Strong Convergence Theorems for a Countable Family of Lipschitzian Mappings

Abstract

We modify the iterative method introduced by Kim and Xu (2006) for a countable family of Lipschitzian mappings by the hybrid method of Takahashi et al. (2008). Our results include recent ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool (2007) and Zegeye and Shahzad (2008, 2010) as special cases.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is said to be Lipschitzian if there exists a positive constant 𝐿 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶. (1.1) In this case, 𝑇 is also said to be 𝐿-Lipschitzian. Clearly, if 𝑇 is 𝐿₁-Lipschitzian and 𝐿₁<𝐿₂, then 𝑇 is 𝐿₂-Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is 𝐿-Lipschitzian with 𝐿≥1. If 𝐿=1, then 𝑇 is known as a nonexpansive mapping. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. If 𝐶 is nonempty bounded closed convex and 𝑇 is a nonexpansive of 𝐶 into itself, then 𝐹(𝑇)≠∅ (see [1]). There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [2] introduced the iteration as follows: a sequence {𝑥_𝑛} defined by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (1.2) where the initial guess element 𝑥₁∈𝐶 is arbitrary and {𝛼_𝑛} is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [3]. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence [4]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2):𝑥₁𝑦=𝑥∈𝐶isarbitrary,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖≤‖𝑥_𝑛,𝑄−𝑧‖_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=1,2,3,…, (1.3) where 𝑃_𝐾 denotes the metric projection from 𝐻 onto a closed convex subset 𝐾 of 𝐻. They prove that if the sequence {𝛼_𝑛} bounded above from one, then {𝑥_𝑛} defined by (1.3) converges strongly to 𝑃_𝐹(𝑇)𝑥. Takahashi et al. [6] modified (1.3) so-called the shrinking projection method for a countable family of nonexpansive mappings {𝑇_𝑛}^∞_𝑛=1 as follows:𝑥₁𝐶=𝑥∈𝐻,₁𝑦=𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶‖𝑦_𝑛−𝑧‖≤‖𝑥_𝑛,𝑥−𝑧‖_𝑛+1=𝑃_𝐶𝑛+1𝑥,𝑛=1,2,3,…, (1.4) and prove that if the sequence {𝛼_𝑛} bounded above from one, then {𝑥_𝑛} defined by (1.4) converges strongly to 𝑃_∩^∞_{𝑛=1𝐹(𝑇𝑛)}𝑥.

Recently, the present authors [7] extended (1.3) to obtain a strong convergence theorem for common fixed points of a countable family of 𝐿_𝑛-Lipschitzian mappings {𝑇_𝑛}^∞_𝑛=1 by 𝑥₁𝑦=𝑥∈𝐶isarbitrary,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=1,2,3,…, (1.5) where 𝜃_𝑛=(1−𝛼_𝑛)(𝐿²_𝑛−1)(diam𝐶)²→0 as 𝑛→∞ and prove that {𝑥_𝑛} defined by (1.5) converges strongly to 𝑃_∩^∞_{𝑛=1𝐹(𝑇𝑛)}𝑥.

In this paper, we establish strong convergence theorems for finding common fixed points of a countable family of Lipschitzian mappings in a real Hilbert space. Moreover, we also apply our results for asymptotically nonexpansive mappings.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Then,‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²,()−2𝑥−𝑦,𝑦‖𝜆𝑥+1−𝜆𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖², (2.1)(2.2) for all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1]. For any 𝑛 points 𝑥₁,𝑥₂,…,𝑥_𝑛 in 𝐻, the following generalized identity holds:^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²=^𝑛_𝑖=1𝜆_𝑖‖𝑥_𝑖‖²−_𝑖<𝑗𝜆_𝑖𝜆_𝑗‖𝑥_𝑖−𝑥_𝑗‖², (2.3) where 𝜆_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝜆_𝑖=1.

We write 𝑥_𝑛→𝑥 (𝑥_𝑛⇀𝑥, resp.) if {𝑥_𝑛} converges strongly (weakly, resp.) to 𝑥. It is also known that 𝐻 satisfies: (1)the Opial's condition [8] that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖ (2.4) holds for every 𝑦∈𝐻 with 𝑦≠𝑥(2)the Kadec-Klee property [9, 10]; that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖ together imply 𝑥_𝑛→𝑥.

Let 𝐶 be a nonempty closed convex subset of 𝐻. Then, for any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖∀𝑦∈𝐶. (2.5) Such a mapping 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is nonexpansive. Furthermore, for 𝑥∈𝐻 and 𝑧∈𝐶, 𝑧=𝑃_𝐶𝑥iff𝑥−𝑧,𝑧−𝑦≥0,∀𝑦∈𝐶. (2.6) To deal with a family of mappings, the following conditions are introduced: let 𝐶 be a subset of a Banach space, let {𝑇_𝑛} and 𝒯 be families of mappings of 𝐶 with ⋂^∞_𝑛=1𝐹(𝑇_𝑛)=𝐹(𝒯)≠∅, where 𝐹(𝒯) is the set of all common fixed points of all mappings in 𝒯. {𝑇_𝑛} is said to satisfy (a)the AKTT-condition [11] if for each bounded subset 𝐵 of 𝐶, ^∞_𝑛=1sup‖𝑇_𝑛+1𝑧−𝑇_𝑛𝑧‖∶𝑧∈𝐵<∞, (2.7)(b)the NST-condition (I) with 𝒯 [12] if for each bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0implieslim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0∀𝑇∈𝒯, (2.8)(c)the NST-condition (II) [12] if for each bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞‖𝑧_𝑛+1−𝑇_𝑛𝑧_𝑛‖=0implieslim_𝑛→∞‖𝑧_𝑛−𝑇_𝑚𝑧_𝑛‖=0∀𝑚∈ℕ, (2.9)(d)NST*-condition with 𝒯 [13] if for each bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0,lim_𝑛→∞‖𝑧_𝑛−𝑧_𝑛+1‖=0, (2.10) imply lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0 for all 𝑇∈𝒯.

