑛𝑥_𝑛−𝑞+𝛽^′_𝑛𝑟^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1,𝑥−𝑞_𝑛−𝑞≤𝛼_𝑛𝑥_𝑛−1−𝑞+𝛽_𝑛𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞+𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝑦_𝑛−𝑞=𝛼_𝑛+𝛽_𝑛𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞+𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝑦_𝑛−𝑞. (17)(18) Substituting (17) into (18) yields that 1−𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝛼^′_𝑛+𝛾^′_𝑛𝑟^(𝑖𝑛)_𝑚𝑛𝑥_𝑛≤𝛼−𝑞_𝑛+𝛽_𝑛𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1+𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝛽^′_𝑛𝑟^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞. (19) Note that 𝑎_𝑛∶=1−𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛(𝛼^′_𝑛+𝛾^′_𝑛𝑟^(𝑖𝑛)_𝑚𝑛)>0. We have 𝑥_𝑛≤𝛼−𝑞^𝑛+𝛽_𝑛𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1+𝛾_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝛽^′_𝑛𝑟^{(𝑖𝑛−1)}_𝑚𝑛−1𝑎_𝑛𝑥_𝑛−1−𝑞, (20) which implies that ‖𝑥_𝑛−𝑞‖≤(1+𝑏_𝑛)‖𝑥_𝑛−1−𝑞‖, where 𝑏_𝑛=𝛾_𝑛𝛾^′_𝑛𝑣^(𝑖𝑛)_𝑚𝑛+𝛾_𝑛𝑢^(𝑖𝑛)_𝑚𝑛+𝛾_𝑛𝛾^′_𝑛𝑢^(𝑖𝑛)_𝑚𝑛𝑣^(𝑖𝑛)_𝑚𝑛+𝛽_𝑛𝑢^{(𝑖𝑛−1)}_𝑚𝑛−1+𝛾_𝑛𝛽^′_𝑛𝑣^{(𝑖𝑛−1)}_𝑚𝑛−1+𝛾_𝑛𝛽^′_𝑛𝑢^(𝑖𝑛)_𝑚𝑛𝑣^{(𝑖𝑛−1)}_𝑚𝑛−1×1−𝛾_𝑛1−𝛽^′_𝑛−𝛾_𝑛𝛾^′_𝑛𝑣^(𝑖𝑛)_𝑚𝑛−𝛾_𝑛𝑢^(𝑖𝑛)_𝑚𝑛1−𝛽^′_𝑛−𝛾_𝑛𝛾^′_𝑛𝑢^(𝑖𝑛)_𝑚𝑛𝑣^(𝑖𝑛)_𝑚𝑛⁻¹. (21) Note that ∑^∞_𝑛=1𝑢^(𝑖𝑛)_𝑚𝑛=∑^∞_𝑖=1∑^∞_𝑛=𝑖(𝑘_𝑛^(𝑖)−1)≤∑^∞_𝑖=1∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞ and ∑^∞_𝑛=1𝑣^(𝑖𝑛)_𝑚𝑛=∑^∞_𝑖=1∑^∞_𝑛=𝑖(𝑟_𝑛^(𝑖)−1)≤∑^∞_𝑖=1∑^∞_𝑛=1(𝑟_𝑛^(𝑖)−1)<∞, which implies that lim_𝑛→∞𝑢^(𝑖𝑛)_𝑚𝑛=lim_𝑛→∞𝑣^(𝑖𝑛)_𝑚𝑛=0. Then, for a given 𝜖₀∈(0,𝛿), there exists a positive 𝑛₀ such that 𝛾_𝑛1−𝛽^′_𝑛+𝛾_𝑛𝛾^′_𝑛𝑣^(𝑖𝑛)_𝑚𝑛+𝛾_𝑛𝑢^(𝑖𝑛)_𝑚𝑛1−𝛽^′_𝑛+𝛾_𝑛𝛾^′_𝑛𝑢^(𝑖𝑛)_𝑚𝑛𝑣^(𝑖𝑛)_𝑚𝑛<1−𝜖₀, (22) as 𝑛≥𝑛₀. Then, it follows from (20) and (22) that 𝑥_𝑛−𝑞≤1+𝑐_𝑛𝑥_𝑛−1−𝑞, (23) where 𝑐_𝑛=(1/𝜖₀)[𝑣^(𝑖𝑛)_𝑚𝑛+𝑢^(𝑖𝑛)_𝑚𝑛+𝑢^(𝑖𝑛)_𝑚𝑛𝑣^(𝑖𝑛)_𝑚𝑛+𝑢^{(𝑖𝑛−1)}_𝑚𝑛−1+𝑣^{(𝑖𝑛−1)}_𝑚𝑛−1+𝑢^(𝑖𝑛)_𝑚𝑛𝑣^{(𝑖𝑛−1)}_𝑚𝑛−1], and so ∑^∞_𝑛=1𝑐_𝑛<∞. Hence, it follows from (23) and Lemma 1 that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists for each 𝑞∈𝐹. The proof is completed.

