Monotone Hybrid Projection Algorithms for an Infinitely Countable Family of Lipschitz Generalized Asymptotically Quasi-Nonexpansive Mappings

Abstract

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

1. Introduction

Let 𝐶 be a nonempty, closed, and convex subset of a real Banach space 𝐸. We denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. Then 𝑇 is said to be

(i)nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶;(ii)asymptotically nonexpansive if there exists a sequence 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ (1.1) for all 𝑥,𝑦∈𝐶 and 𝑛≥1;(iii)asymptotically quasi-nonexpansive if there exists a sequence 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and ‖𝑇^𝑛𝑥−𝑝‖≤𝑘_𝑛‖𝑥−𝑝‖ (1.2) for all 𝑥∈𝐶,𝑝∈𝐹(𝑇) and 𝑛≥1;(iv)generalized asymptotically nonexpansive [1] if there exist nonnegative real sequences {𝑘_𝑛} and {𝑐_𝑛} with 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑐_𝑛=0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖+𝑐_𝑛 (1.3) for all 𝑥,𝑦∈𝐶 and 𝑛≥1;(v)generalized asymptotically quasi-nonexpansive [1] if there exist nonnegative real sequences {𝑘_𝑛} and {𝑐_𝑛} with 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑐_𝑛=0 such that ‖𝑇^𝑛𝑥−𝑝‖≤𝑘_𝑛‖𝑥−𝑝‖+𝑐_𝑛 (1.4) for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇) and 𝑛≥1;(vi)asymptotically nonexpansive in the weak sense [2] if limsup_𝑛→∞sup_𝑦∈𝐶(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)≤0 (1.5) for all 𝑥∈𝐶.(vii)asymptotically nonexpansive in the intermediate sense [3] if limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)≤0; (1.6)(viii)uniformly L-Lipschitzian if there exists a constant 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖ (1.7) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.

It is clear that a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, asymptotically nonexpansive mappings, and nonexpansive mappings. However, the converse of each of above statement may be not true. The example shows that a generalized asymptotically quasi-nonexpansive mapping is not an asymptotically quasi-nonexpansive mapping; see [1]. Note that if 𝑇 is asymptotically nonexpansive in the weak sense, we have that

‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖+𝑐_𝑛 (1.8) for all 𝑥,𝑦∈𝐶, where 𝑐_𝑛=max{0,sup_{𝑥,𝑦∈𝐶}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)} so that lim_𝑛→∞𝑐_𝑛=0. Hence, 𝑇 is a generalized asymptotically nonexpansive mapping.

The mapping 𝑇∶𝐶→𝐶 is said to be demiclosed at 0 if for each sequence {𝑥_𝑛} in 𝐶 converging weakly to 𝑥 and {𝑇𝑥_𝑛} converging strongly to 0, we have 𝑇𝑥=0.

A Banach space 𝐸 is said to satisfy Opial's property, see [4], if for each 𝑥∈𝐸 and each sequence {𝑥_𝑛} weakly convergent to 𝑥, the following condition holds for all 𝑥≠𝑦:

liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖. (1.9)

Let 𝜏 be a Hausdorff linear topology and let 𝐸 satisfy the uniform 𝜏-Opial property. In 1993, Bruck, Kuczumow, and Reich proved that {𝑇^𝑛𝑥} is 𝜏-convergent if and only if {𝑇^𝑛𝑥} is 𝜏-asymptotically regular, that is,

𝑇^𝑛+1𝑥−𝑇^𝑛𝑥→−^𝜏0. (1.10) Moreover, they also proved that the 𝜏-limit of {𝑇^𝑛𝑥} is a fixed point of 𝑇.

In 1953, Mann [5] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping 𝑇 in a Hilbert space 𝐻:

𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇𝑥_𝑛,∀𝑛∈ℕ, (1.11) where the initial point 𝑥₀ is taken in 𝐶 arbitrarily and {𝛼_𝑛} is a sequence in [0,1].

However, we note that Mann's iteration process (1.11) has only weak convergence, in general; for instance, see [6–8].

In 2003, Nakajo and Takahashi [9] proposed the following modification of the Mann iteration for a single nonexpansive mapping 𝑇 in a Hilbert space. They proved the following theorem.