In particular, if 𝒯={𝑇}, then we simply say that {𝑇_𝑛} satisfies the NST-condition (I) with 𝑇 (NST*-condition with 𝑇, resp.) rather than NST-condition (I) with {𝑇} (NST*-condition with {𝑇}, resp.).

Remark 2.1. It follows directly from the definitions above that (i)if {𝑇_𝑛} satisfies the NST-condition (I) with 𝒯, then {𝑇_𝑛} satisfies the NST*-condition with 𝒯(ii)if {𝑇_𝑛} satisfies the NST-condition (II), then {𝑇_𝑛} satisfies the NST*-condition with {𝑇_𝑛}.

Lemma 2.2 . (see [11, Lemma 3.2]). Let 𝐶 be a nonempty closed subset of a Banach space, and let {𝑇_𝑛} be a family of mappings of 𝐶 into itself which satisfies the AKTT-condition, then the mapping 𝑇∶𝐶→𝐶 defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥∀𝑥∈𝐶 (2.11) satisfies lim_𝑛→∞sup‖𝑇𝑧−𝑇_𝑛𝑧‖∶𝑧∈𝐵=0, (2.12) for each bounded subset 𝐵 of 𝐶.

From now on, we will write ({𝑇_𝑛},𝑇) satisfies AKTT-condition if {𝑇_𝑛} satisfies AKTT-condition and 𝑇 is defined by (2.11).

Lemma 2.3 . (see [13, Lemma 2.6]). Let 𝐶 be a nonempty closed subset of a Banach space. Suppose that ({𝑇_𝑛},𝑇) satisfies AKTT-condition and 𝐹(𝑇)=⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Then, {𝑇_𝑛} satisfies the NST-condition (I) with 𝑇. Consequently, {𝑇_𝑛} satisfies the NST*-condition with 𝑇.

Remark 2.4. There are families of mappings {𝑇_𝑛} and 𝒯 such that (1){𝑇_𝑛} satisfies the NST*-condition with 𝒯, and (2){𝑇_𝑛} fails the NST-condition (I) with 𝒯 and the NST-condition (II).

The following example shows that the NST*-condition with 𝒯 is strictly weaker than NST-condition (I) with 𝒯 and the NST-condition (II).

Example 2.5 (see [13, Example 2.9]). Let 𝐻∶=ℝ² and 𝐶∶=[0,1]×[0,1]. Define 𝑇₁,𝑇₂∶𝐶→𝐶 as follows: 𝑇₁𝑥,𝑦=𝑥,1−𝑦,𝑇₂𝑥,𝑦=1−𝑥,𝑦, (2.13) for all (𝑥,𝑦)∈𝐶. Hence, 𝑇₁ and 𝑇₂ are nonexpansive mappings with 𝐹𝑇₁∩𝐹𝑇₂=[]×10,12∩12×[=10,1]2,12≠∅. (2.14) Let 𝑇_𝑛=𝑇_{𝑛−1(mod2)+1}. Then, {𝑇_𝑛} satisfies NST*-condition but it fails NST-condition (I) with 𝒯 and the NST-condition (II).

Lemma 2.6. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛} and {𝑆_𝑛} be two families of 𝑡_𝑛-Lipschitzian and 𝑠_𝑛-Lipschitzian mappings of 𝐶 into itself, respectively. Let {𝑈_𝑛} be a family of mappings of 𝐶 into itself defined by 𝑈_𝑛=𝑇_𝑛𝛽_𝑛𝐼+1−𝛽_𝑛𝑆_𝑛∀𝑛∈ℕ, (2.15) where {𝛽_𝑛} is a sequence in [𝑎,𝑏] for some 𝑎,𝑏∈(0,1) and 𝐼 is an identity mapping. Assume that {𝑡_𝑛} and {𝑠_𝑛} are two sequences such that 𝑡_𝑛→1 and 𝑠_𝑛→1. Then, the following statements hold. (i){𝑈_𝑛} is a family of 𝐿_𝑛-Lipschitzian mappings of 𝐶 into itself, where 𝐿_𝑛=(𝛽_𝑛𝑡²_𝑛+(1−𝛽_𝑛)𝑡²_𝑛𝑠²_𝑛)^1/2 and 𝐿_𝑛→1. (ii)Suppose that 𝒯₁ and 𝒯₂ are families of mappings of 𝐶 into itself such that 𝐹(𝒯₁)=⋂^∞_𝑛=1𝐹(𝑇_𝑛), 𝐹(𝒯₂)=⋂^∞_𝑛=1𝐹(𝑆_𝑛) and 𝐹(𝒯₁)∩𝐹(𝒯₂)≠∅. If {𝑇_𝑛} and {𝑆_𝑛} satisfy the NST*-condition with 𝒯₁ and 𝒯₂, respectively, then {𝑈_𝑛} satisfies the NST*-condition with 𝒯₁∪𝒯₂ and ^∞_𝑛=1𝐹𝑈_𝑛=𝐹𝒯₁∩𝐹𝒯₂. (2.16)

