Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of Strict Pseudocontractions in Banach Spaces

Abstract

We introduce a new modified Ishikawa iterative process and a new W-mapping for comput- ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.

1. Introduction

Let 𝐸 be a real Banach space with norm ‖⋅‖ and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐸^∗ be the dual space of 𝐸, and let ⟨⋅,⋅⟩ denotes the generalized duality pairing between 𝐸 and 𝐸^∗. For 𝑞>1, the generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by 𝐽_𝑞(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖^𝑞,𝑓=‖𝑥‖^𝑞−1, (1.1) for all 𝑥∈𝐸. In particular, if 𝑞=2, the mapping 𝐽=𝐽₂ is called the normalized duality mapping and 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽₂(𝑥) for 𝑥≠0. It is well known that if 𝐸 is smooth, then 𝐽_𝑞 is single-valued, which is denoted by 𝑗_𝑞.

A mapping 𝑇∶𝐶→𝐶 is called 𝑛𝑜𝑛𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑣𝑒 if𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.2) We use 𝐹(𝑇) to denote the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

𝑇 is said to be a 𝜆-strict pseudocontraction in the terminology of Browder and Petryshyn [1] if there exists a constant 𝜆>0 and for some 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≤𝑥−𝑦^𝑞−𝜆(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦^𝑞,∀𝑥,𝑦∈𝐶. (1.3)

𝑇 is said to be a strong pseudocontraction if there exists 𝑘∈(0,1) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≤𝑘‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦∈𝐶. (1.4)

Remark 1.1 (see [2]). Let 𝑇 be a 𝜆-strict pseudocontraction in a Banach space. Let 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). Then, ≤1𝑇𝑥−𝑝1+𝜆^1/𝑞−1𝑥−𝑝. (1.5)

Recall that a self mapping 𝑓∶𝐶→𝐶 is contraction on 𝐶 if there exists a constant 𝛼∈(0,1) and 𝑥,𝑦∈𝐶 such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦. (1.6) We use Π_𝐶 to denote the collection of all contractions on 𝐶. That is, Π_𝐶={𝑓∣𝑓∶𝐶→𝐶𝑎contraction}. Note that each 𝑓∈Π_𝐶 has a unique fixed point in 𝐶.

Very recently, Cai and Hu [3] also proved the strong convergence theorem in Banach spaces. They considered the following iterative algorithm:𝑥₁𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛^𝑁_𝑖=𝑖𝜂_𝑖^(𝑛)𝑇_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥1, (1.7) where 𝑇_𝑖 is a non-self-𝜆_𝑖-strictly pseudocontraction, 𝑓 is a contraction, and 𝐴 is a strongly positive linear bounded operator.

Dong et al. [2] proved the sequence {𝑥_𝑛} converges strongly in Banach spaces under certain appropriate assumptions and used the 𝑊_𝑛 mapping defined by (1.11). Let the sequences {𝑥_𝑛} be generated by𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑊_𝑛𝑦_𝑛,∀𝑛≥0. (1.8)

On the other hand, Katchang and Kumam [4, 5] introduced the following new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces; let the sequences {𝑥_𝑛} be generated by𝑥₀𝑧=𝑥∈𝐶chosenarbitrarily,_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥)+𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0, (1.9) where 𝑓 is a contraction, 𝐴 is a strongly positive linear bounded self-adjoint operator, and 𝑊_𝑛 mapping (see [6, 7]) is defined by𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜆_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜆_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜆_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋮𝑈𝐼,_𝑛,2=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼, (1.10) where 𝑇₁,𝑇₂,… is an infinite family of nonexpansive mappings of 𝐶 into itself and 𝜆₁,𝜆₂,… is real numbers such that 0≤𝜆_𝑛≤1 for every 𝑛∈ℕ. In 2010, Cho [8] considered and proved the strong convergence of the implicit iterative process for an infinite family of strict pseudocontractions in an arbitrary real Banach space.

In this paper, motivated and inspired by Cai and Hu [9], we consider the mapping 𝑊_𝑛 defined by𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝑡_𝑛𝑇_𝑛,𝑛𝑈_𝑛,𝑛+1+1−𝑡_𝑛⋮𝑈𝐼,_𝑛,𝑘=𝑡_𝑘𝑇_𝑛,𝑘𝑈_𝑛,𝑘+1+1−𝑡_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝑡_𝑘−1𝑇_{𝑛,𝑘−1}𝑈_𝑛,𝑘+1−𝑡_𝑘−1⋮𝑈𝐼,_𝑛,2=𝑡₂𝑇_𝑛,2𝑈_𝑛,3+1−𝑡₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝑡₁𝑇_𝑛,1𝑈_𝑛,2+1−𝑡₁𝐼, (1.11) where 𝑡₁,𝑡₂,… are real numbers such that 0≤𝑡_𝑛≤1.𝑇_𝑛,𝑘=𝜃_𝑛,𝑘𝑆_𝑘+(1−𝜃_𝑛,𝑘)𝐼, where 𝑆_𝑘 is a 𝜆_𝑘-strict pseudocontraction of 𝐶 into itself and 𝜃_𝑛,𝑘∈(0,𝜇],𝜇=min{1,{𝑞^𝜆/𝐶_𝑞}^1/𝑞−1}, where 𝜆=inf𝜆_𝑘 for all 𝑘∈ℕ. By Lemma 2.3, we know that 𝑇_𝑛,𝑘 is a nonexpansive mapping, and therefore, 𝑊_𝑛 is a nonexpansive mapping. We note that the 𝑊-mapping (1.10) is a special case of a 𝑊-mapping (1.11) when 𝜃_𝑛,𝑘=𝜃_𝑘 is constant for all 𝑛≥1.