Theorem 1.1. Let 𝐶 be a closed and convex subset of a Hilbert space H and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠∅. Assume that {𝛼_𝑛}^∞_𝑛=0 is a sequence in [0,1] such that 𝛼_𝑛≤1−𝛿 for some 𝛿∈(0,1]. Define a sequence {𝑥_𝑛}^∞_𝑛=0 in C by the following algorithm: 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇𝑥_𝑛,𝐶_𝑛={𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖≤‖𝑥_𝑛𝑄−𝑧‖},_𝑛={𝑧∈𝐶∶⟨𝑥₀−𝑥_𝑛,𝑥_𝑛𝑥−𝑧⟩≥0},_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.12) Then {𝑥_𝑛} defined by (1.12) converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Recently, Kim and Xu [10] extended the result of Nakajo and Takahashi [9] from nonexpansive mappings to asymptotically nonexpansive mappings. They proved the following theorem.

Theorem 1.2. Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H and let 𝑇∶𝐶→𝐶 be an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛} such that 𝑘_𝑛→1 as 𝑛→∞. Assume that {𝛼_𝑛}^∞_𝑛=0 is a sequence in [0,1] such that limsup_𝑛→∞𝛼_𝑛<1. Define a sequence {𝑥_𝑛} in C by the following algorithm: 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛,𝐶_𝑛={𝑧∈𝐶∶‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜃_𝑛𝑄},_𝑛={𝑧∈𝐶∶⟨𝑥₀−𝑥_𝑛,𝑥_𝑛𝑥−𝑧⟩≥0},_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.13) where 𝜃_𝑛=(1−𝛼_𝑛)(𝑘²_𝑛−1)(diamC)²→0, as 𝑛→∞. Then {𝑥_𝑛} defined by (1.13) converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Since 2003, the strong convergence problems of the CQ method for fixed point iteration processes in a Hilbert space or a Banach space have been studied by many authors; see [9–20].

Let {𝑇_𝑖}^∞_𝑖=1 be an infinitely family of uniformly 𝐿_𝑖-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings and let 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖). In this paper, motivated by Kim and Xu [10] and Nakajo and Takahashi [9], we introduce two kinds of new algorithms for finding a common fixed point of a countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings which are modifications of the normal Mann iterative scheme. Our iterative schemes are defined as follows.

Algorithm 1.3. For an initial point 𝑥₀∈𝐶, compute the sequence {𝑥_𝑛} by the iterative process: 𝑦_𝑖,𝑛=𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛,𝐶_𝑖,𝑛={𝑧∈𝐶∶‖𝑦_𝑖,𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛𝐶},_𝑛=^∞_𝑖=1𝐶_𝑖,𝑛,𝑄₀𝑄=𝐶,_𝑛={𝑧∈𝑄_𝑛−1∶⟨𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛𝑥⟩≤0},𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0, (1.14) where 𝜃_𝑖,𝑛=(𝑘²_𝑖,𝑛−1)∇²_𝑛+2𝑘_𝑖,𝑛𝑐_𝑖,𝑛∇_𝑛+𝑐²_𝑖,𝑛, ∇_𝑛=sup_𝑛∈ℕ{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹}<∞.

Algorithm 1.4. For an initial point 𝑥₀∈𝐶, compute the sequence {𝑥_𝑛} by the iterative process: 𝐶_𝑖,0𝑦=𝐶,_𝑖,𝑛=𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛,𝐶_𝑖,𝑛+1={𝑧∈𝐶_𝑖,𝑛∶‖𝑦_𝑖,𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛𝐶},_𝑛+1=^∞_𝑖=1𝐶_𝑖,𝑛+1,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥0, (1.15) where 𝜃_𝑖,𝑛=(𝑘²_𝑖,𝑛−1)∇²_𝑛+2𝑘_𝑖,𝑛𝑐_𝑖,𝑛∇_𝑛+𝑐²_𝑖,𝑛, ∇_𝑛=sup_𝑛∈ℕ{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹}<∞.

2. Preliminaries

In this section, we present some useful lemmas which will be used in our main results.

Lemma 2.1 (see [21]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be three sequences of nonnegative numbers such that 𝑏_𝑛≥1 and 𝑎_𝑛+1≤𝑏_𝑛𝑎_𝑛+𝑐_𝑛,∀𝑛≥1, (2.1) if ∑^∞_𝑛=1𝑐_𝑛<∞ and ∑^∞_𝑛=1(𝑏_𝑛−1)<∞, then lim_𝑛→∞𝑎_𝑛 exists.

