Abstract

This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

1. Introduction

Throughout this paper, we always assume that 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 with inner product and norm denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively, 2𝐻 denoting the family of all the nonempty subsets of 𝐻.

Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2𝐻 a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point 𝑥∈𝐻,𝜃∈𝐵𝑥+𝑀𝑥,(1.1) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (1.1) is denoted by VI(𝐻,𝐵,𝑀).

Recall that 𝑃𝐶 is the metric projection of 𝐻 onto 𝐶; that is, for each 𝑥∈𝐻, there exists the unique point in 𝑃𝐶𝑥∈𝐶 such that ‖𝑥−𝑃𝐶𝑥‖=min𝑦∈𝐶‖𝑥−𝑦‖. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A point 𝑥∈𝐶 is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. We denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. If 𝐶 is nonempty bounded closed convex subset of 𝐻 and 𝑇 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑇) is nonempty (see [1]). Recall that a mapping 𝐴∶𝐶→𝐶 is said to be (i)monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶,(1.2)(ii)𝑘-Lipschitz continuous if there exists a constant 𝑘>0 such that ‖𝐴𝑥−𝐴𝑦‖≤𝑘‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,(1.3) if 𝑘=1, then 𝐴 is a nonexpansive, (iii)pseudocontractive if ‖𝐴𝑥−𝐴𝑦‖2≤‖𝑥−𝑦‖2‖+‖(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦2,∀𝑥,𝑦∈𝐶,(1.4)(iv)𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that ‖𝐴𝑥−𝐴𝑦‖2≤‖𝑥−𝑦‖2‖+𝑘‖(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦2,∀𝑥,𝑦∈𝐶,(1.5) it is obvious that 𝐴 is a nonexpansive if and only if 𝐴 is 0-strictly pseudocontractive, (v)𝛼-strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝑥−𝑦‖2,∀𝑥,𝑦∈𝐶,(1.6)(vi)𝛼-inverse-strongly monotone (or 𝛼-cocoercive) if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖2,∀𝑥,𝑦∈𝐶,(1.7) if 𝛼=1, then 𝐴 is said to be firmly nonexpansive; it is obvious that any 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and (1/𝛼)-Lipschitz continuous.

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [2–5] and the references therein).

In this paper, we study the mapping 𝑊𝑛 defined by𝑈𝑛,𝑛+1𝑈=𝐼,𝑛,𝑛=𝜇𝑛𝑆𝑛𝑈𝑛,𝑛+1+1−𝜇𝑛𝑈𝐼,𝑛,𝑛−1=𝜇𝑛−1𝑆𝑛−1𝑈𝑛,𝑛+1−𝜇𝑛−1⋮𝑈𝐼,𝑛,𝑘=𝜇𝑘𝑆𝑘𝑈𝑛,𝑘+1+1−𝜇𝑘𝑈𝐼,𝑛,𝑘−1=𝜇𝑘−1𝑆𝑘−1𝑈𝑛,𝑘+1−𝜇𝑘−1⋮𝑈𝐼,𝑛,2=𝜇2𝑆2𝑈𝑛,3+1−𝜇2𝑊𝐼,𝑛=𝑈𝑛,1=𝜇1𝑆1𝑈𝑛,2+1−𝜇1𝐼,(1.8) where {𝜇𝑖} is nonnegative real sequence in (0,1), for all 𝑖∈ℕ, 𝑆1,𝑆2,… from a family of infinitely nonexpansive mappings of 𝐶 into itself. It is obvious that 𝑊𝑛 is a nonexpansive mapping of 𝐶 into itself; such a mapping 𝑊𝑛 is called a 𝑊-mapping generated by 𝑆1,𝑆2,…,𝑆𝑛 and 𝜇1,𝜇2,…,𝜇𝑛.

Definition 1.1 (see [6]). Let 𝑀∶𝐻→2𝐻 be a multivalued maximal monotone mapping. Then, the single-valued mapping 𝐽𝑀,𝜆∶𝐻→𝐻 defined by 𝐽𝑀,𝜆(𝑢)=(𝐼+𝜆𝑀)−1(𝑢), for all 𝑢∈𝐻, is called the resolvent operator associated with 𝑀, where 𝜆 is any positive number and 𝐼 is the identity mapping.

Recently, Zhang et al. [6] considered the problem (1.1) and the problem of a fixed point of nonexpansive mapping. To be more precise, they proved the following theorem.

Theorem ZLC. Let 𝐻 be a real Hilbert space, 𝐵∶𝐻→𝐻 an 𝛼-inverse-strongly monotone mapping, 𝑀∶𝐻→2𝐻 a maximal monotone mapping, and 𝑇∶𝐻→𝐻 a nonexpansive mapping. Suppose that the set 𝐹(𝑇)∩VI(𝐻,𝐵,𝑀)≠∅, where VI(𝐻,𝐵,𝑀) is the set of solutions of quasivariational inclusion (1.1). Suppose that 𝑥1=𝑥∈𝐻 and {𝑥𝑛} is the sequence defined by 𝑦𝑛=𝐽𝑀,𝜆𝑥𝑛−𝜆𝐵𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑥+1−𝛼𝑛𝑇𝑦𝑛,(1.9) for all 𝑛∈ℕ, where 𝜆∈(0,2𝛼) and {𝛼𝑛}⊂(0,1) satisfying the following conditions:
(C1) limğ‘›â†’âˆžğ›¼ğ‘›=0 and âˆ‘âˆžğ‘›=1𝛼𝑛=∞,
(C2) âˆ‘âˆžğ‘›=1|𝛼𝑛+1−𝛼𝑛|<∞.
Then, {𝑥𝑛} converges strongly to 𝑃𝐹(𝑇)∩VI(𝐻,𝐵,𝑀)(𝑥).