Throughout this paper, we will assume that inf𝜆_𝑖>0, 0<𝑡_𝑛≤𝑏<1 for all 𝑛∈ℕ and {𝜃_𝑛,𝑘} satisfies(H1)𝜃_𝑛,𝑘∈(0,𝜇],𝜇=min{1,inf_𝑖{𝑞^𝜆𝑖/𝐶_𝑞}^1/𝑞−1} for all 𝑘∈ℕ,(H2)|𝜃_𝑛+1,𝑘−𝜃_𝑛,𝑘|≤𝑎_𝑛 for all 𝑛∈ℕ and 1≤𝑘≤𝑛, where {𝑎_𝑛} satisfies ∑^∞_𝑛=1𝑎_𝑛<∞.

The hypothesis (H2) secures the existence of lim_{𝑛→∞𝜃𝑛,𝑘} for all 𝑘∈ℕ. Set 𝜃_1,𝑘∶=lim_{𝑛→∞𝜃𝑛,𝑘} for all 𝑘∈ℕ. Furthermore, we assume that(H3)𝜃_1,𝑘>0 for all 𝑘∈ℕ.

It is obvious that 𝜃_1,𝑘 satisfy (H1). Using condition (H3), from 𝑇_𝑛,𝑘=𝜃_𝑛,𝑘𝑆_𝑘+(1−𝜃_𝑛,𝑘)𝐼, we define mappings 𝑇_1,𝑘𝑥∶=lim_𝑛→∞𝑇_𝑛,𝑘𝑥=𝜃_1,𝑘𝑆_𝑘𝑥+(1−𝜃_1,𝑘)𝑥 for all 𝑥∈𝐶.

Our results improve and extend the recent ones announced by Cai and Hu [3], Dong et al. [2], Katchang and Kumam [4, 5], and many others.

2. Preliminaries

Recall that 𝑈={𝑥∈𝐸∶‖𝑥‖=1}. A Banach space 𝐸 is said to be uniformly convex if, for any 𝜖∈(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦∈𝑈, ‖𝑥−𝑦‖≥𝜖 implies ‖(𝑥+𝑦)/2‖≤1−𝛿. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [10]). A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑈.

In a smooth Banach space, we define an operator 𝐴 as strongly positive if there exists a constant 𝛾>0 with the property⟨𝐴𝑥,𝐽(𝑥)⟩≥𝛾‖𝑥‖²,‖𝑎𝐼−𝑏𝐴‖=sup_{‖𝑥‖≤1}|⟨(𝑎𝐼−𝑏𝐴)𝑥,𝐽(𝑥)⟩|𝑎∈[0,1],𝑏∈[−1,1], (2.1) where I is the identity mapping and J is the normalized duality mapping.

If 𝐶 and 𝐷 are nonempty subsets of a Banach space 𝐸 such that 𝐶 is a nonempty closed convex and 𝐷⊂𝐶, then a mapping 𝑄∶𝐶→𝐷 is sunny [11, 12] provided that 𝑄(𝑥+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥) for all 𝑥∈𝐶 and 𝑡≥0 whenever 𝑥+𝑡(𝑥−𝑄(𝑥))∈𝐶. A mapping 𝑄∶𝐶→𝐶 is called a retraction if 𝑄²=𝑄. If a mapping 𝑄∶𝐶→𝐶 is a retraction, then 𝑄𝑧=𝑧 for all 𝑧 in the range of 𝑄. A subset 𝐷 of 𝐶 is said to be a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction 𝑄 of 𝐶 onto 𝐷. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [11, 12]: if 𝐸 is a smooth Banach space, then 𝑄∶𝐶→𝐷 is a sunny nonexpansive retraction if and only if there holds the inequality𝑥−𝑄𝑥,𝐽𝑦−𝑄𝑥≤0,∀𝑥∈𝐶,𝑦∈𝐷. (2.2)

We need the following lemmas for proving our main results.