Lemma 2.2 (see [22]). Let 𝑝>1, 𝑟>0 be two fixed numbers. Then a Banach space 𝐸 is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that ‖𝜆𝑥+(1−𝜆)𝑦‖^𝑝≤𝜆‖𝑥‖^𝑝+(1−𝜆)‖𝑦‖^𝑝−𝜔_𝑝(𝜆)𝑔(‖𝑥−𝑦‖) (2.2) for all 𝑥,𝑦∈𝐵_𝑟(0)={𝑥∈𝑋∶‖𝑥‖≤𝑟} and 𝜆∈[0,1] where 𝜔_𝑝(𝜆)=𝜆(1−𝜆)^𝑝+𝜆^𝑝(1−𝜆).

Lemma 2.3. Let 𝐶 be a nonempty subset of a Banach space 𝐸 and let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian mapping. Let {𝑥_𝑛} be a sequence in 𝐶 such that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖=0. Then lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.

Proof. Since 𝑇 is uniformly 𝐿-Lipschitzian, we have ‖𝑥_𝑛−𝑇𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑇^𝑛+1𝑥_𝑛+1‖+‖𝑇^𝑛+1𝑥_𝑛+1−𝑇^𝑛+1𝑥_𝑛‖+‖𝑇^𝑛+1𝑥_𝑛−𝑇𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑇^𝑛+1𝑥_𝑛+1‖+𝐿‖𝑥_𝑛−𝑥_𝑛+1‖+𝐿‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖. (2.3) It follows that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.

The following lemmas give some characterizations and useful properties of the metric projection 𝑃_𝐶 in a Hilbert space.

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ and let 𝐶 be a closed and convex subset of 𝐻. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖,∀𝑦∈𝐶. (2.4)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. We know that 𝑃_𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶.

Lemma 2.4 (see [12]). Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻 and let 𝑃_𝐶 be the metric projection from 𝐻 onto 𝐶. Given 𝑥∈𝐻 and 𝑧∈𝐶, then 𝑧=𝑃_𝐶𝑥 if and only if the following holds: ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶. (2.5)

Lemma 2.5 (see [9]). Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻 and let 𝑃_𝐶∶𝐻→𝐶 be the matric projection from 𝐻 onto 𝐶. Then the following inequality holds: ‖𝑦−𝑃_𝐶𝑥‖²+‖𝑥−𝑃_𝐶𝑥‖²≤‖𝑥−𝑦‖²,∀𝑥∈𝐻,∀𝑦∈𝐶. (2.6)

Lemma 2.6 (see [12]). Let 𝐻 be a real Hilbert space. Then the following equations hold: (i)‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩, for all 𝑥,𝑦∈𝐻;(ii)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖², for all 𝑡∈[0,1] and 𝑥,𝑦∈𝐻.

Lemma 2.7 (see [10]). Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Given 𝑥,𝑦,𝑧∈𝐻 and also given 𝑎∈ℝ, the set {𝑣∈𝐶∶‖𝑦−𝑣‖²≤‖𝑥−𝑣‖²+⟨𝑧,𝑣⟩+𝑎} (2.7) is convex and closed.

Lemma 2.8. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences {𝑘_𝑛}, {𝑐_𝑛} such that 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑐_𝑛=0. Then 𝐹(𝑇) is a closed and convex subset of 𝐶.

Proof. Since 𝑇 is continuous, 𝐹(𝑇) is closed. Next, we show that 𝐹(𝑇) is convex. Let 𝑝₁,𝑝₂∈𝐹(𝑇) and 𝑡∈(0,1). Put 𝑝=𝑡𝑝₁+(1−𝑡)𝑝₂. By Lemma 2.6, we have ‖𝑝−𝑇^𝑛𝑝‖²=‖𝑡(𝑇^𝑛𝑝−𝑝₁)+(1−𝑡)(𝑇^𝑛𝑝−𝑝₂)‖²=𝑡‖𝑇^𝑛𝑝−𝑝₁‖²+(1−𝑡)‖𝑇^𝑛𝑝−𝑝₂‖²−𝑡(1−𝑡)‖𝑝₁−𝑝₂‖²≤𝑡(𝑘_𝑛‖𝑝−𝑝₁‖+𝑐_𝑛)²+(1−𝑡)(𝑘_𝑛‖𝑝−𝑝₂‖+𝑐_𝑛)²−𝑡(1−𝑡)‖𝑝₁−𝑝₂‖²=𝑡(1−𝑡)²𝑘²_𝑛‖𝑝₁−𝑝₂‖²+(1−𝑡)𝑡²𝑘²_𝑛‖𝑝₁−𝑝₂‖²−𝑡(1−𝑡)‖𝑝₁−𝑝₂‖²+2𝑡𝑘_𝑛𝑐_𝑛‖𝑝−𝑝₁‖+𝑡𝑐²_𝑛+2(1−𝑡)𝑘_𝑛𝑐_𝑛‖𝑝−𝑝₂‖+(1−𝑡)𝑐²_𝑛=((1−𝑡)𝑡(𝑘²_𝑛−1))‖𝑝₁−𝑝₂‖²+2𝑡𝑘_𝑛𝑐_𝑛‖𝑝−𝑝₁‖+𝑡𝑐²_𝑛+2(1−𝑡)𝑘_𝑛𝑐_𝑛‖𝑝−𝑝₂‖+(1−𝑡)𝑐²_𝑛. (2.8) This implies that lim_𝑛→∞‖𝑝−𝑇^𝑛𝑝‖=0. Since 𝑇 is continuous, we have lim_𝑛→∞‖𝑇𝑝−𝑇^𝑛+1𝑝‖=0, so that 𝑝=𝑇𝑝. Hence 𝑝∈𝐹(𝑇).

