Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

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, 2^𝐻 denoting the family of all the nonempty subsets of 𝐻.

Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2^𝐻 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 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶; that is, for each 𝑥∈𝐻, there exists the unique point in 𝑃_𝐶𝑥∈𝐶 such that ‖𝑥−𝑃_𝐶𝑥‖=min_𝑦∈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)𝑦‖²,∀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)𝑦‖²,∀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 𝑊_𝑛 defined by𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝜇_𝑛𝑆_𝑛𝑈_𝑛,𝑛+1+1−𝜇_𝑛𝐼,_{𝑛,𝑛−1}𝑈=𝜇_𝑛−1𝑆_𝑛−1⋮𝑈_𝑛,𝑛+1−𝜇_𝑛−1𝐼,_𝑛,𝑘𝑈=𝜇_𝑘𝑆_𝑘𝑈_𝑛,𝑘+1+1−𝜇_𝑘𝐼,_{𝑛,𝑘−1}𝑈=𝜇_𝑘−1𝑆_𝑘−1⋮𝑈_𝑛,𝑘+1−𝜇_𝑘−1𝐼,_𝑛,2𝑊=𝜇₂𝑆₂𝑈_𝑛,3+1−𝜇₂𝐼,_𝑛=𝑈_𝑛,1=𝜇₁𝑆₁𝑈_𝑛,2+1−𝜇₁𝐼,(1.8) where {𝜇_𝑖} 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 𝑊_𝑛 is a nonexpansive mapping of 𝐶 into itself; such a mapping 𝑊_𝑛 is called a 𝑊-mapping generated by 𝑆₁,𝑆₂,…,𝑆_𝑛 and 𝜇₁,𝜇₂,…,𝜇_𝑛.

Definition 1.1 (see [6]). Let 𝑀∶𝐻→2^𝐻 be a multivalued maximal monotone mapping. Then, the single-valued mapping 𝐽_𝑀,𝜆∶𝐻→𝐻 defined by 𝐽_𝑀,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→2^𝐻 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 {𝑥_𝑛} is the sequence defined by 𝑥𝑦_𝑛𝑛+1=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛,=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇𝑦_𝑛,(1.9) for all 𝑛∈ℕ, where 𝜆∈(0,2𝛼) and {𝛼_𝑛}⊂(0,1) satisfying the following conditions:
(C1) lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,
(C2) ∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞.
Then, {𝑥_𝑛} converges strongly to 𝑃_{𝐹(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 {𝑥_𝑛} is the sequence defined by 𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ,(1.10) where 0≤𝛼_𝑛≤𝛼<1. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹(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:𝐶_𝑛𝑦_𝑛=𝑊_𝑛𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ,(1.11) where 𝑥₁=𝑥∈𝐶 and 𝑊_𝑛 is a 𝑊-mapping of 𝐶 into itself generated by {𝑇_𝑛∶𝐶→𝐶} and {𝜇_𝑛}. They prove that, if Ω=⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅, then the sequence {𝑥_𝑛} generated by (1.11) converges strongly to 𝑃_Ω(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:𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−z‖‖,𝑛≥1,_𝑛=𝑧∈𝐶𝐶_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0,𝑛≥1,₀=𝑧∈𝐶∶‖‖𝑄𝑦₀−𝑧‖‖≤‖‖𝑥₀−𝑧₀‖‖,𝑥=𝐶,_𝑛+1=𝑃_{𝐶𝑛∩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≤𝛼_𝑛≤𝛼<1, if 𝐹(𝑇)≠∅, then the sequence {𝑥_𝑛} generated by (1.12) converges strongly to 𝑃_𝐹(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:𝑢_𝑛∈𝐶suchthat𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝐶_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑢_𝑛,𝑄=𝑧∈𝐻∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑥𝑧∈𝐻∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩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 {𝛼_𝑛}⊂[𝛼,1] for some 𝛼∈(0,1) and {𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0, if Ω=𝐹(𝑇)∩EP(𝐹)≠∅, then the sequence {𝑥_𝑛} generated by (1.13) converges strongly to 𝑃_Ω(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:𝑦_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝜖_𝑛𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛,=𝛼_𝑛1−𝛼_𝑛‖‖‖‖𝑊_𝑛𝑥_𝑛−^𝑁_𝑖=1𝐶_𝑛+1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛‖‖‖‖²,𝑄=_𝑛+1𝑧∈𝐶_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝜖_𝑛,=𝑧∈𝐶_𝑛∩𝐶Q_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐻,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(1.15) where 𝑥₁=𝑢∈𝐻 chosen arbitrarily, 𝑀_𝑖∶𝐻→2^𝐻 is a maximal monotone mapping, 𝐵_𝑖∶𝐻→𝐻 is