Remark 6. Because of the importance of the condition that 𝐹 is nonempty, we now give an example satisfying the lemma with the set of common fixed points of {𝑇_𝜆}_𝜆∈Λ and {𝑆_𝜆}_𝜆∈Λ being a non single point set. Let 𝐸∶=ℝ¹, and let 𝐾∶=[−1,1]. Define an infinite family of mappings {𝑇_𝜆}_𝜆≥1∶𝐾→𝐾 by 𝑇_𝜆(𝑥)=1𝜆𝑥^𝜆[],[),,𝑥∈0,1𝑥,𝑥∈−1,0 (24) and an infinite family of mappings {𝑆_𝜆}_𝜆≥1∶𝐾→𝐾 by 𝑆_𝜆(𝑥)=1𝜆sin𝑥^𝜆[],[).,𝑥∈0,1𝑥,𝑥∈−1,0 (25) Then, clearly, {𝑇_𝜆}_𝜆≥1 and {𝑆_𝜆}_𝜆≥1 are two infinite families of asymptotically nonexpansive mappings with 𝐹=[−1,0].

Lemma 7. Let 𝐾, 𝐸, {𝑇_𝜆𝑖}^∞_𝑖=1, {𝑆_𝜆𝑖}^∞_𝑖=1, and {𝑥_𝑛} be the same as those in Lemma 5. If 𝐹≠∅, then for each 𝑖≥1, there exists a subsequence {𝑥_𝑛^(𝑖)} of {𝑥_𝑛} such that lim_𝑛→∞‖𝑥_𝑛^(𝑖)−𝑇_𝜆𝑖𝑥_𝑛^(𝑖)‖=lim_𝑛→∞‖𝑥_𝑛^(𝑖)−𝑆_𝜆𝑖𝑥_𝑛^(𝑖)‖=0.