3. Main Results

First, we prove a weak convergence theorem for a single uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space.

Theorem 3.1. Let 𝐸 be a uniformly convex Banach space 𝐸 which satisfies Opial's property. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸, and 𝑇∶𝐶→𝐶 a uniformly 𝐿-Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences {𝑘_𝑛}, {𝑐_𝑛} such that 𝑘_𝑛≥1, lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝑐_𝑛=0. Assume that 𝐼−𝑇 is demiclosed at 0, where 𝐼 is the identity mapping of 𝐶 and {𝛼_𝑛} is a sequence in [0,1] such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1 and ∑^∞_𝑛=1(𝑘_𝑛−1)<∞,∑^∞_𝑛=1𝑐_𝑛<∞. Let {𝑥_𝑛} be the sequence in 𝐶 generated by the modified Mann iteration process: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛,∀𝑛∈ℕ. (3.1) Then {𝑥_𝑛} converges weakly to a fixed point of 𝐹(𝑇).

Proof. Let 𝑝∈𝐹(𝑇), we have ‖𝑥_𝑛+1−𝑝‖≤𝛼_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)‖𝑇^𝑛𝑥_𝑛−𝑝‖≤𝛼_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)𝑘_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)𝑐_𝑛≤(1+(𝑘_𝑛−1)(1−𝛼_𝑛))‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)𝑐_𝑛. (3.2) Since Σ^∞_𝑛=1(𝑘_𝑛−1)<∞, Σ^∞_𝑛=1𝑐_𝑛<∞, then by Lemma 2.1 and (3.2), we obtain that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ (3.3) exists.
This implies that {𝑇^𝑛𝑥_𝑛−𝑝} is bounded. Put 𝑟=max{sup_𝑛∈𝑁‖𝑇^𝑛𝑥_𝑛−𝑝‖,sup_𝑛∈𝑁‖𝑥_𝑛−𝑝‖}. By Lemma 2.2, there is a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that ‖𝑥_𝑛+1−𝑝‖²=‖𝛼_𝑛(𝑥_𝑛−𝑝)+(1−𝛼_𝑛)(𝑇^𝑛𝑥_𝑛−𝑝)‖²≤𝛼_𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑛)‖𝑇^𝑛𝑥_𝑛−𝑝‖²−𝛼_𝑛(1−𝛼_𝑛)𝑔(‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖)≤𝛼_𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑛)𝑘²_𝑛‖𝑥_𝑛−𝑝‖²+2(1−𝛼_𝑛)𝑘_𝑛𝑐_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)𝑐²_𝑛−𝛼_𝑛(1−𝛼_𝑛)𝑔(‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖). (3.4) It follows that 𝛼_𝑛(1−𝛼_𝑛)𝑔(‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖)≤(𝑘²_𝑛−𝛼_𝑛(𝑘²_𝑛−1))‖𝑥_𝑛−𝑝‖²−‖𝑥_𝑛+1−𝑝‖²+2(1−𝛼_𝑛)𝑘_𝑛𝑐_𝑛‖𝑥_𝑛−𝑝‖+(1−𝛼_𝑛)𝑐²_𝑛. (3.5) By our assumptions and (3.3), we get lim_𝑛→∞𝑔(‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖)=0. Since 𝑔 is continuous strictly increasing with 𝑔(0)=0, we can conclude that lim_𝑛→∞‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖=0. Observe that ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝛼_𝑛‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖→0. It follows from Lemma 2.3 that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} converging weakly to some 𝑞₁∈𝐶. From lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 and 𝐼−𝑇 is demiclosed at 0, we obtain that 𝑇𝑞₁=𝑞₁. That is, 𝑞₁∈𝐹(𝑇). Next, we show that {𝑥_𝑛} converges weakly to 𝑞₁ and take another subsequence {𝑥_𝑚𝑘} of {𝑥_𝑛} converging weakly to some 𝑞₂∈𝐶. Again, as above, we can conclude that 𝑞₂∈𝐹(𝑇). Finally, we show that 𝑞₁=𝑞₂. Assume 𝑞₁≠𝑞₂. Then by Opial's property of 𝐸, we have lim_𝑛→∞‖𝑥_𝑛−𝑞₁‖=lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞₁‖<lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞₂‖=lim_𝑛→∞‖𝑥_𝑛−𝑞₂‖=lim_𝑘→∞‖𝑥_𝑚𝑘−𝑞₂‖<lim_𝑘→∞‖𝑥_𝑚𝑘−𝑞₁‖=lim_𝑛→∞‖𝑥_𝑛−𝑞₁‖, (3.6) which is a contradiction. Therefore 𝑞₁=𝑞₂. This shows that {𝑥_𝑛} converges weakly to 𝑞₁∈𝐹(𝑇).