a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁, and 𝑊_𝑛 is a 𝑊-mapping on 𝐻 generated by {𝑆_𝑛} and {𝜇_𝑛} such that the mapping 𝑆_𝑛∶𝐻→𝐻 defined by 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥 for all 𝑥∈𝐻, where {𝑇_𝑛∶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 𝑀_𝑖≡𝐵_𝑖≡0 for all 𝑖=1,2,…,𝑁 and ∑^𝑁𝑖=1𝛽_𝑖=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:𝑥𝑦₁=𝑢∈𝐶chosenarbitrarily,_𝑛𝜖_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛,𝐶_𝑛+1=𝛼_𝑛1−𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑥_𝑛‖‖²,𝑄=_𝑛+1𝑧∈𝐶_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝜖_𝑛,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(1.16) and if 𝑘=𝛼=0 and setting 𝑇₁≡𝑇, 𝑇_𝑛≡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,_𝑛𝛿_𝑛=𝜎_𝑛𝑇𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛,𝐷_𝑛+1=𝜎_𝑛1−𝜎_𝑛‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²,𝑄=_𝑛+1𝑧∈𝐷_𝑛∩𝑄_n∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝛿_𝑛,=𝑧∈𝐷_𝑛∩𝐷𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐶,_𝑛+1=𝑃_{𝐷𝑛+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)𝑦‖²=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−𝑃_𝐶𝑥,𝑃_𝐶𝑥−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 𝑊_𝑛 as (1.8), let 𝑆_𝑖∶𝐶→𝐶 be a family of infinitely nonexpansive mappings with ⋂^∞𝑖=1𝐹(𝑆_𝑖)≠∅, and let {𝜇_𝑖} be a sequence such that 0<𝜇_𝑖≤𝜇<1, for all 𝑖≥1. Then
(1)𝑊_𝑛 is nonexpansive and 𝐹(𝑊_𝑛)=⋂^𝑛𝑖=1𝐹(𝑆_𝑖) for each 𝑛≥1, (2)for each 𝑥∈𝐶 and for each positive integer 𝑘, lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists, (3)the mapping 𝑊∶𝐶→𝐶 defined by 𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,𝑥∈𝐶,(2.3) is a nonexpansive mapping satisfying 𝐹(𝑊)=⋂^∞𝑖=1𝐹(𝑆_𝑖) and it is called the 𝑊-mapping generated by 𝑆₁,𝑆₂,… and 𝜇₁,𝜇₂,….

Lemma 2.4 (see [6]). The resolvent operator 𝐽_𝑀,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=𝐽_𝑀,𝜆(𝑢−𝜆𝐵𝑢), for all 𝜆>0, that is, VI(𝐻,𝐵,𝑀)=𝐹𝐽_𝑀,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 {𝑇_𝑛∶𝑛∈ℕ} be a sequence of nonexpansive mappings on 𝐶. Suppose that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Let {𝛼_𝑛} be a sequence of positive real numbers such that ∑^∞𝑛=1𝛼_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=^∞_𝑛=1𝛼_𝑛𝑇_𝑛𝑥,(2.5) for 𝑥∈𝐶, is well defined, nonexpansive and 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑇_𝑛) 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 {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑆)𝑥_𝑛} 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 {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖ then 𝑥_𝑛→𝑥.

3. Main Results

Theorem 3.1. Let 𝐻 be a real Hilbert space, 𝑀_𝑖∶𝐻→2^𝐻 a maximal monotone mapping, and 𝐵_𝑖∶𝐻→𝐻 a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let {𝑇_𝑛∶𝐻→𝐻} be an infinite family of 𝑘-strictly pseudocontractive mappings with a fixed point such that 𝑘∈[0,1). Define a mapping 𝑆_𝑛∶𝐻→𝐻 by 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻,(3.1) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1). Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑆_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁𝑖=1VI(𝐻,𝐵_𝑖,𝑀_𝑖))≠∅. For 𝑥₁=𝑢∈𝐻 chosen arbitrarily, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝜖_𝑛𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛,=𝛼_𝑛1−𝛼_𝑛‖‖‖‖𝑊_𝑛𝑥_𝑛−^𝑁_𝑖=1𝐶_𝑛+1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛‖‖‖‖²,𝑄=_𝑛+1𝑧∈𝐶_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝜖_𝑛,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐻,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(3.2) where
(C1) {𝛼_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<1,
(C2) 𝛽_𝑖∈(0,1) and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁,
(C3) ∑^𝑁𝑖=1𝛽_𝑖=1.
Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₁).

Proof. For any 𝑥,𝑦∈𝐻 and for each 𝑖=1,2,…,𝑁, by the 𝜉_𝑖-cocoercivity of 𝐵_𝑖, we have ‖‖𝐼−𝜆_𝑖𝐵_𝑖𝑥−𝐼−𝜆_𝑖𝐵_𝑖𝑦‖‖²=‖‖(𝑥−𝑦)−𝜆_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦‖‖²=+‖𝑥𝜆_2𝑖−𝑦‖²−2𝜆_𝑖⟨𝑥−𝑦,𝐵_𝑖𝑥−𝐵_𝑖𝑦⟩‖‖𝐵_𝑖𝑥−𝐵_𝑖𝑦‖‖²≤‖𝑥−𝑦‖²−2𝜉_𝑖−𝜆_𝑖𝜆_𝑖‖‖𝐵_𝑖𝑥−𝐵_𝑖𝑦‖‖²≤‖𝑥−𝑦‖²,(3.3) which implies that 𝐼−𝜆_𝑖𝐵_𝑖 is nonexpansive. Pick 𝑝∈Ω. Therefore, by Lemma 2.5, we have 𝑝=𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐵_𝑖𝑝,(3.4) for each 𝑖=1,2,…,𝑁. Since 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥, where 𝛼∈[𝑘,1) and {𝑇_𝑛} is a family of 𝑘-strict pseudocontraction, therefore, by Lemma 2.8, we have that 𝑆_𝑛 is nonexpansive and 𝐹(𝑆_𝑛)=𝐹(𝑇_𝑛). It follows from Lemma 2.3(1) that 𝐹(𝑊_𝑛)=⋂^𝑛𝑖=1𝐹(𝑆_𝑖)=⋂^𝑛𝑖=1𝐹(𝑇_𝑖), which implies that 𝑊_𝑛𝑝=𝑝. Therefore, by (C3), (3.4), Lemma 2.1, and the nonexpansiveness of 𝑊_𝑛,𝐽_{𝑀𝑖,𝜆𝑖}, and 𝐼−𝜆_𝑖𝐵_𝑖, we have ‖‖𝑦_𝑛−𝑝‖‖²=‖‖‖‖𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛−𝑝‖‖‖‖²=‖‖‖‖𝛼_𝑛𝑊_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛^𝑁_𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛−𝑝‖‖‖‖²=𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑝‖‖²+1−𝛼_𝑛‖‖‖‖^𝑁_𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛−𝐽_{𝑀𝑖,𝜆𝑖}𝑝−𝜆_𝑖𝐵_𝑖𝑝‖‖‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖‖‖𝑊_𝑛𝑥_𝑛−^𝑁_𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛‖‖‖‖²≤𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛^𝑁_𝑖=1𝛽_𝑖‖‖𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛−𝑝−𝜆_𝑖𝐵_𝑖𝑝‖‖²−𝜖_𝑛≤𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²−𝜖_𝑛=‖‖𝑥_𝑛−𝑝‖‖²−𝜖_𝑛,(3.5) for all 𝑛∈ℕ. Firstly, we prove that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ. It is obvious that 𝐶₁∩𝑄₁ is closed and, by mathematical induction, that 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛≥2, that is 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛∈ℕ. Since ‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−𝜖_𝑛 is equivalent to ‖‖𝑦_𝑛−𝑥_𝑛‖‖²+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−z⟩+𝜖_𝑛≤0,(3.6) for all 𝑛∈ℕ, therefore, for any 𝑧₁,𝑧₂∈𝐶_𝑛+1∩𝑄_𝑛+1⊂𝐶_𝑛∩𝑄_𝑛 and 𝜖∈(0,1), we have ‖‖𝑦_𝑛−𝑥_𝑛‖‖²+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂+𝜖_𝑛=+𝜖(1‖‖−𝑦_𝑛𝜖−)𝑥^𝑛‖‖‖‖𝑦²_𝑛+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₁⟩+𝜖_𝑛≤0,−𝑥_𝑛‖‖²+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₂⟩+𝜖_𝑛(3.7) for all 𝑛∈ℕ, and we have 𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂,𝑥₁−𝑥_𝑛=𝜖⟨𝑥_𝑛−𝑧₁,𝑥₁−𝑥_𝑛⟩+(1−𝜖)⟨𝑥_𝑛−𝑧₂,𝑥₁−𝑥_𝑛⟩≥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 𝑥_𝑘 is given and 𝐶_𝑘∩𝑄_𝑘 is convex for some 𝑘≥2. It follows by putting 𝑛=𝑘 in (3.6), (3.7), and (3.8) that 𝐶_𝑘+1∩𝑄_𝑘+1 is convex. Therefore, by mathematical induction, we have that 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛≥2, that is, 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛∈ℕ. Hence, we obtain that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ.