Lemma 2.1 (see [13]). In a Banach space E, the following holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑥,𝑦∈𝐸, (2.3) where 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2.2 (see [14]). Let 𝐸 be a real 𝑞-uniformly smooth Banach space, then there exists a constant 𝐶_𝑞>0 such that 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝑗_𝑞𝑥+𝐶_𝑞𝑦^𝑞,∀𝑥,𝑦∈𝐸. (2.4) In particular, if 𝐸 be a real 2-uniformly smooth Banach space with the best smooth constant 𝐾, then the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+2𝐾𝑦²,∀𝑥,𝑦∈𝐸. (2.5)

The relation between the 𝜆-strict pseudocontraction and the nonexpansive mapping can be obtained from the following lemma.

Lemma 2.3 (see [15]). Let 𝐶 be a nonempty convex subset of a real 𝑞-uniformly smooth Banach space 𝐸 and 𝑆∶𝐶→𝐶 a 𝜆-strict pseudocontraction. For 𝛼∈(0,1), one defines 𝑇𝑥=(1−𝛼)𝑥+𝛼𝑆𝑥. Then, as 𝛼∈(0,𝜇],𝜇=min{1,{𝑞^𝜆/𝐶_𝑞}^1/𝑞−1}, 𝑇∶𝐶→𝐶 is nonexpansive such that 𝐹(𝑇)=𝐹(𝑆), where 𝐶_𝑞 is the constant in Lemma 2.2.

Concerning 𝑊_𝑛, we have the following lemmas, which are important to prove the main results.

Lemma 2.4 (see [2]). Let 𝐶 be a nonempty closed convex subset of a 𝑞-uniformly smooth and strictly convex Banach space E. Let 𝑆_𝑖,𝑖=1,2,…, be a 𝜆_𝑖-strict pseudocontraction from 𝐶 into itself such that ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅, and let inf𝜆_𝑖>0. Let 𝑡_𝑛,𝑛=1,2,…, be real numbers such that 0<𝑡_𝑛≤𝑏<1 for any 𝑛≥1. Assume that the sequence {𝜃_𝑛,𝑘} satisfies (H₁) and (H₂). Then, for every 𝑥∈𝐶 and 𝑘∈ℕ, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

Using Lemma 2.4, we define the mappings 𝑈_1,𝑘 and 𝑊∶𝐶→𝐶 as follows: 𝑈_1,𝑘𝑥∶=lim_𝑛→∞𝑈_𝑛,𝑘𝑥,𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥, (2.6) for all 𝑥∈𝐶. Such 𝑊 is called the 𝑊-mapping generated by 𝑆₁,𝑆₂,…,𝑡₁,𝑡₂,… and 𝜃_𝑛,𝑘,forall𝑛∈ℕ and 1≤𝑘≤𝑛.

Lemma 2.5 (see [2]). Let {𝑥_𝑛} be a bounded sequence in a 𝑞-uniformly smooth and strictly convex Banach space 𝐸. Under the assumptions of Lemma 2.4, it holds lim_𝑛→∞𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛=0. (2.7)

Lemma 2.6 (see [2]). Let 𝐶 be a nonempty closed convex subset of a 𝑞-uniformly smooth and strictly convex Banach space E. Let 𝑆_𝑖,𝑖=1,2,…, be a 𝜆_𝑖-strict pseudocontraction from 𝐶 into itself such that ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅, and let inf𝜆_𝑖>0. Let 𝑡_𝑛,𝑛=1,2,…, be real numbers such that 0<𝑡_𝑛≤𝑏<1 for any 𝑛≥1. Assume that the sequence {𝜃_𝑛,𝑘} satisfies (H₁)–(H₃). Then, 𝐹(𝑊)=∩^∞_𝑛=1𝐹(𝑆_𝑛).

Lemma 2.7 (see [16]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0, (2.8) where {𝛼_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=1𝛼_𝑛=∞, (ii)limsup_𝑛→∞𝛿_𝑛/𝛼_𝑛≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.8 (see [17]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.9 (see [3]). Assume that 𝐴 is a strong positive linear bounded operator on a smooth Banach space 𝐸 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then, ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

3. Main Results

In this section, we prove a strong convergence theorem.