Nakajo and Takahashi [7] introduced an iterative scheme for finding a fixed point of a nonexpansive mapping by a hybrid method which is called that shrinking projection method (or 𝐶𝑄 method) as in the following theorem.

Theorem NT. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑇)≠∅. Suppose that 𝑥1=𝑥∈𝐶 and {𝑥𝑛} is the sequence defined by 𝑦𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛,𝐶𝑛=‖‖𝑦𝑧∈𝐶∶𝑛‖‖≤‖‖𝑥−𝑧𝑛‖‖,𝑄−𝑧𝑛=𝑧∈𝐶∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝑥⟩≥0𝑛+1=𝑃𝐶𝑛∩𝑄𝑛𝑥1,∀𝑛∈ℕ,(1.10) where 0≤𝛼𝑛≤𝛼<1. Then, {𝑥𝑛} converges strongly to 𝑃𝐹(𝑇)(𝑥1).

In the same way, Kikkawa and Takahashi [8] introduced an iterative scheme for finding a common fixed point of an infinite family of nonexpansive mappings as follows:𝑦𝑛=𝑊𝑛𝑥𝑛,𝐶𝑛=‖‖𝑦𝑧∈𝐶∶𝑛‖‖≤‖‖𝑥−𝑧𝑛‖‖,𝑄−𝑧𝑛=𝑧∈𝐶∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝑥⟩≥0𝑛+1=𝑃𝐶𝑛∩𝑄𝑛𝑥1,∀𝑛∈ℕ,(1.11) where 𝑥1=𝑥∈𝐶 and 𝑊𝑛 is a 𝑊-mapping of 𝐶 into itself generated by {𝑇𝑛∶𝐶→𝐶} and {𝜇𝑛}. They prove that, if ⋂Ω=âˆžğ‘›=1𝐹(𝑇𝑛)≠∅, then the sequence {𝑥𝑛} generated by (1.11) converges strongly to 𝑃Ω(𝑥1).

Recently, Su and Qin [9] modified the shrinking projection method for finding a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Nakajo and Takahashi [7] as follows:𝑦𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛,𝐶𝑛=𝑧∈𝐶𝑛−1∩𝑄𝑛−1∶‖‖𝑦𝑛‖‖≤‖‖𝑥−𝑧𝑛‖‖𝑄−z,𝑛≥1,𝑛=𝑧∈𝐶𝑛−1∩𝑄𝑛−1∶⟨𝑥𝑛−𝑧,𝑥0−𝑥𝑛𝐶⟩≥0,𝑛≥1,0=‖‖𝑦𝑧∈𝐶∶0‖‖≤‖‖𝑥−𝑧0‖‖,𝑄−𝑧0𝑥=𝐶,𝑛+1=𝑃𝐶𝑛∩𝑄𝑛𝑥0,∀𝑛∈ℕ∪{0},(1.12) where 𝑥0=𝑥∈𝐶 and 𝑇 is a nonexpansive mapping of 𝐶 into itself. They prove that, under the parameter 0≤𝛼𝑛≤𝛼<1, if 𝐹(𝑇)≠∅, then the sequence {𝑥𝑛} generated by (1.12) converges strongly to 𝑃𝐹(𝑇)(𝑥0).

On the other hand, Tada and Takahashi [10] introduced an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of a fixed point problem of a nonexpansive mapping as follows:𝑢𝑛𝑢∈𝐶suchthat𝐹𝑛+1,𝑦𝑟𝑛⟨𝑦−𝑢𝑛,𝑢𝑛−𝑥𝑛𝑦⟩≥0,∀𝑦∈𝐶,𝑛=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑢𝑛,𝐶𝑛=‖‖𝑦𝑧∈𝐻∶𝑛‖‖≤‖‖𝑥−𝑧𝑛‖‖,𝑄−𝑧𝑛=𝑧∈𝐻∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝑥⟩≥0𝑛+1=𝑃𝐶𝑛∩𝑄𝑛𝑥1,∀𝑛∈ℕ,(1.13) where 𝑥1=𝑥∈𝐻, 𝑇 is a nonexpansive mapping of 𝐶 into 𝐻 and 𝐹 is a bifunction from 𝐶×𝐶 into ℝ. They prove that, under the sequences {𝛼𝑛}⊂[𝛼,1] for some 𝛼∈(0,1) and {𝑟𝑛}⊂[𝑟,∞) for some 𝑟>0, if Ω=𝐹(𝑇)∩EP(𝐹)≠∅, then the sequence {𝑥𝑛} generated by (1.13) converges strongly to 𝑃Ω(𝑥1) such that EP(𝐹) is the set of solutions of equilibrium problem defined by EP(𝐹)={𝑥∈𝐶∶𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶}.(1.14)

In this paper, we introduce an iterative scheme (1.15) for finding a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems by the shrinking projection method in Hilbert spaces as follows:𝑦𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛,𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖‖‖𝑊𝑛𝑥𝑛−𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖2,𝐶𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝜖𝑛,𝑄𝑛+1=𝑧∈𝐶𝑛∩Q𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐶⟩≥01=𝑄1𝑥=𝐻,𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(1.15) where 𝑥1=𝑢∈𝐻 chosen arbitrarily, 𝑀𝑖∶𝐻→2𝐻 is a maximal monotone mapping, 𝐵𝑖∶𝐻→𝐻 is a 𝜉𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁, and 𝑊𝑛 is a 𝑊-mapping on 𝐻 generated by {𝑆𝑛} and {𝜇𝑛} such that the mapping 𝑆𝑛∶𝐻→𝐻 defined by 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇𝑛𝑥 for all 𝑥∈𝐻, where {𝑇𝑛∶𝐻→𝐻} is an infinite family of 𝑘-strictly pseudocontractive mappings with a fixed point.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if 𝑘=0, then the iterative scheme (1.15) is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.