Next, we prove that Ω⊂𝐶_𝑛∩𝑄_𝑛 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=𝑃_{𝐶2∩𝑄2}(𝑥₁). Suppose that 𝑥_𝑘∈𝐶_𝑘∩𝑄_𝑘 is given such that 𝑥_𝑘=𝑃_{𝐶𝑘∩𝑄𝑘}(𝑥₁), and 𝑝∈Ω⊂𝐶_𝑘∩𝑄_𝑘 for some 𝑘≥2. Therefore, by (3.2) and (3.5), we have 𝑝∈𝐶_𝑘+1. Since 𝑥_𝑘=𝑃_{𝐶𝑘∩𝑄𝑘}(𝑥₁), therefore, by Lemma 2.2, we have ⟨𝑥_𝑘−𝑧,𝑥₁−𝑥_𝑘⟩≥0(3.9) for all 𝑧∈𝐶_𝑘∩𝑄_𝑘. Thus, by (3.2), we have 𝑝∈𝑄_𝑘+1, and so 𝑝∈𝐶_𝑘+1∩𝑄_𝑘+1. Hence, we have Ω⊂𝐶_𝑘+1∩𝑄_𝑘+1. Since 𝐶_𝑘+1∩𝑄_𝑘+1 is a nonempty closed convex subset of 𝐻, there exists a unique element 𝑥_𝑘+1∈𝐶_𝑘+1∩𝑄_𝑘+1 such that 𝑥_𝑘+1=𝑃_{𝐶𝑘+1∩𝑄𝑘+1}(𝑥₁). Therefore, by mathematical induction, we obtain Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛≥2, and so Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ, and we can define 𝑥_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}(𝑥₁) for all 𝑛∈ℕ. Hence, we obtain that the iteration (3.2) is well defined.
Next, we prove that {𝑥_𝑛} is bounded. Since 𝑥_𝑛=𝑃_{𝐶𝑛∩𝑄𝑛}(𝑥₁) for all 𝑛∈ℕ, we have ‖‖𝑥_𝑛−𝑥₁‖‖≤‖‖𝑧−𝑥₁‖‖,(3.10) for all 𝑧∈𝐶_𝑛∩𝑄_𝑛. It follows by 𝑝∈Ω⊂𝐶_𝑛∩𝑄_𝑛 that ‖𝑥_𝑛−𝑥₁‖≤‖𝑝−𝑥₁‖ for all 𝑛∈ℕ. This implies that {𝑥_𝑛} is bounded, and so is {𝑦_𝑛}.
Next, we prove that ‖𝑦_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. Since 𝑥_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}(𝑥₁)∈𝐶_𝑛+1∩𝑄_𝑛+1⊂𝐶_𝑛∩𝑄_𝑛, therefore, by (3.10), we have ‖𝑥_𝑛−𝑥₁‖≤‖𝑥_𝑛+1−𝑥₁‖ for all 𝑛∈ℕ. This implies that {‖𝑥_𝑛−𝑥₁‖} is a bounded nondecreasing sequence and there exists the limit of ‖𝑥_𝑛−𝑥₁‖, that is, lim_𝑛→∞‖‖𝑥_𝑛−𝑥₁‖‖=𝑚,(3.11) for some 𝑚≥0. Since 𝑥_𝑛+1∈𝑄_𝑛+1, therefore, by (3.2), we have 𝑥_𝑛−𝑥_𝑛+1,𝑥₁−x_𝑛≥0.(3.12) It follows by (3.12) that ‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²=‖‖𝑥_𝑛−𝑥₁+𝑥₁−𝑥_𝑛+1‖‖²=+‖‖𝑥2_𝑛−𝑥_𝑛𝑥₁‖‖²+2⟨𝑥_𝑛−𝑥₁,𝑥₁−𝑥_𝑛⟩−𝑥₁,𝑥_𝑛−𝑥_𝑛+1+‖‖𝑥_𝑛+1−𝑥₁‖‖²≤‖‖𝑥_𝑛+1−𝑥₁‖‖²−‖‖𝑥_𝑛−𝑥₁‖‖².(3.13) Therefore, by (3.11), we have ‖‖𝑥_𝑛−𝑥_𝑛+1‖‖⟶0as𝑛⟶∞.(3.14) Since 𝑥_𝑛+1∈𝐶_𝑛+1, therefore, by (3.2), we have ‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²−𝜖_𝑛≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖².(3.15) It follows by (3.15) that ‖‖𝑦_𝑛−𝑥_𝑛‖‖≤≤‖‖‖‖𝑦𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=2‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖−𝑥_𝑛+1‖‖.(3.16) Therefore, by (3.14), we obtain ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.17)