Proof. By Lemma 5, we may assume that lim_𝑛→∞‖𝑥_𝑛−𝑞‖=𝑑 for a given 𝑞∈𝐹, that is, lim_𝑛→∞𝑑_𝑛=𝑑, (26) where 𝑑_𝑛=‖(1−𝛾_𝑛)[𝛼_𝑛(𝑥_𝑛−1−𝑞)/(1−𝛾_𝑛)+𝛽_𝑛((𝑇^∗_𝑛−1)^𝑚𝑛−1𝑥_𝑛−1−𝑞)/(1−𝛾_𝑛)]+𝛾_𝑛((𝑇^∗_𝑛)^𝑚𝑛𝑦_𝑛−𝑞)‖. It follows from (17) that 𝑇^∗_𝑛^𝑚𝑛𝑦_𝑛−𝑞≤𝑘^(𝑖𝑛)_𝑚𝑛𝑦_𝑛−𝑞≤𝑘^(𝑖𝑛)_𝑚𝑛𝛼^′_𝑛+𝛾^′_𝑛𝑟^(𝑖𝑛)_𝑚𝑛𝑥_𝑛−𝑞+𝛽^′_𝑛𝑟^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞=𝑘^(𝑖𝑛)_𝑚𝑛𝑥_𝑛−𝑞+𝛽^′_𝑛𝑥_𝑛−𝑞−𝑥_𝑛−1−𝑞+𝛾^′_𝑛𝑣^(𝑖𝑛)_𝑚𝑛𝑥_𝑛−𝑞+𝛽^′_𝑛𝑣^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞. (27) Taking limsup on both sides in (27) yields that limsup_𝑛→∞𝑇^∗_𝑛^𝑚𝑛𝑦_𝑛−𝑞≤limsup_𝑛→∞𝑦_𝑛−𝑞≤𝑑. (28) Next, it follows from (26) that limsup_𝑛→∞𝑒_𝑛≤limsup_𝑛→∞𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−1+𝛽−𝑞_𝑛1−𝛾_𝑛𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞=limsup_𝑛→∞𝛽1+^𝑛1−𝛾_𝑛𝑢^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞=𝑑, (29) where 𝑒_𝑛=‖𝛼_𝑛(𝑥_𝑛−1−𝑞)+𝛽_𝑛((𝑇^∗_𝑛−1)^𝑚𝑛−1𝑥_𝑛−1−𝑞)‖/(1−𝛾_𝑛). It then follows from (26), (28), (29), and Lemma 2 that lim_𝑛→∞𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−1+𝛽_𝑛1−𝛾_𝑛𝑇^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1−𝑇^∗_𝑛^𝑚𝑛𝑦_𝑛=0, (30) which, in addition to (3), implies that lim_𝑛→∞𝑥_𝑛−(𝑇^∗_𝑛)^𝑚𝑛𝑦_𝑛=0. (31) Now, we show that ‖𝑦_𝑛−𝑞‖→𝑑 as 𝑛→∞. It follows from (18) that 𝛼_𝑛+𝛽_𝑛𝛾_𝑛𝑥_𝑛−𝑞−𝑥_𝑛−1−𝑞+𝑥_𝑛−𝛽−𝑞_𝑛𝛾_𝑛𝑢^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞≤𝑘^(𝑖𝑛)_𝑚𝑛𝑦_𝑛−𝑞. (32) Taking liminf on both sides in the inequality above yields that 𝑑=liminf_𝑛→∞𝑥_𝑛−𝑞≤liminf_𝑛→∞𝑦_𝑛−𝑞. (33) Combining (28) with (33), we have lim_𝑛→∞‖𝑦_𝑛−𝑞‖=𝑑. Then, by ways similar to the preceding ones, it is easily shown that lim_𝑛→∞𝛽^′_𝑛1−𝛼^′_𝑛𝑆^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1+𝛾^′_𝑛1−𝛼^′_𝑛𝑆^∗_𝑛^𝑚𝑛𝑥_𝑛−𝑥_𝑛=0, (34) which means that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (35) Set 𝑑^′_𝑛=‖(1−𝛽_𝑛)[𝛼_𝑛(𝑥_𝑛−1−𝑞)/(1−𝛽_𝑛)+𝛾_𝑛((𝑇^∗_𝑛)^𝑚𝑛𝑦_𝑛−𝑞)/(1−𝛽_𝑛)]+𝛽_𝑛((𝑇^∗_𝑛−1)^𝑚𝑛−1𝑥_𝑛−1−𝑞)‖. Since lim_𝑛→∞𝑥_𝑛−𝑞=lim_𝑛→∞𝑑^′_𝑛=𝑑, (36) then