Proof. (i) We first observe that ‖𝑈_𝑛𝑥−𝑈_𝑛𝑦‖²≤𝑡²_𝑛‖𝛽_𝑛𝑥−𝑦+1−𝛽_𝑛𝑆_𝑛𝑥−𝑆_𝑛𝑦‖²≤𝑡²_𝑛𝛽_𝑛‖𝑥−𝑦‖²+1−𝛽_𝑛‖𝑆_𝑛𝑥−𝑆_𝑛𝑦‖²≤𝛽_𝑛𝑡²_𝑛‖𝑥−𝑦‖²+1−𝛽_𝑛𝑡²_𝑛𝑠²_𝑛‖𝑥−𝑦‖²=𝐿²_𝑛‖𝑥−𝑦‖², (2.17) for all 𝑥,𝑦∈𝐶. That is, 𝑈_𝑛 is 𝐿_𝑛-Lipschitzian. Since 𝑡_𝑛→1 and 𝑠_𝑛→1, it follows that 𝐿_𝑛→1.
(ii) Let {𝑧_𝑛} be a bounded sequence in 𝐶 such that lim_𝑛→∞‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖=lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Let 𝑝∈𝐹(𝒯₁)∩𝐹(𝒯₂), and let 𝑀=sup{‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖,‖𝑧_𝑛−𝑝‖∶𝑛∈ℕ}. Then ‖𝑧_𝑛−𝑝‖²≤‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+‖𝑈_𝑛𝑧_𝑛−𝑝‖²=‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖²+2𝑧_𝑛−𝑈_𝑛𝑧_𝑛𝑈_𝑛𝑧_𝑛−𝑝+‖𝑈_n𝑧_𝑛−𝑝‖²≤3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+‖𝑇_𝑛𝛽_𝑛𝑧_𝑛+1−𝛽_𝑛𝑆_𝑛𝑧_𝑛−𝑇_𝑛𝑝‖²≤3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+𝑡²_𝑛‖𝛽_𝑛𝑧_𝑛−𝑝+1−𝛽_𝑛𝑆_𝑛𝑧_𝑛−𝑝‖²=3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+𝛽_𝑛𝑡²_𝑛‖𝑧_𝑛−𝑝‖²+1−𝛽_𝑛𝑡²_𝑛‖𝑆_𝑛𝑧_𝑛−𝑝‖²−𝛽_𝑛1−𝛽_𝑛𝑡²_𝑛‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖²≤3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+𝛽_𝑛𝑡²_𝑛‖𝑧_𝑛−𝑝‖²+1−𝛽_𝑛𝑡²_𝑛𝑠²_𝑛‖𝑧_𝑛−𝑝‖²−𝑎(1−𝑏)‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖²=3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+𝐿²_𝑛‖𝑧_𝑛−𝑝‖²−𝑎(1−𝑏)‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖², (2.18) for all 𝑛∈ℕ. In particular, 𝑎(1−𝑏)‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖²≤3𝑀‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+𝐿²_𝑛−1‖𝑧_𝑛−𝑝‖². (2.19) So, we get lim_𝑛→∞‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖=0. (2.20) Since {𝑆_𝑛} satisfies the NST*-condition with 𝒯₂, we have lim_𝑛→∞‖𝑧_𝑛−𝑆𝑧_𝑛‖=0∀𝑆∈𝒯₂. (2.21) Since ‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖≤‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+‖𝑈_𝑛𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖≤‖𝑧_𝑛−𝑈_𝑛𝑧_𝑛‖+1−𝛽_𝑛𝑡_𝑛‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖, (2.22) it follows that lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0. (2.23) Since {𝑇_𝑛} satisfies the NST*-condition with 𝒯₁, we have lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0∀𝑇∈𝒯₁. (2.24) It is easy to see that 𝐹(𝒯₁)∩𝐹(𝒯₂)⊂⋂^∞_𝑛=1𝐹(𝑈_𝑛). To see the reverse inclusion, let 𝑧∈⋂^∞_𝑛=1𝐹(𝑈_𝑛) follow the first part of the proof above but now let 𝑧_𝑛≡𝑧. Then, 𝑧∈𝐹(𝒯₁)∩𝐹(𝒯₂)=𝐹(𝒯₁∪𝒯₂). Hence, {𝑈_𝑛} satisfies the NST*-condition with 𝒯₁∪𝒯₂.