Theorem 3.1. Let 𝐸 be a real 𝑞-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 to 𝐸^∗. Let 𝐶 be a nonempty closed and convex subset of 𝐸 which is also a sunny nonexpansive retraction of 𝐸 such that 𝐶+𝐶⊂𝐶. Let 𝐴 be a strongly positive linear bounded operator on 𝐸 with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼, and let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1). Let 𝑆_𝑖,𝑖=1,2,…, be 𝜆_𝑖-strict pseudocontractions from 𝐶 into itself such that ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅ and inf𝜆_𝑖>0. Assume that the sequences {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} in (0,1) satisfy the following conditions:(i)∑^∞_𝑛=0𝛼_𝑛=∞; and lim_𝑛→∞𝛼_𝑛=0,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,(iii)lim_𝑛→∞|𝛾_𝑛+1−𝛾_𝑛|=0,(iv)lim_𝑛→∞|𝛿_𝑛+1−𝛿_𝑛|=0,(v)𝛿_𝑛(1+𝛾_𝑛)−2𝛾_𝑛>𝑎 for some 𝑎∈(0,1),and the sequence {𝜃_𝑛,𝑘} satisfies (H₁)–(H₃). Then, the sequence {𝑥_𝑛} generated by 𝑥₀𝑧∈𝐶chosenarbitrarily,_𝑛=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛,𝑦_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0 (3.1) converges strongly to 𝑥^∗∈∩^∞_𝑛=1𝐹(𝑆_𝑛), which solves the following variational inequality: 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑝−𝑥^∗≤0,∀𝑓∈Π_𝐶,𝑝∈^∞_𝑛=1𝐹𝑆_𝑛. (3.2)