Remark 3.2. (1) In [3, Theorem 2], Bruck et al. proved that if 𝑇∶𝐶→𝐶 is asymptotically nonexpansive in the weak sense and 𝑇 is 𝜏-asymptotically regular, then Picard’s iterated sequence {𝑇^𝑛𝑥} is 𝜏-convergent to a fixed point of 𝑇. Since every asymptotically nonexpansive in the weak sense is a generalized asymptotically nonexpansive mapping and if its fixed point set is nonempty, then it is a generalized quasiasymptotically nonexpansive mapping. So we can apply Theorem 3.1 with a mapping 𝑇 which is asymptotically nonexpansive in the weak sense when its fixed point set is nonempty and obtain that the sequence {𝑥_𝑛} generated by the modified Mann iteration process 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛,∀𝑛∈ℕ (3.7) converges weakly to a fixed point of 𝑇 without asymptotically regularity condition of 𝑇.
(2) In [3, Theorem 4], Bruck et al. showed that if 𝐶 is a bounded convex subset of a uniformly convex Banach space and 𝐶 is sequentially 𝜏-compact, 𝑇∶𝐶→𝐶 is asymptotically nonexpansive in the intermediate sense and ∑^∞_𝑖=1𝑐_𝑛𝑖<∞ where 𝑐_𝑛=max{0,sup_{𝑥,𝑦∈𝐶}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)} for some sequence of nonnegative integers {𝑛_𝑖}, then the sequence {𝑥_𝑖} generated by 𝑥_𝑖+1=𝛼_𝑖𝑥_𝑖+(1−𝛼_𝑖)𝑇^𝑛𝑖𝑥_𝑖 (3.8) is 𝜏-convergent to a fixed point of 𝑇. Note that every asymptotically nonexpansive mapping in the intermediate sense is a generalized asymptotically nonexpansive mapping. Hence, Theorem 3.1 can be applied to the class of asymptotically nonexpansive mappings in the intermediate sense to obtain weak convergence of the sequence {𝑥_𝑛} generated by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛,∀𝑛∈ℕ (3.9) without the boundedness and compactness conditions on 𝐶.

Note that the modified Mann's iteration in Theorem 3.1 has only weak convergence.

Question 1. How can we modify the modified Mann's iteration in order to obtain strong convergence?

In the following theorem, we introduce a monotone hybrid method with the modified Mann's iteration to obtain a strong convergence theorem for an infinite family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings.

Theorem 3.3. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be an infinitely countable family of uniformly 𝐿_𝑖-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of 𝐶 into itself with nonnegative real sequences {𝑘_𝑖,𝑛}, {𝑐_𝑖,𝑛} such that 𝑘_𝑖,𝑛≥1, lim_𝑛→∞𝑘_𝑖,𝑛=1, lim_𝑛→∞𝑐_𝑖,𝑛=0, for all 𝑖∈𝑁. Assume that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and the sequence 𝛼_𝑖,𝑛∈[0,1), for all 𝑖,𝑛∈ℕ. Then the sequence {𝑥_𝑛} generated by Algorithm 1.3 converges strongly to 𝑃_𝐹𝑥₀.