Furthermore, if 𝑀𝑖≡𝐵𝑖≡0 for all 𝑖=1,2,…,𝑁 and ∑𝑁𝑖=1𝛽𝑖=1, then the iterative scheme (1.15) is reduced to extend and improve the results of Kikkawa and Takahashi [8] for finding a common fixed point of an infinite family of 𝑘-strictly pseudocontractive mappings as follows:𝑥1𝑦=𝑢∈𝐶chosenarbitrarily,𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑥𝑛,𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖𝑊𝑛𝑥𝑛−𝑥𝑛‖‖2,𝐶𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝜖𝑛,𝑄𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐶⟩≥01=𝑄1𝑥=𝐶,𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(1.16) and if 𝑘=𝛼=0 and setting 𝑇1≡𝑇, 𝑇𝑛≡𝐼 for all 𝑛=2,3,…, then the iterative scheme (1.16) is reduced to find a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Su and Qin [9] as follows:𝑥1𝑦=𝑢∈𝐶chosenarbitrarily,𝑛=ğœŽğ‘›ğ‘‡ğ‘¥ğ‘›+1âˆ’ğœŽğ‘›î€¸ğ‘¥ğ‘›,𝛿𝑛=ğœŽğ‘›î€·1âˆ’ğœŽğ‘›î€¸â€–â€–ğ‘‡ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–2,𝐷𝑛+1=𝑧∈𝐷𝑛∩𝑄n∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝛿𝑛,𝑄𝑛+1=𝑧∈𝐷𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐷⟩≥01=𝑄1𝑥=𝐶,𝑛+1=𝑃𝐷𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ.(1.17)

We suggest and analyze the iterative scheme (1.15) under some appropriate conditions imposed on the parameters. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

2. Preliminaries

We collect the following lemmas which will be used in the proof of the main results in the next section.

Lemma 2.1 (see [11]). Let 𝐻 be a Hilbert space. For any 𝑥,𝑦∈𝐻 and 𝜆∈ℝ, one has ‖‖𝜆𝑥+(1−𝜆)𝑦2=𝜆‖𝑥‖2+(1−𝜆)‖𝑦‖2−𝜆(1−𝜆)‖𝑥−𝑦‖2.(2.1)

Lemma 2.2 (see [1]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Then the following inequality holds: ⟨𝑥−𝑃𝐶𝑥,𝑃𝐶𝑥−𝑦⟩≥0,∀𝑥∈𝐻,𝑦∈𝐶.(2.2)

Lemma 2.3 (see [5]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, define mapping 𝑊𝑛 as (1.8), let 𝑆𝑖∶𝐶→𝐶 be a family of infinitely nonexpansive mappings with â‹‚âˆžğ‘–=1𝐹(𝑆𝑖)≠∅, and let {𝜇𝑖} be a sequence such that 0<𝜇𝑖≤𝜇<1, for all 𝑖≥1. Then
(1)𝑊𝑛 is nonexpansive and 𝐹(𝑊𝑛⋂)=𝑛𝑖=1𝐹(𝑆𝑖) for each 𝑛≥1, (2)for each 𝑥∈𝐶 and for each positive integer 𝑘, limğ‘›â†’âˆžğ‘ˆğ‘›,𝑘𝑥 exists, (3)the mapping 𝑊∶𝐶→𝐶 defined by 𝑊𝑥∶=limğ‘›â†’âˆžğ‘Šğ‘›ğ‘¥=limğ‘›â†’âˆžğ‘ˆğ‘›,1𝑥,𝑥∈𝐶,(2.3) is a nonexpansive mapping satisfying ⋂𝐹(𝑊)=âˆžğ‘–=1𝐹(𝑆𝑖) and it is called the 𝑊-mapping generated by 𝑆1,𝑆2,… and 𝜇1,𝜇2,….

Lemma 2.4 (see [6]). The resolvent operator 𝐽𝑀,𝜆 associated with 𝑀 is single valued and nonexpansive for all 𝜆>0.

Lemma 2.5 (see [6]). 𝑢∈𝐻 is a solution of quasivariational inclusion (1.1) if and only if 𝑢=𝐽𝑀,𝜆(𝑢−𝜆𝐵𝑢), for all 𝜆>0, that is, 𝐽VI(𝐻,𝐵,𝑀)=𝐹𝑀,𝜆(𝐼−𝜆𝐵),∀𝜆>0.(2.4)

Lemma 2.6 (see [12]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇𝑛∶𝑛∈ℕ} be a sequence of nonexpansive mappings on 𝐶. Suppose that â‹‚âˆžğ‘›=1𝐹(𝑇𝑛)≠∅. Let {𝛼𝑛} be a sequence of positive real numbers such that âˆ‘âˆžğ‘›=1𝛼𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=âˆžî“ğ‘›=1𝛼𝑛𝑇𝑛𝑥,(2.5) for 𝑥∈𝐶, is well defined, nonexpansive and ⋂𝐹(𝑆)=âˆžğ‘›=1𝐹(𝑇𝑛) holds.