Lemma 2.7. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛𝑖}^∞_𝑛=1 be families of 𝐿_𝑛𝑖-Lipschitzian mappings of 𝐶 into itself for 𝑖=1,2,…,𝑟, respectively. Let {𝑇_𝑛} be a family of mappings of 𝐶 into itself defined by 𝑇_𝑛=^𝑟_𝑖=1𝛽_𝑛𝑖𝑇_𝑛𝑖∀𝑛∈ℕ, (2.25) where {𝛽_𝑛𝑖}^∞_𝑛=1 are sequences in [𝑎,𝑏] for some 𝑎,𝑏∈(0,1) satisfying ∑^𝑟_𝑖=1𝛽_𝑛𝑖=1 for all 𝑛∈ℕ. Assume that {𝐿_𝑛𝑖}^∞_𝑛=1 are sequences such that 𝐿_𝑛𝑖→1 as 𝑛→∞ for all 𝑖=1,2,…,𝑟. Then, the following statements hold. (i){𝑇_𝑛} is a family of 𝐿_𝑛-Lipschitzian mappings of 𝐶 into itself, where 𝐿_𝑛=(∑^𝑟_𝑖=1𝛽_𝑛𝑖𝐿²_𝑛𝑖)^1/2 and 𝐿_𝑛→1. (ii)Suppose that 𝒯_𝑖 are families of mappings of 𝐶 into itself such that 𝐹(𝒯_𝑖)=⋂^∞_𝑛=1𝐹(𝑇_𝑛𝑖) for 𝑖=1,2,…,𝑟 and ⋂^𝑟_𝑖=1𝐹(𝒯_𝑖)≠∅. If {𝑇_𝑛𝑖} satisfies the NST*-condition with 𝒯_𝑖 for all 𝑖=1,2,…,𝑟, then {𝑇_𝑛} satisfies the NST*-condition with ⋃^𝑟_𝑖=1𝒯_𝑖 and ^∞_𝑛=1𝐹𝑇_𝑛=^𝑟_𝑖=1𝐹𝒯_𝑖. (2.26)

Proof. (i) From (2.3), we have ‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖²=^𝑟_𝑖=1𝛽_𝑛𝑖(𝑇_𝑛𝑖𝑥−𝑇_𝑛𝑖𝑦)²≤^𝑟_𝑖=1𝛽_𝑛𝑖‖𝑇_𝑛𝑖𝑥−𝑇_𝑛𝑖𝑦‖²≤^𝑟_𝑖=1𝛽_𝑛𝑖𝐿²_𝑛𝑖‖𝑥−𝑦‖²=𝐿²_𝑛‖𝑥−𝑦‖², (2.27) for all 𝑥,𝑦∈𝐶. That is, 𝑇_𝑛 is 𝐿_𝑛-Lipschitzian. Since 𝐿_𝑛𝑖→1 for 𝑖=1,2,…,𝑟 and ∑^𝑟_𝑖=1𝛽_𝑛𝑖=1, it follows that 𝐿_𝑛→1.
(ii) Let {𝑧_𝑛} be a bounded sequence in 𝐶 such that lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. Let 𝑝∈⋂^𝑟_𝑖=1𝐹(𝒯_𝑖), and let 𝑀=sup{‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖,‖𝑧_𝑛−𝑝‖∶𝑛∈ℕ}; it follows from (2.3) that ‖𝑧_𝑛−𝑝‖²≤‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+‖𝑇_𝑛𝑧_𝑛−𝑝‖²=‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖²+2𝑧_𝑛−𝑇_𝑛𝑧_𝑛𝑇_𝑛𝑧_𝑛−𝑝+‖𝑇_𝑛𝑧_𝑛−𝑝‖²≤3𝑀‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+^𝑟_𝑖=1𝛽_𝑛𝑖(𝑇_𝑛𝑖𝑧_𝑛−𝑝)²=3𝑀‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+^𝑟_𝑖=1𝛽_𝑛𝑖‖𝑇_𝑛𝑖𝑧_𝑛−𝑝‖²−_𝑖<𝑗𝛽_𝑛𝑖𝛽_𝑛𝑗‖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑗𝑧_𝑛‖²≤3𝑀‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+^𝑟_𝑖=1𝛽_𝑛𝑖𝐿²_𝑛𝑖‖𝑧_𝑛−𝑝‖²−𝑎²_𝑖<𝑗‖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑗𝑧_𝑛‖²=3𝑀‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+𝐿²_𝑛‖𝑧_𝑛−𝑝‖²−𝑎²_𝑖<𝑗‖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑗𝑧_𝑛‖². (2.28) So, by (i), we get lim_𝑛→∞‖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑗𝑧_𝑛‖=0∀𝑖,𝑗∈{1,2,…,𝑟}. (2.29) For each 𝑘=1,2,…,𝑟, we have ‖𝑧_𝑛−𝑇_𝑛𝑘𝑧_𝑛‖≤‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+‖𝑇_𝑛𝑧_𝑛−𝑇_𝑛𝑘𝑧_𝑛‖=‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+^𝑟_𝑖=1𝛽_𝑛𝑖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑘𝑧_𝑛≤‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖+^𝑟_𝑖=1𝛽_𝑛𝑖𝑇_𝑛𝑖𝑧_𝑛−𝑇_𝑛𝑘𝑧_𝑛⟶0. (2.30) Since each family {𝑇_𝑛𝑘}^∞_𝑛=1 satisfies the NST*-condition with 𝒯_𝑘, lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0∀𝑇∈^𝑟_𝑖=1𝒯_𝑖. (2.31) It is easy to see that ⋂^𝑟_𝑖=1𝐹(𝒯_𝑖)⊂⋂^∞_𝑛=1𝐹(𝑇_𝑛). To see the reverse inclusion, let 𝑧∈⋂^∞_𝑛=1𝐹(𝑇_𝑛). Follow the first part of the proof above but now let 𝑧_𝑛≡𝑧. Then, 𝑧∈⋂^𝑟_𝑖=1𝐹(𝒯_𝑖)=𝐹(∪^𝑟_𝑖=1𝒯_𝑖). Hence, {𝑇_𝑛} satisfies the NST*-condition with ⋃^𝑟_𝑖=1𝒯_𝑖.