Proof. We split the proof into six steps.
Step 1. Show that 𝑃_𝐹𝑥₀ is well defined for every 𝑥₀∈𝐶.
By Lemma 2.8, we obtain that 𝐹(𝑇_𝑖) is a closed and convex subset of 𝐶 for every 𝑖∈ℕ. Hence, 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖) is a nonempty, closed, and convex subset of 𝐶; consequently, 𝑃_𝐹𝑥₀ is well defined for every 𝑥₀∈𝐶.Step 2. Show that 𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀ is well defined.
From the definition of 𝐶_𝑛 and 𝑄_𝑛, it is obvious that 𝑄_𝑛 is closed and convex for each 𝑛∈ℕ. It follows from Lemma 2.7 that 𝐶_𝑖,𝑛 is closed and convex for all 𝑖,𝑛∈ℕ. This implies that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ. Next, we will show that 𝐹⊂𝐶_𝑛∩𝑄_𝑛, for all 𝑛≥0. First, we prove that 𝐹⊂𝐶_𝑖,𝑛 for all 𝑛≥0 and 𝑖∈ℕ. Since 𝑇_𝑖 is a generalized asymptotically quasi-nonexpansive mapping for all 𝑖∈ℕ, we have that for any 𝑝∈𝐹, ‖𝑦_𝑖,𝑛−𝑝‖²=‖𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛−𝑝‖²=𝛼_𝑖,𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑝‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²≤𝛼_𝑖,𝑛‖𝑥_𝑛−𝑝‖²+(1−𝛼_𝑖,𝑛)(𝑘_𝑖,𝑛‖𝑥_𝑛−𝑝‖+𝑐_𝑖,𝑛)²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²=‖𝑥_𝑛−𝑝‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)((𝑘²_𝑖,𝑛−1)‖𝑥_𝑛−𝑝‖²+2𝑘_𝑖,𝑛𝑐_𝑖,𝑛‖𝑥_𝑛−𝑝‖+𝑐²_𝑖,𝑛)≤‖𝑥_𝑛−𝑝‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛. (3.10) Hence, 𝑝∈𝐶_𝑖,𝑛 for all 𝑛≥0 and 𝑖∈ℕ. This proves that 𝐹⊂𝐶_𝑖,𝑛 for all 𝑛≥0 and 𝑖∈ℕ. Hence, 𝐹⊂𝐶_𝑛 for all 𝑛≥0.
As shown in Marino and Xu [12], by induction, we can show that 𝐹⊂𝑄_𝑛 for all 𝑛≥0. Hence 𝐹⊂𝐶_𝑛∩𝑄_𝑛, for all 𝑛≥0, and so 𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀ is well defined.Step 3. Show that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists.
From 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀ and 𝑥_𝑛+1∈𝑄_𝑛, we have ‖𝑥_𝑛−𝑥₀‖≤‖𝑥_𝑛+1−𝑥₀‖,∀𝑛≥0. (3.11) On the other hand, as 𝐹⊂𝑄_𝑛, we obtain ‖𝑥_𝑛−𝑥₀‖≤‖𝑧−𝑥₀‖,∀𝑛≥0,∀𝑧∈𝐹. (3.12) So we have that the sequence {𝑥_𝑛} is bounded and nondecreasing. Therefore lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists.Step 4. Show that 𝑥_𝑛→𝑞, where 𝑞∈𝐶.
For 𝑚>𝑛, by the definition of 𝑄_𝑛, we see that 𝑄_𝑚⊂𝑄_𝑛. Noting that 𝑥_𝑚=𝑃_𝑄𝑚𝑥₀ and 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀, by Lemma 2.5, we conclude that ‖𝑥_𝑚−𝑥_𝑛‖²≤‖𝑥_𝑚−𝑥₀‖²−‖𝑥_𝑛−𝑥₀‖². (3.13) It follows from Step 3 that {𝑥_𝑛} is a Cauchy. So, we can assume that 𝑥_𝑛→𝑞 as 𝑛→∞ for some 𝑞∈𝐶. In particular, we have that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.14)Step 5. Show that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0, for all 𝑖∈ℕ.
Let 𝑖,𝑛∈ℕ. Since 𝑥_𝑛+1∈𝐶_𝑛, it follows from (3.10) that ‖𝑦_𝑖,𝑛−𝑥_𝑛+1‖²≤‖𝑥_𝑛−𝑥_𝑛+1‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛. (3.15) Moreover, by Lemma 2.6, we have ‖𝑦_𝑖,𝑛−𝑥_𝑛+1‖²=‖𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛+1‖²=𝛼_𝑖,𝑛‖𝑥_𝑛−𝑥_𝑛+1‖²+(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛+1‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖². (3.16) It follows from (3.15) and (3.16) that (1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛+1‖²≤(1−𝛼_𝑖,𝑛)‖𝑥_𝑛−𝑥_𝑛+1‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛. (3.17) Since 𝜃_𝑖,𝑛→0 as 𝑛→∞ for all 𝑖∈ℕ, we have by (3.14) and (3.17) that lim_𝑛→∞‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛+1‖=0. (3.18) This implies that ‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖≤‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖→0 (3.19) for all 𝑖∈ℕ. By Lemma 2.3 and (3.14), we get that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for all 𝑖∈ℕ.Step 6. Show that 𝑞=𝑃_𝐹𝑥₀.
Since 𝑥_𝑛→𝑞 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for all 𝑖∈ℕ, we have 𝑞=𝑇_𝑖𝑞 for all 𝑖∈ℕ. Hence 𝑞∈𝐹. By Lemma 2.4, we obtain ⟨𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛⟩≤0 (3.20) for all 𝑧∈𝑄_𝑛−1∩𝐶_𝑛−1. Since 𝐹⊂𝑄_𝑛−1∩𝐶_𝑛−1, we have ⟨𝑧−𝑞,𝑥₀−𝑞⟩≤0 (3.21) for all 𝑧∈𝐹. Again by Lemma 2.4, we obtain that 𝑞=𝑃_𝐹𝑥₀. This completes the proof.