Lemma 2.7 (see [13]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 a nonexpansive mapping. Then 𝐼−𝑆 is demiclosed at zero. That is, whenever {𝑥𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑆)𝑥𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑆)𝑥=𝑦.

Lemma 2.8 (see [14]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a 𝑘-strict pseudocontraction. Define 𝑆∶𝐶→𝐶 by 𝑆𝑥=𝛼𝑥+(1−𝛼)𝑇𝑥 for each 𝑥∈𝐶. Then, as 𝛼∈[𝑘,1), S is nonexpansive such that 𝐹(𝑆)=𝐹(𝑇).

Lemma 2.9 (see [1]). Every Hilbert space 𝐻 has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {𝑥𝑛}⊂𝐻 with 𝑥𝑛⇀𝑥 and ‖𝑥𝑛‖→‖𝑥‖ then 𝑥𝑛→𝑥.

3. Main Results

Theorem 3.1. Let 𝐻 be a real Hilbert space, 𝑀𝑖∶𝐻→2𝐻 a maximal monotone mapping, and 𝐵𝑖∶𝐻→𝐻 a 𝜉𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let {𝑇𝑛∶𝐻→𝐻} be an infinite family of 𝑘-strictly pseudocontractive mappings with a fixed point such that 𝑘∈[0,1). Define a mapping 𝑆𝑛∶𝐻→𝐻 by 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇𝑛𝑥,∀𝑥∈𝐻,(3.1) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1). Let 𝑊𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑆𝑛} and {𝜇𝑛} such that {𝜇𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that ⋂Ω∶=(âˆžğ‘›=1𝐹(𝑇𝑛⋂))∩(𝑁𝑖=1VI(𝐻,𝐵𝑖,𝑀𝑖))≠∅. For 𝑥1=𝑢∈𝐻 chosen arbitrarily, suppose that {𝑥𝑛} is generated iteratively by 𝑦𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛,𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖‖‖𝑊𝑛𝑥𝑛−𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖2,𝐶𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝜖𝑛,𝑄𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐶⟩≥01=𝑄1𝑥=𝐻,𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(3.2) where
(C1) {𝛼𝑛}⊂[ğ‘Ž,𝑏] such that 0<ğ‘Ž<𝑏<1,
(C2) 𝛽𝑖∈(0,1) and 𝜆𝑖∈(0,2𝜉𝑖] for each 𝑖=1,2,…,𝑁,
(C3) ∑𝑁𝑖=1𝛽𝑖=1.
Then, the sequences {𝑥𝑛} and {𝑦𝑛} converge strongly to 𝑤=𝑃Ω(𝑥1).