limsup_𝑛→∞𝑇^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1−𝑞≤limsup_𝑛→∞𝑘^{(𝑖𝑛−1)}_𝑚𝑛−1𝑥_𝑛−1−𝑞≤𝑑,limsup_𝑛→∞𝑒^′_𝑛≤limsup_𝑛→∞𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−1+𝛾−𝑞_𝑛1−𝛽_𝑛𝑘^(𝑖𝑛)_𝑚𝑛𝑦_𝑛−𝑞=𝑑, (37)(38) where 𝑒^′_𝑛=‖𝛼_𝑛(𝑥_𝑛−1−𝑞)+𝛾_𝑛((𝑇^∗_𝑛)^𝑚𝑛𝑦_𝑛−𝑞)‖/(1−𝛽_𝑛). It then follows from (36)–(38) and Lemma 2 that lim_𝑛→∞𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛1−𝛽_𝑛𝑇^∗_𝑛^𝑚𝑛𝑦_𝑛−𝑇^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1=0, (39) which implies that lim_𝑛→∞𝑥_𝑛−𝑇^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1=0. (40) Then, it follows from (31), (35), and (40) that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛−1=0. (41) Similarly, we have lim_𝑛→∞𝑦_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑥_𝑛=0,lim_𝑛→∞𝑦_𝑛−𝑆^∗_𝑛−1^𝑚𝑛−1𝑥_𝑛−1=0. (42) Next, for any 𝑖≥1, we consider the corresponding subsequence {𝑥_𝑙^(𝑖)}_{𝑙∈Γ𝑖} of {𝑥_𝑛}, where 𝑙∈Γ_𝑖∶={𝑙∶𝑙=𝑖+(𝑗−1)𝑗/2,𝑗≥𝑖,𝑗∈ℕ}. For example, by the definition of Γ₁, we have Γ₁={1,2,4,7,11,16,…} and 𝑖₁=𝑖₂=𝑖₄=𝑖₇=𝑖₁₁=𝑖₁₆=⋯=1. For simplicity, {𝑥_𝑙^(𝑖)}_{𝑙∈Γ𝑖},{𝑦_𝑙^(𝑖)}_{𝑙∈Γ𝑖}, {𝑇_𝑙^∗(𝑖)}_{𝑙∈Γ𝑖}, and {𝑗_𝑙^(𝑖)}_{𝑙∈Γ𝑖} are written as {𝑥^′_𝑛},{𝑦^′_𝑛}, {𝑇^′_𝑛} and {𝑚_𝑛}, respectively. Note that {𝑚_𝑛}_{𝑛∈Γ𝑖}={𝑖,𝑖+1,𝑖+2,…}, that is, 𝑘^(𝑖𝑙)₁=𝑘₁^(𝑖),𝑚_𝑛−1=𝑚_𝑛−1, and 𝑇^′_𝑛=𝑇_𝜆𝑖=𝑇^′_𝑛−1 whenever 𝑙∈Γ_𝑖. Then, we have 𝑥^′_𝑛−𝑇^′_𝑛𝑥_𝑛≤𝑥^′_𝑛−𝑇^′_𝑛^𝑚𝑛𝑦^′_𝑛+𝑇^′_𝑛^𝑚𝑛𝑦^′_𝑛−𝑇^′_𝑛^𝑚𝑛𝑥^′_𝑛+𝑇^′_𝑛^𝑚𝑛𝑥^′_𝑛−𝑇^′_𝑛𝑥^′_𝑛≤𝑥^′_𝑛−𝑇^′_𝑛^𝑚𝑛𝑦^′_𝑛+𝑘_𝑚^(𝑖)_𝑛𝑦^′_𝑛−𝑥^′_𝑛+𝑘₁^(𝑖)𝑇^′_𝑛^𝑚𝑛−1𝑥^′_𝑛−𝑥^′_𝑛≤𝑥^′_𝑛−𝑇^′_𝑛^𝑚𝑛𝑦^′_𝑛+𝑘_𝑚^(𝑖)_𝑛𝑦^′_𝑛−𝑥^′_𝑛+𝑘₁^(𝑖)×𝑇^′_𝑛^𝑚𝑛−1𝑥^′_𝑛−𝑇^′_𝑛−1^𝑚𝑛−1𝑥^′_𝑛−1+𝑇^′_𝑛−1^𝑚𝑛−1𝑥^′_𝑛−1−𝑥^′_𝑛≤𝑥^′_𝑛−𝑇^′_𝑛^𝑚𝑛𝑦^′_𝑛+𝑘_𝑚^(𝑖)_𝑛𝑦^′_𝑛−𝑥^′_𝑛+𝑘₁^(𝑖)𝑘_𝑚^(𝑖)_𝑛−1𝑥^′_𝑛−𝑥^′_𝑛−1+𝑇^′_𝑛−1^𝑚𝑛−1𝑥^′_𝑛−1−𝑥^′_𝑛. (43) It hence follows from (31), (35), (40), and (41) that lim_𝑛→∞‖𝑥^′_𝑛−𝑇^′_𝑛𝑥^′_𝑛‖=0. That is, for each 𝑖≥1, there exists a subsequence {𝑥_𝑛^(𝑖)} of {𝑥_𝑛} such that lim_𝑛→∞‖𝑥_𝑛^(𝑖)−𝑇_𝑛^∗(𝑖)𝑥_𝑛^(𝑖)‖=0. Since 𝑇_𝑛^∗(𝑖)=𝑇_𝜆𝑖, we have, for each 𝑖≥1, lim_𝑛→∞𝑥_𝑛^(𝑖)−𝑇_𝜆𝑖𝑥_𝑛^(𝑖)=0. (44) Similarly, it can be shown that, for each 𝑖≥1, lim_𝑛→∞𝑦_𝑛^(𝑖)−𝑆_𝜆𝑖𝑦_𝑛^(𝑖)=0. (45) This completes the proof.