Theorem 3.4. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be an infinitely countable family of uniformly 𝐿_𝑖-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of 𝐶 into itself with nonnegative real sequences {𝑘_𝑖,𝑛}, {𝑐_𝑖,𝑛} such that 𝑘_𝑖,𝑛≥1, lim_𝑛→∞𝑘_𝑖,𝑛=1, lim_𝑛→∞𝑐_𝑖,𝑛=0, for all 𝑖∈ℕ. Assume that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and the sequence {𝛼_𝑖,𝑛}⊂[0,1), for all 𝑖,𝑛∈ℕ. Then the sequence {𝑥_𝑛} generated by Algorithm 1.4 converges strongly to 𝑃_𝐹𝑥₀.

Proof. We divide our proof into four steps.
Step 1. Show that 𝐶_𝑛 is closed and convex and 𝐹⊂𝐶_𝑛 for all 𝑛≥1.
It follows from Lemma 2.7 that 𝐶_𝑖,𝑛 is closed and convex for all 𝑖,𝑛∈ℕ. This implies that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ. Next, we will show that 𝐹⊂𝐶_𝑖,𝑛 for all 𝑛∈ℕ. For 𝑛=1, 𝐹⊂𝐶_𝑖,1=𝐶. Assume that 𝐹⊂𝐶_𝑖,𝑛 for 𝑛∈ℕ. It follows from (3.10) and the definition of 𝐶_𝑖,𝑛+1 that 𝐹⊂𝐶_𝑖,𝑛+1.Step 2. Show that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists.
From 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀, 𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛+1∈𝐶_𝑛, for all 𝑛≥0, we have ‖𝑥_𝑛−𝑥₀‖≤‖𝑥_𝑛+1−𝑥₀‖,∀𝑛≥0. (3.22) On the other hand, as 𝐹⊂𝐶_𝑛, we obtain ‖𝑥_𝑛−𝑥₀‖≤‖𝑧−𝑥₀‖,∀𝑛≥0,∀𝑧∈𝐹. (3.23) So we have that the sequence {𝑥_𝑛} is bounded and nondecreasing. Therefore lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists.Step 3. Show that 𝑥_𝑛→𝑞, where 𝑞∈𝐶.
For 𝑚>𝑛, by the definition of 𝐶_𝑛, we see that 𝑥_𝑚=𝑃_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛. By Lemma 2.5, we obtain that ‖𝑥_𝑚−𝑥_𝑛‖²≤‖𝑥_𝑚−𝑥₀‖²−‖𝑥_𝑛−𝑥₀‖². (3.24) From Step 2, we obtain that {𝑥_𝑛} is Cauchy. Hence 𝑥_𝑛→𝑞 as 𝑛→∞ for some 𝑞∈𝐶 and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. By using the same proof as in Step 5 of Theorem 3.3, we can show that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0, for all 𝑖∈ℕ.Step 4. Show that 𝑞=𝑃_𝐹𝑥₀.
Since 𝑥_𝑛→𝑞 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0, for all 𝑖∈ℕ, we have 𝑞=𝑇_𝑖𝑞, for all 𝑖∈ℕ. Hence 𝑞∈𝐹. Since 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀, by Lemma 2.4, we have ⟨𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛⟩≤0 (3.25) for all 𝑧∈𝐶_𝑛, and hence, ⟨𝑧−𝑞,𝑥₀−𝑞⟩≤0 (3.26) for all 𝑧∈𝐹. This shows that 𝑞=𝑃_𝐹𝑥₀, which completes the proof.