Proof. For any 𝑥,𝑦∈𝐻 and for each 𝑖=1,2,…,𝑁, by the 𝜉𝑖-cocoercivity of 𝐵𝑖, we have ‖‖𝐼−𝜆𝑖𝐵𝑖𝑥−𝐼−𝜆𝑖𝐵𝑖𝑦‖‖2=‖‖(𝑥−𝑦)−𝜆𝑖𝐵𝑖𝑥−𝐵𝑖𝑦‖‖2=‖𝑥−𝑦‖2−2𝜆𝑖⟨𝑥−𝑦,𝐵𝑖𝑥−𝐵𝑖𝑦⟩+𝜆2𝑖‖‖𝐵𝑖𝑥−𝐵𝑖𝑦‖‖2≤‖𝑥−𝑦‖2−2𝜉𝑖−𝜆𝑖𝜆𝑖‖‖𝐵𝑖𝑥−𝐵𝑖𝑦‖‖2≤‖𝑥−𝑦‖2,(3.3) which implies that 𝐼−𝜆𝑖𝐵𝑖 is nonexpansive. Pick 𝑝∈Ω. Therefore, by Lemma 2.5, we have 𝑝=𝐽𝑀𝑖,𝜆𝑖𝐼−𝜆𝑖𝐵𝑖𝑝,(3.4) for each 𝑖=1,2,…,𝑁. Since 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇𝑛𝑥, where 𝛼∈[𝑘,1) and {𝑇𝑛} is a family of 𝑘-strict pseudocontraction, therefore, by Lemma 2.8, we have that 𝑆𝑛 is nonexpansive and 𝐹(𝑆𝑛)=𝐹(𝑇𝑛). It follows from Lemma 2.3(1) that 𝐹(𝑊𝑛⋂)=𝑛𝑖=1𝐹(𝑆𝑖⋂)=𝑛𝑖=1𝐹(𝑇𝑖), which implies that 𝑊𝑛𝑝=𝑝. Therefore, by (C3), (3.4), Lemma 2.1, and the nonexpansiveness of 𝑊𝑛,𝐽𝑀𝑖,𝜆𝑖, and 𝐼−𝜆𝑖𝐵𝑖, we have ‖‖𝑦𝑛‖‖−𝑝2=‖‖‖‖𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖−𝑝2=‖‖‖‖𝛼𝑛𝑊𝑛𝑥𝑛+−𝑝1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖−𝑝2=𝛼𝑛‖‖𝑊𝑛𝑥𝑛−𝑊𝑛𝑝‖‖2+1−𝛼𝑛‖‖‖‖𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛−𝐽𝑀𝑖,𝜆𝑖𝑝−𝜆𝑖𝐵𝑖𝑝‖‖‖‖2−𝛼𝑛1−𝛼𝑛‖‖‖‖𝑊𝑛𝑥𝑛−𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖2≤𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+1−𝛼𝑛𝑁𝑖=1𝛽𝑖‖‖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛−𝑝−𝜆𝑖𝐵𝑖𝑝‖‖2−𝜖𝑛≤𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2−𝜖𝑛=‖‖𝑥𝑛‖‖−𝑝2−𝜖𝑛,(3.5) for all 𝑛∈ℕ. Firstly, we prove that 𝐶𝑛∩𝑄𝑛 is closed and convex for all 𝑛∈ℕ. It is obvious that 𝐶1∩𝑄1 is closed and, by mathematical induction, that 𝐶𝑛∩𝑄𝑛 is closed for all 𝑛≥2, that is 𝐶𝑛∩𝑄𝑛 is closed for all 𝑛∈ℕ. Since ‖𝑦𝑛−𝑧‖2≤‖𝑥𝑛−𝑧‖2−𝜖𝑛 is equivalent to ‖‖𝑦𝑛−𝑥𝑛‖‖2+2⟨𝑦𝑛−𝑥𝑛,𝑥𝑛−z⟩+𝜖𝑛≤0,(3.6) for all 𝑛∈ℕ, therefore, for any 𝑧1,𝑧2∈𝐶𝑛+1∩𝑄𝑛+1⊂𝐶𝑛∩𝑄𝑛 and 𝜖∈(0,1), we have ‖‖𝑦𝑛−𝑥𝑛‖‖2𝑦+2𝑛−𝑥𝑛,𝑥𝑛−𝜖𝑧1+(1−𝜖)𝑧2+𝜖𝑛‖‖𝑦=𝜖𝑛−𝑥𝑛‖‖2+2⟨𝑦𝑛−𝑥𝑛,𝑥𝑛−𝑧1⟩+𝜖𝑛‖‖𝑦+(1−𝜖)𝑛−𝑥𝑛‖‖2+2⟨𝑦𝑛−𝑥𝑛,𝑥𝑛−𝑧2⟩+𝜖𝑛≤0,(3.7) for all 𝑛∈ℕ, and we have 𝑥𝑛−𝜖𝑧1+(1−𝜖)𝑧2,𝑥1−𝑥𝑛=𝜖⟨𝑥𝑛−𝑧1,𝑥1−𝑥𝑛⟩+(1−𝜖)⟨𝑥𝑛−𝑧2,𝑥1−𝑥𝑛⟩≥0,(3.8) for all 𝑛∈ℕ. Since 𝐶1∩𝑄1 is convex, and by putting 𝑛=1 in (3.6), (3.7), and (3.8), we have that 𝐶2∩𝑄2 is convex. Suppose that 𝑥𝑘 is given and 𝐶𝑘∩𝑄𝑘 is convex for some 𝑘≥2. It follows by putting 𝑛=𝑘 in (3.6), (3.7), and (3.8) that 𝐶𝑘+1∩𝑄𝑘+1 is convex. Therefore, by mathematical induction, we have that 𝐶𝑛∩𝑄𝑛 is convex for all 𝑛≥2, that is, 𝐶𝑛∩𝑄𝑛 is convex for all 𝑛∈ℕ. Hence, we obtain that 𝐶𝑛∩𝑄𝑛 is closed and convex for all 𝑛∈ℕ.
Next, we prove that Ω⊂𝐶𝑛∩𝑄𝑛 for all 𝑛∈ℕ. It is obvious that 𝑝∈Ω⊂𝐻=𝐶1∩𝑄1. Therefore, by (3.2) and (3.5), we have 𝑝∈𝐶2 and note that 𝑝∈𝐻=𝑄2, and so 𝑝∈𝐶2∩𝑄2. Hence, we have Ω⊂𝐶2∩𝑄2. Since 𝐶2∩𝑄2 is a nonempty closed convex subset of 𝐻, there exists a unique element 𝑥2∈𝐶2∩𝑄2 such that 𝑥2=𝑃𝐶2∩𝑄2(𝑥1). Suppose that 𝑥𝑘∈𝐶𝑘∩𝑄𝑘 is given such that 𝑥𝑘=𝑃𝐶𝑘∩𝑄𝑘(𝑥1), and 𝑝∈Ω⊂𝐶𝑘∩𝑄𝑘 for some 𝑘≥2. Therefore, by (3.2) and (3.5), we have 𝑝∈𝐶𝑘+1. Since 𝑥𝑘=𝑃𝐶𝑘∩𝑄𝑘(𝑥1), therefore, by Lemma 2.2, we have ⟨𝑥𝑘−𝑧,𝑥1−𝑥𝑘⟩≥0(3.9) for all 𝑧∈𝐶𝑘∩𝑄𝑘. Thus, by (3.2), we have 𝑝∈𝑄𝑘+1, and so 𝑝∈𝐶𝑘+1∩𝑄𝑘+1. Hence, we have Ω⊂𝐶𝑘+1∩𝑄𝑘+1. Since 𝐶𝑘+1∩𝑄𝑘+1 is a nonempty closed convex subset of 𝐻, there exists a unique element 𝑥𝑘+1∈𝐶𝑘+1∩𝑄𝑘+1 such that 𝑥𝑘+1=𝑃𝐶𝑘+1∩𝑄𝑘+1(𝑥1). Therefore, by mathematical induction, we obtain Ω⊂𝐶𝑛∩𝑄𝑛 for all 𝑛≥2, and so Ω⊂𝐶𝑛∩𝑄𝑛 for all 𝑛∈ℕ, and we can define 𝑥𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1(𝑥1) for all 𝑛∈ℕ. Hence, we obtain that the iteration (3.2) is well defined.