Proof. By (i), we may assume, without loss of generality, that 𝛼_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹ for all 𝑛. Since 𝐴 is a strongly positive bounded linear operator on 𝐸 and by (2.1), we have ‖𝐴‖=sup{|⟨𝐴𝑥,𝐽(𝑥)⟩|∶𝑥∈𝐸,‖𝑥‖=1}. (3.3) Observe that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑥,𝐽(𝑥)=1−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑥,𝐽(𝑥)⟩≥1−𝛽_𝑛−𝛼_𝑛‖𝐴‖≥0,∀𝑥∈𝐸. (3.4) This shows that (1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴 is positive. It follows that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴=sup1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑥,𝐽(𝑥)∶𝑥∈𝐸,‖𝑥‖=1=sup1−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑥,𝐽(𝑥)⟩∶𝑥∈𝐸,‖𝑥‖=1≤1−𝛽_𝑛−𝛼_𝑛𝛾. (3.5)
First, we show that {𝑥_𝑛} is bounded. Let 𝑝∈∩^∞_𝑛=1𝐹(𝑆_𝑛). By the definition of {𝑧_𝑛},{𝑦_𝑛}, and {𝑥_𝑛}, we have 𝑧_𝑛−𝑝=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛−𝑝≤𝛿_𝑛𝑥_𝑛−𝑝+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛−𝑝≤𝛿_𝑛𝑥_𝑛−𝑝+1−𝛿_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝, (3.6) and from this, we have 𝑦_𝑛−𝑝=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛−𝑝≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛−𝑝≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑧_𝑛−𝑝≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (3.7)
It follows that 𝑥_𝑛+1−𝑝=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛=𝛼−𝑝_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝛼_𝑛𝛾𝑦_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+1−𝛼_𝑛𝛾𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+1−𝛼_𝑛𝛾𝑥_𝑛=−𝑝1−𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑝+𝛾−𝛾𝛼𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝𝑥𝛾−𝛾𝛼≤max₁,𝑝−𝑝𝛾𝑓−𝐴𝑝.𝛾−𝛾𝛼 (3.8) By induction on 𝑛, we obtain ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₀−𝑝‖,‖𝛾𝑓(𝑝)−𝐴𝑝‖/𝛾−𝛾𝛼‖} for every 𝑛≥0 and 𝑥₀∈𝐶, then {𝑥_𝑛} is bounded. So, {𝑦_𝑛},{𝑧_𝑛},{𝐴𝑦_𝑛},{𝑊_𝑛𝑥_𝑛}, {𝑊_𝑛𝑧_𝑛}, and {𝑓(𝑥_𝑛)} are also bounded.
Next, we claim that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. Let 𝑥∈𝐶 and 𝑝∈∩^∞_𝑛=1𝐹(𝑆_𝑛). Fix 𝑘∈ℕ for any 𝑛∈ℕ with 𝑛≥𝑘, and since 𝑇_𝑛,𝑘 and 𝑈_𝑛,𝑘 are nonexpansive, we have ‖𝑇_𝑛,𝑘𝑥−𝑝‖≤‖𝑥−𝑝‖ and ‖𝑈_𝑛,𝑘𝑥−𝑝‖≤‖𝑥−𝑝‖, respectively. From (1.5), it follows that ‖𝑆_𝑘𝑥−𝑝‖≤(1+(1/𝜆_𝑘^1/𝑞−1))sup_𝑛‖𝑥−𝑝‖. We can set 𝑀₁=inf_𝑖12+𝜆_𝑖^1/𝑞−1sup_𝑛𝑥_𝑛𝑀−𝑝<∞,₂=inf_𝑖12+𝜆_𝑖^1/𝑞−1sup_𝑛𝑧_𝑛−𝑝<∞. (3.9)
From (1.11), we have 𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛=𝑈_𝑛+1,1𝑥_𝑛−𝑈_𝑛,1𝑥_𝑛=𝑡₁𝑇_𝑛+1,1𝑈_𝑛+1,2𝑥_𝑛+1−𝑡₁𝑥_𝑛−𝑡₁𝑇_𝑛,1𝑈_𝑛,2𝑥_𝑛−1−𝑡₁𝑥_𝑛=𝑡₁𝑇_𝑛+1,1𝑈_𝑛+1,2𝑥_𝑛−𝑇_𝑛,1𝑈_𝑛,2𝑥_𝑛=𝑡₁𝜃_𝑛+1,1𝑆₁+1−𝜃_𝑛+1,1𝑈_𝑛+1,2𝑥_𝑛−𝑇_𝑛,1𝑈_𝑛,2𝑥_𝑛=𝑡₁𝜃_𝑛,1𝑆₁+1−𝜃_𝑛,1𝑈_𝑛+1,2𝑥_𝑛−𝑇_𝑛,1𝑈_𝑛,2𝑥_𝑛+𝜃_𝑛+1,1−𝜃_𝑛,1𝑆₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛+1,2𝑥_𝑛≤𝑡₁𝑇_𝑛,1𝑈_𝑛+1,2𝑥_𝑛−𝑇_𝑛,1𝑈_𝑛,2𝑥_𝑛+𝑡₁𝜃_𝑛+1,1−𝜃_𝑛,1S₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛+1,2𝑥_𝑛≤𝑡₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛,2𝑥_𝑛+𝑡₁𝜃_𝑛+1,1−𝜃_𝑛,1𝑀₁≤𝑡₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛,2𝑥_𝑛+𝑡₁𝑎_𝑛𝑀₁⋮≤^𝑛_𝑖=1𝑡_𝑖𝑈_{𝑛+1,𝑛+1}𝑥_𝑛−𝑈_𝑛,𝑛+1𝑥_𝑛+𝑎_𝑛𝑀₁^𝑛𝑗_𝑗=1𝑖=1𝑡_𝑖≤^𝑛_𝑖=1𝑡_𝑖𝑡_𝑛+1𝑇_{𝑛+1,𝑛+1}𝑥_𝑛+1−𝑡_𝑛+1𝑥_𝑛−𝑥_𝑛+𝑎_𝑛𝑀₁𝑏≤1−𝑏_𝑛+1𝑖=1𝑡_𝑖𝑇_{𝑛+1,𝑛+1}𝑥_𝑛−𝑥_𝑛+𝑎_𝑛𝑀₁𝑏≤𝑏1−𝑏_𝑛+1+𝑎_𝑛𝑏𝑀1−𝑏¹, (3.10) for all 𝑛≥0. Similarly, we also have ‖𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛‖≤(𝑏^𝑛+1+𝑎_𝑛(𝑏/(1−𝑏)))𝑀₂ for all 𝑛≥0. We compute that 𝑧_𝑛+1−𝑧_𝑛=𝛿_𝑛+1𝑥_𝑛+1+1−𝛿_𝑛+1𝑊_𝑛+1𝑥_𝑛+1−𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛≤1−𝛿_𝑛+1𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛+1𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑥_𝑛−𝑊_𝑛+1𝑥_𝑛+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛≤1−𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝑏^𝑛+1+𝑎_𝑛𝑏𝑀1−𝑏¹, (3.11) and 𝑦_𝑛+1−𝑦_𝑛=𝛾_𝑛+1𝑥_𝑛+1+1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+1−𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛=𝛾_𝑛+1𝑥_𝑛+1+1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+1−1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛−𝛾_𝑛𝑥_𝑛−1−𝛾_𝑛𝑊_𝑛𝑧_𝑛−1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝛾_𝑛+1𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛=1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+𝛾_𝑛−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛≤1−𝛾_𝑛+1𝑊_𝑛+1𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑥_𝑛−𝑊_𝑛+1𝑧_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛≤1−𝛾_𝑛+1𝑧_𝑛+1−𝑧_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛≤1−𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝑏^𝑛+1+𝑎_𝑛𝑏𝑀1−𝑏¹+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑏^𝑛+1+𝑎_𝑛𝑏𝑀1−𝑏²≤𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑥_𝑛+2𝑏^𝑛+1+𝑎_𝑛𝑏1−𝑏𝑀, (3.12) where 𝑀=inf_𝑖(2+(1/𝜆_𝑖^1/𝑞−1))sup_𝑛(‖𝑥_𝑛−𝑝‖+‖𝑧_𝑛−𝑝‖)<∞. Observe that we put 𝑙_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/1−𝛽_𝑛, then 𝑥_𝑛+1=1−𝛽_𝑛𝑙_𝑛+𝛽_𝑛𝑥_𝑛,∀𝑛≥0. (3.13) Now, we have 𝑙_𝑛+1−𝑙_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐴𝑦_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛1−𝛽_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+11−𝛽_𝑛+1+1−𝛽_𝑛+1𝑦_𝑛+11−𝛽_𝑛+1−𝛼_𝑛+1𝐴𝑦_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛1−𝛽_𝑛−1−𝛽_𝑛𝑦_𝑛1−𝛽_𝑛+𝛼_𝑛𝐴𝑦_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑦_𝑛−𝛾𝑓𝑥_𝑛+𝑦_𝑛+1−𝑦_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑦_𝑛−𝛾𝑓𝑥_𝑛+𝑦_𝑛+1−𝑦_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑦_𝑛−𝛾𝑓𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛+1𝑧_𝑛−𝑥_𝑛+2𝑏^𝑛+1+𝑎_𝑛𝑏1−𝑏𝑀. (3.14) Therefore, we have 𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑦_𝑛−𝛾𝑓𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑊_𝑛+1𝑥_𝑛−𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛+1z_𝑛−𝑥_𝑛+2𝑏^𝑛+1+𝑎_𝑛𝑏1−𝑏𝑀. (3.15) From the conditions (i)–(iv), (H2), 0<𝑏<1 and the boundedness of {𝑥_𝑛}, {𝑓(𝑥_𝑛)}, {𝐴𝑦_𝑛}, {𝑊_𝑛𝑥_𝑛}, and {𝑊_𝑛𝑧_𝑛}, we obtain limsup_𝑛→∞𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.16) It follows from Lemma 2.8 that lim_𝑛→∞‖𝑙_𝑛−𝑥_𝑛‖=0. Noting (3.13), we see that 𝑥_𝑛+1−𝑥_𝑛=1−𝛽_𝑛𝑙_𝑛−𝑥_𝑛⟶0, (3.17) as 𝑛→∞. Therefore, we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.18) We also have ‖𝑦_𝑛+1−𝑦_𝑛‖→0 and ‖𝑧_𝑛+1−𝑧_𝑛‖→0 as 𝑛→∞. Observing that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑦_𝑛, (3.19) it follows that 