Since a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, we have the following.

Corollary 3.5. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be an infinitely countable family of uniformly 𝐿_𝑖-Lipschitzian and generalized asymptotically nonexpansive mappings of 𝐶 into itself with nonnegative real sequences {𝑘_𝑖,𝑛}, {𝑐_𝑖,𝑛} such that 𝑘_𝑖,𝑛≥1, lim_𝑛→∞𝑘_𝑖,𝑛=1, lim_𝑛→∞𝑐_𝑖,𝑛=0, for all 𝑖∈𝑁. Assume that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and the sequence {𝛼_𝑖,𝑛}⊂[0,1), for all 𝑖,𝑛∈ℕ. Let a sequence {𝑥_𝑛} be generated by the following manner: 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑖,𝑛=𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛,𝐶_𝑖,𝑛={𝑧∈𝐶∶‖𝑦_𝑖,𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛𝐶},_𝑛=^∞_𝑖=1𝐶_𝑖,𝑛,𝑄₀𝑄=𝐶,_𝑛={𝑧∈𝑄_𝑛−1∶⟨𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛𝑥⟩≤0},𝑛≥1,_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0, (3.27) where 𝜃_𝑖,𝑛=(𝑘²_𝑖,𝑛−1)∇²_𝑛+2𝑘_𝑖,𝑛𝑐_𝑖,𝑛∇_𝑛+𝑐²_𝑖,𝑛, ∇_𝑛=sup_𝑛∈ℕ{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹}<∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥₀.

Corollary 3.6. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞_𝑖=1 be an infinitely countable family of uniformly 𝐿_𝑖-Lipschitzian and generalized asymptotically nonexpansive mappings of 𝐶 into itself with nonnegative real sequences {𝑘_𝑖,𝑛}, {𝑐_𝑖,𝑛} such that 𝑘_𝑖,𝑛≥1, lim_𝑛→∞𝑘_𝑖,𝑛=1, lim_𝑛→∞𝑐_𝑖,𝑛=0, for all 𝑖∈𝑁. Assume that 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and the sequence {𝛼_𝑖,𝑛}⊂[0,1), for all 𝑖,𝑛∈ℕ. Let a sequence {𝑥_𝑛} be generated by the following manner: 𝑥₀𝐶∈𝐶chosenarbitrarily,_𝑖,0𝑦=𝐶,_𝑖,𝑛=𝛼_𝑖,𝑛𝑥_𝑛+(1−𝛼_𝑖,𝑛)𝑇^𝑛_𝑖𝑥_𝑛,𝐶_𝑖,𝑛+1={𝑧∈𝐶_𝑖,𝑛∶‖𝑦_𝑖,𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−𝛼_𝑖,𝑛(1−𝛼_𝑖,𝑛)‖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛‖²+(1−𝛼_𝑖,𝑛)𝜃_𝑖,𝑛𝐶},_𝑛+1=^∞_𝑖=1𝐶_𝑖,𝑛+1,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥0, (3.28) where 𝜃_𝑖,𝑛=(𝑘²_𝑖,𝑛−1)∇²_𝑛+2𝑘_𝑖,𝑛𝑐_𝑖,𝑛∇_𝑛+𝑐²_𝑖,𝑛, ∇_𝑛=sup_𝑛∈ℕ{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹}<∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥₀.

Remark 3.7. (i) Corollaries 3.5 and 3.6 improve and extend the main result in [10].
(ii) If we take 𝑇_𝑖=𝑇, 𝑘_𝑖,𝑛=1 and 𝑐_𝑖,𝑛=0 for all 𝑖,𝑛∈ℕ where 𝑇 is a nonexpansive mapping, then Corollary 3.5 reduces to [9, Theorem 3.4].

Acknowledgments

The authors would like to thank the referees for valuable suggestions on the manuscript. The authors were supported by the Center of Excellence in Mathematics, the Thailand Research Fund, the Commission on Higher Education, and the Graduate School of Chiang Mai University for the financial support.