Next, we prove that {𝑥𝑛} is bounded. Since 𝑥𝑛=𝑃𝐶𝑛∩𝑄𝑛(𝑥1) for all 𝑛∈ℕ, we have ‖‖𝑥𝑛−𝑥1‖‖≤‖‖𝑧−𝑥1‖‖,(3.10) for all 𝑧∈𝐶𝑛∩𝑄𝑛. It follows by 𝑝∈Ω⊂𝐶𝑛∩𝑄𝑛 that ‖𝑥𝑛−𝑥1‖≤‖𝑝−𝑥1‖ for all 𝑛∈ℕ. This implies that {𝑥𝑛} is bounded, and so is {𝑦𝑛}.
Next, we prove that ‖𝑦𝑛−𝑥𝑛‖→0 as ğ‘›â†’âˆž. Since 𝑥𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1(𝑥1)∈𝐶𝑛+1∩𝑄𝑛+1⊂𝐶𝑛∩𝑄𝑛, therefore, by (3.10), we have ‖𝑥𝑛−𝑥1‖≤‖𝑥𝑛+1−𝑥1‖ for all 𝑛∈ℕ. This implies that {‖𝑥𝑛−𝑥1‖} is a bounded nondecreasing sequence and there exists the limit of ‖𝑥𝑛−𝑥1‖, that is, limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›âˆ’ğ‘¥1‖‖=𝑚,(3.11) for some 𝑚≥0. Since 𝑥𝑛+1∈𝑄𝑛+1, therefore, by (3.2), we have 𝑥𝑛−𝑥𝑛+1,𝑥1−x𝑛≥0.(3.12) It follows by (3.12) that ‖‖𝑥𝑛−𝑥𝑛+1‖‖2=‖‖𝑥𝑛−𝑥1+𝑥1−𝑥𝑛+1‖‖2=‖‖𝑥𝑛−𝑥1‖‖2+2⟨𝑥𝑛−𝑥1,𝑥1−𝑥𝑛⟩𝑥+2𝑛−𝑥1,𝑥𝑛−𝑥𝑛+1+‖‖𝑥𝑛+1−𝑥1‖‖2≤‖‖𝑥𝑛+1−𝑥1‖‖2−‖‖𝑥𝑛−𝑥1‖‖2.(3.13) Therefore, by (3.11), we have ‖‖𝑥𝑛−𝑥𝑛+1‖‖⟶0asğ‘›âŸ¶âˆž.(3.14) Since 𝑥𝑛+1∈𝐶𝑛+1, therefore, by (3.2), we have ‖‖𝑦𝑛−𝑥𝑛+1‖‖2≤‖‖𝑥𝑛−𝑥𝑛+1‖‖2−𝜖𝑛≤‖‖𝑥𝑛−𝑥𝑛+1‖‖2.(3.15) It follows by (3.15) that ‖‖𝑦𝑛−𝑥𝑛‖‖≤‖‖𝑦𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑥𝑛‖‖‖‖𝑥=2𝑛−𝑥𝑛+1‖‖.(3.16) Therefore, by (3.14), we obtain ‖‖𝑦𝑛−𝑥𝑛‖‖⟶0asğ‘›âŸ¶âˆž.(3.17)
Since {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} which converges weakly to 𝑤. Next, we prove that 𝑤∈Ω. Define the sequence of mappings {𝑄𝑛∶𝐻→𝐻} and the mapping 𝑄∶𝐻→𝐻 by 𝑄𝑛𝑥=𝛼𝑛𝑊𝑛𝑥+1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝐼−𝜆𝑖𝐵𝑖𝑥,∀𝑥∈𝐻,𝑄𝑥=limğ‘›â†’âˆžğ‘„ğ‘›ğ‘¥,(3.18) for all 𝑛∈ℕ. Therefore, by (C1) and Lemma 2.3(3), we have 𝑄𝑥=𝑐𝑊𝑥+(1−𝑐)𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝐼−𝜆𝑖𝐵𝑖𝑥,∀𝑥∈𝐻,(3.19) where ğ‘Žâ‰¤ğ‘=limğ‘›â†’âˆžğ›¼ğ‘›â‰¤ğ‘. From (C3) and Lemma 2.3(3), we have that 𝑊 and ∑𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖(𝐼−𝜆𝑖𝐵𝑖) are nonexpansive. Therefore, by (C2), (C3), Lemmas 2.3(3), 2.5, 2.6, and 2.8, we have 𝐹(𝑄)=𝐹(𝑊)∩𝐹𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝐼−𝜆𝑖𝐵𝑖=îƒ©âˆžî™ğ‘–=1𝐹𝑆𝑖∩𝑁𝑖=1𝐹𝐽𝑀𝑖,𝜆𝑖𝐼−𝜆𝑖𝐵𝑖=îƒ©î€¸î€¸âˆžî™ğ‘–=1𝐹𝑇𝑖∩𝑁𝑖=1VI𝐻,𝐵𝑖,𝑀𝑖,(3.20) that is, 𝐹(𝑄)=Ω. From (3.17), we have ‖𝑦𝑛𝑖−𝑥𝑛𝑖‖→0asğ‘–â†’âˆž. Thus, from (3.2) and (3.18), we get ‖𝑄𝑥𝑛𝑖−𝑥𝑛𝑖‖→0asğ‘–â†’âˆž. It follows from 𝑥𝑛𝑖⇀𝑤 and Lemma 2.7 that 𝑤∈𝐹(𝑄), that is, 𝑤∈Ω.
Since Ω is a nonempty closed convex subset of 𝐻, there exists a unique 𝑤∈Ω such that 𝑤=𝑃Ω(𝑥1). Next, we prove that 𝑥𝑛→𝑤 as ğ‘›â†’âˆž. Since 𝑤=𝑃Ω(𝑥1), we have ‖𝑥1−𝑤‖≤‖𝑥1−𝑧‖ for all 𝑧∈Ω, and it follows that ‖‖𝑥1‖‖≤‖‖𝑥−𝑤1−𝑤‖‖.(3.21) Since 𝑤∈Ω⊂𝐶𝑛∩𝑄𝑛, therefore, by (3.10), we have ‖‖𝑥1−𝑥𝑛‖‖≤‖‖𝑥1‖‖.−𝑤(3.22) Therefore, by (3.21), (3.22), and the weak lower semicontinuity of norm, we have ‖‖𝑥1‖‖≤‖‖𝑥−𝑤1−𝑤‖‖≤liminfğ‘–â†’âˆžâ€–â€–ğ‘¥1−𝑥𝑛𝑖‖‖≤limsupğ‘–â†’âˆžâ€–â€–ğ‘¥1−𝑥𝑛𝑖‖‖≤‖‖𝑥1‖‖.−𝑤(3.23) It follows that ‖‖𝑥1‖‖−𝑤=limğ‘–â†’âˆžâ€–â€–ğ‘¥1−𝑥𝑛𝑖‖‖=‖‖𝑥1−𝑤‖‖.(3.24) Since 𝑥𝑛𝑖⇀𝑤 as ğ‘–â†’âˆž, therefore, we have 𝑥1−𝑥𝑛𝑖⇀𝑥1−𝑤asğ‘–âŸ¶âˆž.(3.25) Hence, from (3.24), (3.25), the Kadec-Klee property, and the uniqueness of 𝑤=𝑃Ω(𝑥1), we obtain 𝑥𝑛𝑖⟶𝑤=𝑤asğ‘–âŸ¶âˆž.(3.26) It follows that {𝑥𝑛} converges strongly to 𝑤, and so is {𝑦𝑛}. This completes the proof.