1−𝛽_𝑛𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛. (3.20) By the conditions (i), (ii), (3.18), and the boundedness of {𝑥_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐴𝑦_𝑛}, we obtain lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.21) Consider 𝑦_𝑛−𝑊_𝑛𝑧_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛−𝑊_𝑛𝑧_𝑛=𝛾_𝑛𝑥_𝑛−𝑊_𝑛𝑧_𝑛,𝑧_𝑛−𝑥_𝑛=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛−𝑥_𝑛=1−𝛿_𝑛𝑊_𝑛𝑥_𝑛−𝑥_𝑛. (3.22) It follows that 𝑥_𝑛−𝑊_𝑛𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑊_𝑛𝑧_𝑛+𝑊_𝑛𝑧_𝑛−𝑊_𝑛𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛾_𝑛𝑥_𝑛−𝑊_𝑛𝑧_𝑛+𝑧_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛾_𝑛𝑥_𝑛−𝑊_𝑛𝑥_𝑛+𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑧_𝑛+𝑧_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛾_𝑛𝑥_𝑛−𝑊_𝑛𝑥_𝑛+1+𝛾_𝑛𝑧_𝑛−𝑥_𝑛=𝑥_𝑛−𝑦_𝑛+𝛾_𝑛𝑥_𝑛−𝑊_𝑛𝑥_𝑛+1+𝛾_𝑛1−𝛿_𝑛𝑊_𝑛𝑥_𝑛−𝑥_𝑛. (3.23) This implies that 𝛿_𝑛1+𝛾_𝑛−2𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛. (3.24) From the condition (v) and (3.21), we get lim_𝑛→∞𝑊_𝑛𝑥_𝑛−𝑥_𝑛=0. (3.25) On the other hand, 𝑊𝑥_𝑛−𝑥_𝑛≤𝑊𝑥_𝑛−𝑊_𝑛𝑥_𝑛+𝑊_𝑛𝑥_𝑛−𝑥_𝑛. (3.26) From the boundedness of {𝑥_𝑛} and using (2.7), we have ‖𝑊𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖→0 as 𝑛→∞. It follows that lim_𝑛→∞𝑊𝑥_𝑛−𝑥_𝑛=0. (3.27)
Next, we prove that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗≤0, (3.28) where 𝑥^∗=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of contraction 𝑥↦𝑡𝛾𝑓(𝑥)+(1−𝑡𝐴)𝑊𝑥. Noticing that 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝛾𝑓(𝑥_𝑡)+(1−𝑡𝐴)𝑊𝑥_𝑡, it follows that 𝑥_𝑡−𝑥_𝑛=(𝐼−𝑡𝐴)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛. (3.29) It follows from Lemma 2.1 that𝑥_𝑡−𝑥_𝑛²=(𝐼−𝑡𝐴)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛²≤1−𝛾𝑡²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−𝛾𝑡²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤1−𝛾𝑡²𝑊𝑥_𝑡−𝑊𝑥_𝑛²+2𝑊𝑥_𝑛−𝑥_𝑛,𝐽𝑊𝑥_𝑡−𝑥_𝑛+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤1−𝛾𝑡²𝑥_𝑡−𝑥_𝑛²+2𝑊𝑥_𝑛−𝑥_𝑛𝑊𝑥_𝑡−𝑥_𝑛+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤1−𝛾𝑡²𝑥_𝑡−𝑥_𝑛²+2𝑊𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−2𝛾𝑡+𝛾𝑡²𝑥_𝑡−𝑥_𝑛²+21−𝛾𝑡²𝑊𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛+2𝑡𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛, (3.30) where 𝑓_𝑛(𝑡)=21−𝛾𝑡²𝑊𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛⟶0as𝑛⟶∞. (3.31) Since 𝐴 is linearly strong and positive and using (2.1), we have 𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=𝐴𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≥𝛾𝑥_𝑡−𝑥_𝑛². (3.32) Substituting (3.32) in (3.30), we have 𝐴𝑥_𝑡−𝛾𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝛾²t2−𝛾𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡)+𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤𝛾𝑡2−1𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛+1𝑓2𝑡^𝑛(𝑡)+𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=𝛾𝑡2𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛+1𝑓2𝑡^𝑛(𝑡). (3.33) Letting 𝑛→∞ in (3.33) and noting (3.31) yield that limsup_𝑛→∞𝐴𝑥_𝑡−𝛾𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₃, (3.34) where 𝑀₃>0 is a constant such that 𝑀₃≥𝛾⟨𝐴𝑥_𝑡−𝐴𝑥_𝑛,𝐽(𝑥_𝑡−𝑥_𝑛)⟩ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 from (3.34), we have limsup_𝑡→0limsup_𝑛→∞𝐴𝑥_𝑡−𝛾𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (3.35) On the other hand, we have 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗=𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗−𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥_𝑡+𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥_𝑡−𝛾𝑓𝑥^∗−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝛾𝑓𝑥^∗−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡−𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡=𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗−𝐽𝑥_𝑛−𝑥_𝑡+𝐴𝑥_𝑡−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥_𝑡+𝛾𝑓𝑥^∗−𝛾𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡, (3.36) which implies that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗≤limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗−𝐽𝑥_𝑛−𝑥_𝑡+‖𝐴‖𝑥_𝑡−𝑥^∗limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝛾𝛼𝑥^∗−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝛾𝑓𝑥_𝑡−𝐴𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (3.37)
Noticing that 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝐶, it follows from (3.35) that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗=limsup_𝑡→0limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛−𝑥^∗≤0. (3.38)
Therefore, we obtain that (3.28) holds.
Finally, we prove that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Now, from Lemma 2.1, we have 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗2=1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗2≤1−𝛽_𝑛−𝛼_𝑛𝛾²𝑦_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=1−𝛽_𝑛−𝛼_𝑛𝛾²𝑦_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛽_𝑛𝑥_𝑛−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=1−𝛽_𝑛−𝛼_𝑛𝛾²𝑦_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛽_𝑛𝑥_𝑛−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛽_𝑛−𝛼_𝑛𝛾²𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝛾𝛼𝑥_𝑛+1−𝑥^∗2+𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛+1−𝑥^∗2+𝑥_𝑛−𝑥^∗2=1−𝛽_𝑛−𝛼_𝑛𝛾²+𝛼_𝑛𝛾𝛼+𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝛾𝛼+𝛽_𝑛𝑥_𝑛+1−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗, (3.39) and consequently, 𝑥_𝑛+1−𝑥^∗2≤1−𝛽_𝑛−𝛼_𝑛𝛾²+𝛼_𝑛𝛾𝛼+𝛽_𝑛1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=1−2𝛼_𝑛𝛾−𝛾𝛼1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝛽²_𝑛+2𝛽_𝑛𝛼_𝑛𝛾+𝛼²_𝑛𝛾²1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=1−2𝛼_𝑛𝛾−𝛾𝛼1−𝛼_𝑛𝛾𝛼−𝛽_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾−𝛾𝛼1−𝛼_𝑛𝛾𝛼−𝛽_𝑛×𝛽²_𝑛+2𝛽_𝑛𝛼_𝑛𝛾+𝛼²_𝑛𝛾²2𝛼_𝑛𝑀𝛾−𝛾𝛼⁴+1𝑥𝛾−𝛾𝛼𝛾𝑓^∗−𝐴𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗, (3.40) where 𝑀₄ is an appropriate constant such that M₄≥sup_𝑛≥0‖𝑥_𝑛−𝑥^∗‖². Setting 𝑐_𝑛=2𝛼_𝑛(𝛾−𝛾𝛼)/(1−𝛼_𝑛𝛾𝛼−𝛽_𝑛) and 𝑏_𝑛=(𝛽²_𝑛+2𝛽_𝑛𝛼_𝑛𝛾+𝛼²_𝑛𝛾²)/(2𝛼_𝑛(𝛾−𝛾𝛼))𝑀₄+(1/(𝛾−𝛾𝛼))⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽(𝑥_𝑛+1−𝑥^∗)⟩, then we have 𝑥_𝑛+1−𝑥^∗2≤1−𝑐_𝑛𝑥_𝑛−𝑥^∗2+𝑐_𝑛𝑏_𝑛. (3.41) By (3.28), (i) and applying Lemma 2.7 to (3.41), we have 𝑥_𝑛→𝑥^∗ as 𝑛→∞. This completes the proof.