Lemma 2.8. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let {𝑇_𝑛} be a family of 𝐿_𝑛-Lipschitzian mappings of 𝐶 into itself with 𝐿_𝑛→1 and ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. If {𝑇_𝑛} satisfies the NST*-condition with 𝒯, where 𝒯 is a family of mappings of 𝐶 into itself such that 𝐹(𝒯)=⋂^∞_𝑛=1𝐹(𝑇_𝑛), then 𝐹(𝒯) is closed and convex.

Proof. It follows from the continuity of 𝑇_𝑛 that 𝐹(𝑇_𝑛) is closed and so is 𝐹(𝒯)=⋂^∞_𝑛=1𝐹(𝑇_𝑛). Now, we prove that 𝐹(𝒯) is convex. To this end, let 𝑥,𝑦∈𝐹(𝒯). Put 𝑧=𝑡𝑥+(1−𝑡)𝑦, where 𝑡∈(0,1). From (2.2), we have ‖𝑧−𝑇_𝑛𝑧‖²=𝑡‖𝑥−𝑇_𝑛𝑧‖²+(1−𝑡)‖𝑦−𝑇_𝑛𝑧‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²≤𝑡𝐿²_𝑛‖𝑥−𝑧‖²+(1−𝑡)𝐿²_𝑛‖𝑦−𝑧‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²=𝑡(1−𝑡)𝐿²_𝑛−1‖𝑥−𝑦‖². (2.32) So, we get lim_𝑛→∞‖𝑧−𝑇_𝑛𝑧‖=0. (2.33) Since {𝑇_𝑛} satisfies the NST*-condition with 𝒯, we have ‖𝑧−𝑇𝑧‖=0 for all 𝑇∈𝒯. Then, 𝑧∈𝐹(𝒯) and so 𝐹(𝒯) is convex.

Remark 2.9. The conclusions of Lemmas 2.6, 2.7, and 2.8 remain true if we replace a Hilbert space with a uniformly convex Banach space. Recall a Banach space 𝑋 is uniformly convex if for any 𝜀>0, there exists 𝛿>0 such that ‖𝑥‖=‖𝑦‖=1 and ‖𝑥−𝑦‖≥𝜀 imply ‖(𝑥+𝑦)/2‖≤1−𝛿.

3. Main Results

In this section, using the method introduced by Takahashi et al. [6], we obtain a strong convergence theorem for a countable family of Lipschitzian mappings.

Recall that a mapping 𝑇∶𝐶→𝐶 is closed (demiclosed, resp.) at 𝑦 if whenever {𝑥_𝑛} is a sequence in 𝐶 satisfying 𝑥_𝑛→𝑥 (𝑥_𝑛⇀𝑥, resp.) and 𝑇𝑥_𝑛→𝑦, then 𝑥∈𝐶 and 𝑇𝑥=𝑦.

Theorem 3.1. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛} be a family of 𝐿_𝑛-Lipschitzian mappings of 𝐶 into itself with a common fixed point. Assume that {𝛼_𝑛} is a sequence in [0,𝑏] for some 𝑏∈(0,1). For 𝑥₁=𝑥∈𝐻 and 𝐶₁=𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥,𝑛=1,2,3,…, (3.1) where 𝜃_𝑛=1−𝛼_𝑛𝐿²_𝑛−1(diam𝐶)²⟶0as𝑛⟶∞. (3.2) Suppose that 𝒯 is a family of mappings of 𝐶 into itself such that 𝐹(𝒯)=⋂^∞_𝑛=1𝐹(𝑇_𝑛) and 𝐼−𝑇 is closed at 0 for all 𝑇∈𝒯. If {𝑇_𝑛} satisfies the NST*-condition with 𝒯, then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝒯)𝑥.