Remark 3.2. The iteration (3.2) is the difference with some well known results as the following.
(1)The sequence {𝑥𝑛} is the projection sequence of 𝑥1 onto 𝐶𝑛∩𝑄𝑛 for all 𝑛∈ℕ such that 𝐶1∩𝑄1⊃𝐶2∩𝑄2⊃⋯⊃𝐶𝑛∩𝑄𝑛⊃⋯⊃Ω.(3.27)(2)The proof of 𝑤∈Ω is simple by the demiclosedness principle because the sequence {𝑦𝑛} is a linear nonexpansive mapping form of the mappings 𝑊𝑛 and 𝐽𝑀𝑖,𝜆𝑖(𝐼−𝜆𝑖𝐵𝑖).(3)Solving a common fixed point for an infinite family of strictly pseudocontractive mappings and a system of cocoercive quasivariational inclusions problems by iteration is obtained.

4. Applications

Theorem 4.1. Let 𝐻 be a real Hilbert space, 𝑀𝑖∶𝐻→2𝐻 a maximal monotone mapping, and 𝐵𝑖∶𝐻→𝐻 a 𝜉𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let {𝑇𝑛∶𝐻→𝐻} be an infinite family of nonexpansive mappings. Define a mapping 𝑆𝑛∶𝐻→𝐻 by 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇𝑛𝑥,∀𝑥∈𝐻,(4.1) for all 𝑛∈ℕ, where 𝛼∈[0,1). Let 𝑊𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑆𝑛} and {𝜇𝑛} such that {𝜇𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that ⋂Ω∶=(âˆžğ‘›=1𝐹(𝑇𝑛⋂))∩(𝑁𝑖=1VI(𝐻,𝐵𝑖,𝑀𝑖))≠∅. For 𝑥1=𝑢∈𝐻 chosen arbitrarily, suppose that {𝑥𝑛} is generated iteratively by 𝑦𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛,𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖‖‖𝑊𝑛𝑥𝑛−𝑁𝑖=1𝛽𝑖𝐽𝑀𝑖,𝜆𝑖𝑥𝑛−𝜆𝑖𝐵𝑖𝑥𝑛‖‖‖‖2,𝐶𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝜖𝑛,𝑄𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐶⟩≥01=𝑄1𝑥=𝐻,𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(4.2) where
(C1) {𝛼𝑛}⊂[ğ‘Ž,𝑏] such that 0<ğ‘Ž<𝑏<1,
(C2) 𝛽𝑖∈(0,1) and 𝜆𝑖∈(0,2𝜉𝑖] for each 𝑖=1,2,…,𝑁,
(C3) ∑𝑁𝑖=1𝛽𝑖=1.
Then the sequences {𝑥𝑛} and {𝑦𝑛} converge strongly to 𝑤=𝑃Ω(𝑥1).

Proof. It is concluded from Theorem 3.1 immediately, by putting 𝑘=0.