Corollary 3.2. Let 𝐸 be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 to 𝐸^∗. Let 𝐶 be a nonempty closed and convex subset of 𝐸 which is also a sunny nonexpansive retraction of 𝐸 such that 𝐶+𝐶⊂𝐶. Let 𝐴 be a strongly positive linear bounded operator on 𝐸 with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼, and let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1). Let 𝑆_𝑖,𝑖=1,2,…, be 𝜆_𝑖-strict pseudocontractions from 𝐶 into itself such that ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅ and inf𝜆_𝑖>0. Assume that the sequences {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} in (0,1) satisfy the following conditions:(i)∑^∞_𝑛=0𝛼_𝑛=∞; and lim_𝑛→∞𝛼_𝑛=0,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,(iii)lim_𝑛→∞|𝛾_𝑛+1−𝛾_𝑛|=0,(iv)lim_𝑛→∞|𝛿_𝑛+1−𝛿_𝑛|=0,(v)𝛿_𝑛(1+𝛾_𝑛)−2𝛾_𝑛>𝑎 for some 𝑎∈(0,1),and the sequence {𝜃_𝑛} satisfies (H₁). Then, the sequence {𝑥_𝑛} generated by 𝑥₀𝑧∈𝐶chosenarbitrarily,_𝑛=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛,𝑦_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0 (3.42) converges strongly to 𝑥^∗∈∩^∞_𝑛=1𝐹(𝑆_𝑛), which solves the following variational inequality: 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑝−𝑥^∗≤0,∀𝑓∈Π_𝐶,𝑝∈^∞_𝑛=1𝐹𝑆_𝑛. (3.43)