Remark 8. The key point of the proof of Lemma 7 lies in the use of a specific way of choosing the indexes of the involved mappings, which makes the generalization of finite families of nonlinear mappings to infinite ones possible.

Theorem 9. Let 𝐾, 𝐸, {𝑇_𝜆𝑖}^∞_𝑖=1, {𝑆_𝜆𝑖}^∞_𝑖=1, and {𝑥_𝑛} be the same as those in Lemma 5. If 𝐹≠∅ and there exist a 𝑇_𝜆𝑖0∈{𝑇_𝜆𝑖}^∞_𝑖=1 or an 𝑆_𝜆𝑖0∈{𝑆_𝜆𝑖}^∞_𝑖=1 and a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑟)>0 for all 𝑟∈(0,∞) such that 𝑓(𝑑(𝑥_𝑛,𝐹))≤‖𝑥_𝑛−𝑇_𝜆𝑖0𝑥_𝑛‖ or 𝑓(𝑑(𝑥_𝑛,𝐹))≤‖𝑥_𝑛−𝑆_𝜆𝑖0𝑥_𝑛‖ for all 𝑛≥1, then {𝑥_𝑛} converges strongly to some point of 𝐹.

Proof. By Lemma 7, there exists a subsequence {𝑥^(𝑖0)_𝑛} of {𝑥_𝑛} such that lim_𝑛→∞‖𝑥^(𝑖0)_𝑛−𝑇_𝜆𝑖0𝑥^(𝑖0)_𝑛‖=lim_𝑛→∞‖𝑥^(𝑖0)_𝑛−𝑆_𝜆𝑖0𝑥^(𝑖0)_𝑛‖=0. Since 𝑓𝑑𝑥^(𝑖0)_𝑛,𝐹≤𝑥^(𝑖0)_𝑛−𝑇_𝜆𝑖0𝑥^(𝑖0)_𝑛, (46) or 𝑓𝑑𝑥^(𝑖0)_𝑛,𝐹≤𝑥^(𝑖0)_𝑛−𝑆_𝜆𝑖0𝑥^(𝑖0)_𝑛, (47) by taking limsup as 𝑛→∞ on both sides in the inequality above, we have lim_𝑛→∞𝑓𝑑𝑥^(𝑖0)_𝑛,𝐹=0, (48) which implies lim_𝑛→∞𝑑(𝑥^(𝑖0)_𝑛,𝐹)=0 by the definition of the function 𝑓.
Now, we will show that {𝑥^(𝑖0)_𝑛} is a Cauchy sequence. By Lemma 5, there exists a constant 𝑀>0 such that ‖𝑥_𝑛−𝑞‖≤𝑀‖𝑥_𝑚−𝑞‖ as 𝑛>𝑚. And for any 𝜖>0, there exists a positive integer 𝑁 such that 𝑑(𝑥^(𝑖0)_𝑛,𝐹)<𝜖/2𝑀 for all 𝑛≥𝑁. Then, for any 𝑞∈𝐹 and 𝑛,𝑚≥𝑁, we have 𝑥^(𝑖0)_𝑛−𝑥^(𝑖0)_𝑚≤𝑥^(𝑖0)_𝑛−𝑞+𝑥^(𝑖0)_𝑚−𝑞≤2𝑀𝑥^(𝑖0)_𝑁−𝑞. (49) Taking the infimum in the above inequalities for all 𝑞∈𝐹 yields that 𝑥^(𝑖0)_𝑛−𝑥^(𝑖0)_𝑚≤2𝑀𝑑𝑥^(𝑖0)_𝑁,𝐹<𝜖, (50) which implies that {𝑥^(𝑖0)_𝑛} is a Cauchy sequence. Therefore, there exists a 𝑝∈𝐾 such that 𝑥^(𝑖0)_𝑛→𝑝 as 𝑛→∞ since 𝐸 is complete. Furthermore, lim_𝑛→∞𝑑(𝑥^(𝑖0)_𝑛,𝐹)=0 shows that 𝑑(𝑝,𝐹)=0, which implies that 𝑝∈𝐹 since 𝐹 is closed. It follows from the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖ that 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

Remark 10. The result of Theorem 9 extends that of Wang et al. [10] whose related research is restricted to the situation of two asymptotically nonexpansive mappings.

Acknowledgments

The authors are greatly grateful to the referees for their useful suggestions by which the contents of this paper are improved. This study is supported by the National Natural Science Foundation of China (Grant no. 11061037).