Theorem 4.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and {𝑇𝑛∶𝐶→𝐶} an infinite family of 𝑘-strictly pseudocontractive mappings with a fixed point such that 𝑘∈[0,1). Define a mapping 𝑆𝑛∶𝐶→𝐶 by 𝑆𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇𝑛𝑥,∀𝑥∈𝐶,(4.3) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1). Let 𝑊𝑛∶𝐶→𝐶 be a 𝑊-mapping generated by {𝑆𝑛} and {𝜇𝑛} such that {𝜇𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that ⋂Ω∶=âˆžğ‘›=1𝐹(𝑇𝑛)≠∅. For 𝑥1=𝑢∈𝐶 chosen arbitrarily, suppose that {𝑥𝑛} is generated iteratively by 𝑦𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑥𝑛,𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖𝑊𝑛𝑥𝑛−𝑥𝑛‖‖2,𝐶𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝜖𝑛,𝑄𝑛+1=𝑧∈𝐶𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐶⟩≥01=𝑄1𝑥=𝐶,𝑛+1=𝑃𝐶𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(4.4) where {𝛼𝑛}⊂[ğ‘Ž,𝑏] such that 0<ğ‘Ž<𝑏<1. Then the sequences {𝑥𝑛} and {𝑦𝑛} converge strongly to 𝑤=𝑃Ω(𝑥1).

Proof. It is concluded from Theorem 3.1 immediately, by putting 𝑀𝑖≡𝐵𝑖≡0 for all 𝑖=1,2,…,𝑁.

Theorem 4.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a nonexpansive mapping. Assume that 𝐹(𝑇)≠∅. For 𝑥1=𝑢∈𝐶 chosen arbitrarily, suppose that {𝑥𝑛} is generated iteratively by 𝑦𝑛=ğœŽğ‘›ğ‘‡ğ‘¥ğ‘›+1âˆ’ğœŽğ‘›î€¸ğ‘¥ğ‘›ğ›¿ğ‘›=ğœŽğ‘›î€·1âˆ’ğœŽğ‘›î€¸â€–â€–ğ‘‡ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–2,𝐷𝑛+1=𝑧∈𝐷𝑛∩𝑄𝑛∶‖‖𝑦𝑛‖‖−𝑧2≤‖‖𝑥𝑛‖‖−𝑧2−𝛿𝑛,𝑄𝑛+1=𝑧∈𝐷𝑛∩𝑄𝑛∶⟨𝑥𝑛−𝑧,𝑥1−𝑥𝑛,𝐷⟩≥01=𝑄1𝑥=𝐶,𝑛+1=𝑃𝐷𝑛+1∩𝑄𝑛+1𝑥1,∀𝑛∈ℕ,(4.5) where {ğœŽğ‘›}⊂[ğ‘Ž,𝑏] such that 0<ğ‘Ž<𝑏<1. Then the sequences {x𝑛} and {𝑦𝑛} converge strongly to 𝑤=𝑃𝐹(𝑇)(𝑥1).

Proof. It is concluded from Theorem 4.2, by putting 𝛼=0. Setting 𝑇1≡𝑇, 𝑇𝑛≡𝐼 for all 𝑛=2,3,… and leting 𝜇𝑛⊂(0,𝜇] for some 𝜇∈(0,1), therefore, from the definition of 𝑆𝑛 in Theorem 4.2, we have 𝑆1=𝑇1=𝑇 and 𝑆𝑛=𝐼 for all 𝑛=2,3,…. Since 𝑊𝑛 is a 𝑊-mapping generated by {𝑆𝑛} and {𝜇𝑛}, therefore, by the definition of 𝑈𝑛,𝑖 and 𝑊𝑛 in (1.8), we have 𝑈𝑛,𝑖=𝐼 for all 𝑖=2,3,… and 𝑊𝑛=𝑈𝑛,1=𝜇1𝑆1𝑈𝑛,2+(1−𝜇1)𝐼=𝜇1𝑇+(1−𝜇1)𝐼. Hence, by Theorem 4.2, we obtain 𝑦𝑛=𝛼𝑛𝑊𝑛𝑥𝑛+1−𝛼𝑛𝑥𝑛=𝛼𝑛𝜇1𝑇𝑥𝑛+1−𝜇1𝑥𝑛+1−𝛼𝑛𝑥𝑛=ğœŽğ‘›ğ‘‡ğ‘¥ğ‘›+1âˆ’ğœŽğ‘›î€¸ğ‘¥ğ‘›,(4.6) where ğœŽğ‘›âˆ¶=𝛼𝑛𝜇1. Since, the same as in the proof of Theorem 3.1, we have that 𝐷𝑛∩𝑄𝑛 is a nonempty closed convex subset of 𝐶 for all 𝑛∈ℕ and by Theorem 4.2, we have 𝜖𝑛=𝛼𝑛1−𝛼𝑛‖‖𝑊𝑛𝑥𝑛−𝑥𝑛‖‖2=𝛼𝑛1−𝛼𝑛‖‖𝜇1𝑇𝑥𝑛+(1−𝜇1)𝑥𝑛−𝑥𝑛‖‖2=𝛼𝑛𝜇1𝜇1−𝜇1𝛼𝑛‖‖𝑇𝑥𝑛−𝑥𝑛‖‖2=ğœŽğ‘›î€·ğœ‡1âˆ’ğœŽğ‘›î€¸â€–â€–ğ‘‡ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–2â‰¤ğœŽğ‘›î€·1âˆ’ğœŽğ‘›î€¸â€–â€–ğ‘‡ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–2=𝛿𝑛,(4.7) for all 𝑛∈ℕ. It follows that 𝐷𝑛⊂𝐶𝑛 for all 𝑛∈ℕ, where 𝐶𝑛 is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.