Corollary 3.3. Let 𝐸 be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 to 𝐸^∗. Let 𝐶 be a nonempty closed and convex subset of 𝐸 which is also a sunny nonexpansive retraction of 𝐸 such that 𝐶+𝐶⊂𝐶. Let 𝐴 be a strongly positive linear bounded operator on 𝐸 with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼, and let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1). Let 𝑆_𝑖,𝑖=1,2,…, be a nonexpansive mapping from C into itself such that ∩^∞_𝑛=1𝐹(𝑆_𝑛)≠∅ and inf𝜆_𝑖>0. Assume that the sequences {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} in (0,1) satisfy the following conditions:(i)∑^∞_𝑛=0𝛼_𝑛=∞; and lim_𝑛→∞𝛼_𝑛=0,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (iii)lim_𝑛→∞|𝛾_𝑛+1−𝛾_𝑛|=0,(iv)lim_𝑛→∞|𝛿_𝑛+1−𝛿_𝑛|=0,(v)𝛿_𝑛(1+𝛾_𝑛)−2𝛾_𝑛>𝑎 for some 𝑎∈(0,1).Then, the sequence {𝑥_𝑛} generated by 𝑥₀𝑧∈𝐶chosenarbitrarily,_𝑛=𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝑊_𝑛𝑥_𝑛,𝑦_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0 (3.44) converges strongly to 𝑥^∗∈∩^∞_𝑛=1𝐹(𝑆_𝑛), which solves the following variational inequality: 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽𝑝−𝑥^∗≤0,∀𝑓∈Π_𝐶,𝑝∈^∞_𝑛=1𝐹𝑆_𝑛. (3.45)

Remark 3.4. Theorem 3.1, Corollaries 3.2, and 3.3, improve and extend the corresponding results of Cai and Hu [3], Dong et al. [2], and Katchang and Kumam [4, 5] in the following senses.(i)For the mappings, we extend the mappings from an infinite family of nonexpansive mappings to an infinite family of strict pseudocontraction mappings.(ii)For the algorithms, we propose new modified Ishikawa iterative algorithms, which are different from the ones given in [2–5] and others.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT) and the Faculty of Science KMUTT for financial support. Furthermore, they also would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support (NRU-CSEC Project no. 54000267). Finally, they are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work.