Proof. By Lemma 2.8, we have 𝐹(𝒯) is closed and convex. We now prove that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ by induction. It is obvious that 𝐶₁=𝐶 is closed and convex. Assume that 𝐶_𝑘 is closed and convex for some 𝑘∈ℕ. For 𝑧∈𝐶_𝑘, we know that ‖𝑦_𝑘−𝑧‖²≤‖𝑥_𝑘−𝑧‖²+𝜃_𝑘 (3.3) is equivalent to 2⟨𝑥_𝑘−𝑦_𝑘,𝑧⟩≤‖𝑥_𝑘‖²−‖𝑦_𝑘‖²+𝜃_𝑘. (3.4) It follows that 𝐶_𝑘+1 is closed and convex. Next, we show that 𝐹(𝒯)⊂𝐶_𝑛∀𝑛∈ℕ. (3.5) It is clear that 𝐹(𝒯)⊂𝐶₁=𝐶. Suppose that 𝐹(𝒯)⊂𝐶_𝑘 for some 𝑘∈ℕ. Then, for 𝑝∈𝐹(𝒯), ‖𝑦_𝑘−𝑝‖²=‖𝛼_𝑘𝑥_𝑘+1−𝛼_𝑛𝑇_𝑘𝑥_𝑘−𝑝‖²≤𝛼_𝑘‖𝑥_𝑘−𝑝‖²+1−𝛼_𝑘‖𝑇_𝑘𝑥_𝑘−𝑝‖²≤𝛼_𝑘‖𝑥_𝑘−𝑝‖²+1−𝛼_𝑘𝐿²_𝑘‖𝑥_𝑘−𝑝‖²=‖𝑥_𝑘−𝑝‖²+1−𝛼_𝑘𝐿²_𝑘−1‖𝑥_𝑘−𝑝‖²≤‖𝑥_𝑘−𝑝‖²+𝜃_𝑘, (3.6) we have 𝑝∈𝐶_𝑘+1. Therefore, we obtain (3.5). Now, the sequence {𝑥_𝑛} is well defined. As 𝑥_𝑛=𝑃_𝐶𝑛𝑥, ‖𝑥_𝑛−𝑥‖≤‖𝑧−𝑥‖∀𝑧∈𝐶_𝑛,∀𝑛∈ℕ. (3.7) In particular, since 𝐹(𝒯)⊂𝐶_𝑛, ‖𝑥_𝑛−𝑥‖≤‖𝑝−𝑥‖∀𝑝∈𝐹(𝒯),∀𝑛∈ℕ. (3.8) On the other hand, from 𝑥_𝑛=𝑃_𝐶𝑛𝑥 and 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛, we have ‖𝑥_𝑛−𝑥‖≤‖𝑥_𝑛+1−𝑥‖∀𝑛∈ℕ. (3.9) Therefore, {‖𝑥_𝑛−𝑥‖} is nondecreasing and bounded. So, lim_𝑛→∞‖𝑥_𝑛−𝑥‖exists. (3.10) Noticing again that 𝑥_𝑛=𝑃_𝐶𝑛𝑥 and for any positive integer 𝑘, 𝑥_𝑛+𝑘∈𝐶_{𝑛+𝑘−1}⊂𝐶_𝑛, we have ⟨𝑥_𝑛−𝑥_𝑛+𝑘,𝑥−𝑥_𝑛⟩≥0. (3.11) It follows from (2.1) that ‖𝑥_𝑛+𝑘−𝑥_𝑛‖²=‖𝑥_𝑛+𝑘−𝑥−𝑥_𝑛−𝑥‖²=‖𝑥_𝑛+𝑘−𝑥‖²−‖𝑥_𝑛−𝑥‖²−2⟨𝑥_𝑛+𝑘−𝑥_𝑛,𝑥_𝑛−𝑥⟩≤‖𝑥_𝑛+𝑘−𝑥‖²−‖𝑥_𝑛−𝑥‖². (3.12) It then follows from the existence of lim_𝑛→∞‖𝑥_𝑛−𝑥‖² that {𝑥_𝑛} is a Cauchy sequence. Moreover, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.13) We now assume that 𝑥_𝑛→𝑤 for some 𝑤∈𝐶. Now, since 𝑥_𝑛+1∈𝐶_𝑛+1 and 𝐶_𝑛+1⊂𝐶_𝑛, ‖𝑦_𝑛−𝑥_𝑛+1‖²≤‖𝑥_𝑛−𝑥_𝑛+1‖²+𝜃_𝑛 which implies that ‖𝑦_𝑛−𝑥_𝑛+1‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+𝜃_𝑛⟶0. (3.14) From 𝛼_𝑛≤𝑏<1, we get ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛1‖=1−𝛼_𝑛‖𝑦_𝑛−𝑥_𝑛‖≤11−𝑏‖𝑦_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛−𝑥_𝑛+1‖⟶0. (3.15) Since {𝑇_𝑛} satisfies the NST*-condition with 𝒯, we have lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0∀𝑇∈𝒯. (3.16) Since 𝐼−𝑇 is closed at 0 for all 𝑇∈𝒯, we have (𝐼−𝑇)𝑤=0. This implies that 𝑤∈𝐹(𝒯). Furthermore, by (3.8), ‖𝑤−𝑥‖=lim_𝑛→∞‖𝑥_𝑛−𝑥‖≤‖𝑝−𝑥‖∀𝑝∈𝐹(𝒯). (3.17) Hence, 𝑤=𝑃_𝐹(𝒯)𝑥. This completes the proof.

Lemma 3.2 . (see [9, Theorem 10.4]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then, 𝐼−𝑇 is demiclosed at 0.

It is not difficult to see from the proof of Theorem 3.1 that the boundedness of 𝐶 can be discarded if {𝑇_𝑛} is a family of nonexpansive mappings. So, we immediately obtain the following theorem.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛} and 𝒯 be two families of nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑛=1𝐹(𝑇_n)=𝐹(𝒯)≠∅ and suppose that {𝑇_𝑛} satisfies the NST*-condition with 𝒯. Assume that {𝛼_𝑛} is a sequence in [0,𝑏] for some 𝑏∈(0,1). Then, the sequence {𝑥_𝑛} in 𝐶 defined by (1.4) converges strongly to 𝑃_𝐹(𝒯)𝑥.

Remark 3.4. Theorem 3.3 includes [6, Theorem 3.3] as a special case since the NST-condition (I) with 𝒯 implies the NST*-condition with 𝒯.

Theorem 3.5. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛} be a family of 𝐿_𝑛-Lipschitzian mappings of 𝐶 into itself with a common fixed point. Suppose that 𝒯 is a family of mappings from 𝐶 into itself such that 𝐹(𝒯)=⋂^∞_𝑛=1𝐹(𝑇_𝑛) and 𝐼−𝑇 is demiclosed at 0 for all 𝑇∈𝒯. Assume that {𝛼_𝑛} is a sequence in [0,𝑏] for some 𝑏∈(0,1). If {𝑇_𝑛} satisfies the NST*-condition with 𝒯, then the sequence {𝑥_𝑛} in 𝐶 defined by (1.5) converges strongly to 𝑃_𝐹(𝒯)𝑥.

Proof. The proof is analogous to the proof of [7, Theorem 10] and Theorem 3.1, so it is omitted.

4. Deduced Results

Let 𝐶 be a subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is said to be an asymptotically nonexpansive if there exists a sequence {𝑘_𝑛} of real numbers such that 𝑘_𝑛∈[1,∞), 𝑘_𝑛→1, and ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶,𝑛∈ℕ. (4.1) The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] as an important generalization of the class of nonexpansive mappings. They proved that if 𝐶 is nonempty bounded closed convex and 𝑇 is an asymptotically nonexpansive self-mapping of 𝐶, then 𝑇 has a fixed point.

In this section, we use the NST*-condition to obtain recent results proved by Kim and Xu [15], Plubtieng and Ungchittrakool [16], and Zegeye and Shahzad [17, 18]. We start with the following auxiliary result.

Lemma 4.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝑇 be an asymptotically nonexpansive mappings of 𝐶 into itself with a sequence {𝑘_𝑛} in [1,∞) satisfying 𝑘_𝑛→1 and 𝐹(𝑇)≠∅. Then, {𝑇^𝑛} is a family of 𝑘_𝑛-Lipschitzian mappings of 𝐶 into itself and satisfies the NST*-condition with 𝑇.

Proof. We note that {𝑇^𝑛} is a family of 𝑘_𝑛-Lipschitzian mappings of 𝐶 into itself. Let {𝑧_𝑛} be a bounded sequence in 𝐶 such that lim_𝑛→∞‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖=0,lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=0. (4.2) Since ‖𝑧_𝑛+1−𝑇𝑧_𝑛+1‖≤‖𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛+1‖+‖𝑇^𝑛+1𝑧_𝑛+1−𝑇𝑧_𝑛+1‖≤‖𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛+1‖+𝑘₁‖𝑇^𝑛𝑧_𝑛+1−𝑧_𝑛+1‖≤‖𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛+1‖+𝑘₁‖𝑇^𝑛𝑧_𝑛+1−𝑇^𝑛𝑧_𝑛‖+‖𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖+‖𝑧_𝑛+1−𝑧_𝑛‖≤‖𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛+1‖+𝑘₁𝑘_𝑛+1‖𝑧_𝑛+1−𝑧_𝑛‖+𝑘₁‖𝑇^𝑛𝑧_𝑛−𝑧_𝑛‖, (4.3) it follows that lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0. (4.4) It is easy to see that 𝐹(𝑇)⊂⋂^∞_𝑛=1𝐹(𝑇^𝑛). To see the reverse inclusion, let 𝑧∈⋂^∞_𝑛=1𝐹(𝑇^𝑛) following from the first part of the proof above, but now let 𝑧_𝑛≡𝑧. Then, 𝑧∈⋂^∞_𝑛=1𝐹(𝑇^𝑛), and hence ⋂^∞_𝑛=1𝐹(𝑇^𝑛)⊂𝐹(𝑇). This implies that {𝑇^𝑛} satisfies the NST*-condition with 𝑇.

Lemma 4.2 . (see [19]). Let 𝐶 be a nonempty bounded closed convex subset of a Hilbert space 𝐻, and let 𝑇 be an asymptotically nonexpansive mappings of 𝐶 into itself. Then, 𝐼−𝑇 is demiclosed at 0.

Using Theorem 3.1 and Lemmas 2.6 and 4.1, we have the following result.

Theorem 4.3. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻, and let 𝑆, 𝑇 be two asymptotically nonexpansive mappings of 𝐶 into itself with sequences {𝑠_𝑛} and {𝑡_𝑛}, respectively, and 𝐹(𝑆)∩𝐹(𝑇)≠∅. Assume that {𝛼_𝑛} is a sequence in [0,𝑏] and {𝛽_𝑛} is a sequence in [𝑎,𝑏] for some 𝑎,𝑏∈(0,1). For 𝑥₁=𝑥∈𝐻 and 𝐶₁=𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑧_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆^𝑛𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥,𝑛=1,2,3,…, (4.5) where 𝜃_𝑛=1−𝛼_𝑛𝑡²_𝑛−1+1−𝛽_𝑛𝑡²_𝑛𝑠²_𝑛−1(diam𝐶)²⟶0as𝑛⟶∞. (4.6) Then, {𝑥_𝑛} converges strongly to 𝑃_{𝐹(𝑆)∩𝐹(𝑇)}𝑥.

Using Theorem 3.5 and Lemmas 2.6 and 4.1, we have the following result.

Theorem 4.4 . (see [16, Theorem 3.1]). Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻, and let 𝑆, 𝑇 be two asymptotically nonexpansive mappings of 𝐶 into itself with sequences {𝑠_𝑛} and {𝑡_𝑛}, respectively, and 𝐹(𝑆)∩𝐹(𝑇)≠∅. Assume that {𝛼_𝑛} is a sequence in [0,𝑏] and {𝛽_𝑛} is a sequence in [𝑎,𝑏] for some 𝑎,𝑏∈(0,1). For 𝑥₁=𝑥∈𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑧_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆^𝑛𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=1,2,3,…, (4.7) where 𝜃_𝑛=1−𝛼_𝑛𝑡²_𝑛−1+1−𝛽_𝑛𝑡²_𝑛𝑠²_𝑛−1(diam𝐶)²⟶0as𝑛⟶∞. (4.8) Then, {𝑥_𝑛} converges strongly to 𝑃_{𝐹(𝑆)∩𝐹(𝑇)}𝑥.

Using Theorem 3.1 and Lemmas 2.7 and 4.1, we have the following result.

Theorem 4.5. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^𝑟_𝑖=1 be a finite family of asymptotically nonexpansive mappings of 𝐶 into itself with sequences {𝑘_𝑛𝑖} for 𝑖=1,2,…,𝑟, respectively, and suppose that ⋂^𝑟_𝑖=1𝐹(𝑇_𝑖)≠∅. Assume that {𝛼_𝑛𝑖}^∞_𝑛=1 are sequences in [0,1) such that 𝛼_𝑛0≤𝑏<1, 𝛼_𝑛𝑖≥𝑎>0 for some 𝑎,𝑏∈(0,1) and ∑^𝑟_𝑖=0𝛼_𝑛𝑖=1 for all 𝑛∈ℕ. For 𝑥₁=𝑥∈𝐻 and 𝐶₁=𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=𝛼_𝑛0𝑥_𝑛+𝛼_𝑛1𝑇^𝑛₁𝑥_𝑛+𝛼_𝑛2𝑇^𝑛₂𝑥_𝑛+⋯+𝛼_𝑛𝑟𝑇^𝑛_𝑟𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥,𝑛=1,2,3,…, (4.9) where 𝜃_𝑛=𝛼_𝑛1𝑘²_𝑛1−1+𝛼_𝑛2𝑘²_𝑛2−1+⋯+𝛼_𝑛𝑟𝑘²_𝑛𝑟−1(diam𝐶)²⟶0as𝑛⟶∞. (4.10) Then, {𝑥_𝑛} converges strongly to 𝑃_∩^𝑟_{𝑖=1𝐹(𝑇𝑖)}𝑥.

Using Theorem 3.5 and Lemmas 2.7 and 4.1, we have the following two results which were proved by Zegeye and Shahzad [17, 18].

Theorem 4.6. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^𝑟_𝑖=1 be a finite family of asymptotically nonexpansive mappings of 𝐶 with sequences {𝑘_𝑛𝑖} for 𝑖=1,2,…,𝑟, respectively, and suppose that ⋂^𝑟_𝑖=1𝐹(𝑇_𝑖)≠∅. Assume that {𝛼_𝑛𝑖}^∞_𝑛=1 are sequences in [0,1) such that 𝛼_𝑛0≤𝑏<1, 𝛼_𝑛𝑖≥𝑎>0 for some 𝑎,𝑏∈(0,1) and ∑^𝑟_𝑖=0𝛼_𝑛𝑖=1 for all 𝑛∈ℕ. For 𝑥₁=𝑥∈𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=𝛼_𝑛0𝑥_𝑛+𝛼_𝑛1𝑇^𝑛₁𝑥_𝑛+𝛼_𝑛2𝑇^𝑛₂𝑥_𝑛+⋯+𝛼_𝑛𝑟𝑇^𝑛_𝑟𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=1,2,3,…, (4.11) where 𝜃_𝑛=𝛼_𝑛1𝑘²_𝑛1−1+𝛼_𝑛2𝑘²_𝑛2−1+⋯+𝛼_𝑛𝑟𝑘²_𝑛𝑟−1(diam𝐶)²⟶0as𝑛⟶∞. (4.12) Then, {𝑥_𝑛} converges strongly to 𝑃_∩^𝑟_{𝑖=1𝐹(𝑇𝑖)}𝑥.

Theorem 4.7. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let 𝔗_𝑖={𝑇_𝑖(𝑡)∶𝑡∈ℝ⁺,𝑖=1,2,…,𝑟} be a finite family of asymptotically nonexpansive semigroups such that 𝐹=⋂^𝑟_𝑖=1𝐹(𝔗_𝑖)≠∅. Assume that {𝛼_𝑛𝑖}^∞_𝑛=1 are sequences in [0,1) such that 𝛼_𝑛0≤𝑏<1,𝛼_𝑛𝑖≥𝑎>0, (4.13) for some 𝑎,𝑏∈(0,1) and ∑^𝑟_𝑖=0𝛼_𝑛𝑖=1 for all 𝑛∈ℕ. Let {𝑡_𝑛𝑖,𝑖=1,2,…,𝑟} be finite positive and divergent real sequences. For 𝑥₁=𝑥∈𝐶, one defines a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=𝛼_𝑛0𝑥_𝑛+𝛼_𝑛11𝑡_𝑛1^𝑡𝑛1₀𝑇₁(𝑢)𝑥_𝑛𝑑𝑢+𝛼_𝑛21𝑡_𝑛2^𝑡𝑛2₀𝑇₂(𝑢)𝑥_𝑛𝑑𝑢+⋯+𝛼_𝑛𝑟1𝑡_𝑛𝑟^𝑡𝑛𝑟₀𝑇_𝑟(𝑢)𝑥_𝑛𝐶𝑑𝑢,_𝑛=𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=1,2,3,…, (4.14) where 𝜃_𝑛=𝛼_𝑛1𝐿²_𝑢1−1+𝛼_𝑛2𝐿²_𝑢2−1+⋯+𝛼_𝑛𝑟𝐿²_𝑢𝑟−1(diam𝐶)²⟶0as𝑛⟶∞, (4.15) with 𝐿_𝑢𝑖=(1/𝑡_𝑛𝑖)∫^𝑡𝑛𝑖₀𝐿^𝑇𝑖_𝑖𝑑𝑢. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥.

Acknowledgments

The authors would like to thank the referee for the insightful comments and suggestions. The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and Khon Kaen University under Grant no. RMU5380039.