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} which converges weakly to 𝑤. Next, we prove that 𝑤∈Ω. Define the sequence of mappings {𝑄_𝑛∶𝐻→𝐻} and the mapping 𝑄∶𝐻→𝐻 by 𝑄_𝑛𝑥=𝛼_𝑛𝑊_𝑛𝑥+1−𝛼_𝑛^𝑁_𝑖=1𝑄𝑥=𝛽li_𝑖m𝐽_{𝑀𝑖,𝜆𝑖𝑛→∞}𝐼−𝜆_𝑖𝐵_𝑖𝑥,∀𝑥∈𝐻,𝑄_𝑛𝑥,(3.18) for all 𝑛∈ℕ. Therefore, by (C1) and Lemma 2.3(3), we have 𝑄𝑥=𝑐𝑊𝑥+(1−𝑐)^𝑁_𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐵_𝑖𝑥,∀𝑥∈𝐻,(3.19) where 𝑎≤𝑐=lim_𝑛→∞𝛼_𝑛≤𝑏. From (C3) and Lemma 2.3(3), we have that 𝑊 and ∑^𝑁𝑖=1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}(𝐼−𝜆_𝑖𝐵_𝑖) are nonexpansive. Therefore, by (C2), (C3), Lemmas 2.3(3), 2.5, 2.6, and 2.8, we have 𝐹(𝑄)=𝐹(𝑊)∩𝐹^𝑁_𝑖=1=_∞𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐵_𝑖_𝑖=1𝐹𝑆_𝑖∩^𝑁_𝑖=1=𝐹𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐵_𝑖_∞_𝑖=1𝐹𝑇_𝑖∩^𝑁_𝑖=1VI𝐻,𝐵_𝑖,𝑀_𝑖,(3.20) that is, 𝐹(𝑄)=Ω. From (3.17), we have ‖𝑦_𝑛𝑖−𝑥_𝑛𝑖‖→0as𝑖→∞. Thus, from (3.2) and (3.18), we get ‖𝑄𝑥_𝑛𝑖−𝑥_𝑛𝑖‖→0as𝑖→∞. It follows from 𝑥_𝑛𝑖⇀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=𝑃_Ω(𝑥₁). Next, we prove that 𝑥_𝑛→𝑤 as 𝑛→∞. Since 𝑤=𝑃_Ω(𝑥₁), we have ‖𝑥₁−𝑤‖≤‖𝑥₁−𝑧‖ for all 𝑧∈Ω, and it follows that ‖‖𝑥₁−𝑤‖‖≤‖‖𝑥₁−𝑤‖‖.(3.21) Since 𝑤∈Ω⊂𝐶_𝑛∩𝑄_𝑛, therefore, by (3.10), we have ‖‖𝑥₁−𝑥_𝑛‖‖≤‖‖𝑥₁−𝑤‖‖.(3.22) Therefore, by (3.21), (3.22), and the weak lower semicontinuity of norm, we have ‖‖𝑥₁−𝑤‖‖≤≤l‖‖im𝑥₁i−nf_𝑖→∞𝑤‖‖≤limsup_𝑖→∞‖‖𝑥₁−𝑥_𝑛𝑖‖‖≤‖‖𝑥₁‖‖𝑥₁−𝑥_𝑛𝑖‖‖−𝑤‖‖.(3.23) It follows that ‖‖𝑥₁−𝑤‖‖=lim_𝑖→∞‖‖𝑥₁−𝑥_𝑛𝑖‖‖=‖‖𝑥₁−𝑤‖‖.(3.24) Since 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞, therefore, we have 𝑥₁−𝑥_𝑛𝑖⇀𝑥₁−𝑤as𝑖⟶∞.(3.25) Hence, from (3.24), (3.25), the Kadec-Klee property, and the uniqueness of 𝑤=𝑃_Ω(𝑥₁), we obtain 𝑥_𝑛𝑖⟶𝑤=𝑤as𝑖⟶∞.(3.26) It follows that {𝑥_𝑛} converges strongly to 𝑤, and so is {𝑦_𝑛}. 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 {𝑥_𝑛} is the projection sequence of 𝑥₁ onto 𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ such that 𝐶₁∩𝑄₁⊃𝐶₂∩𝑄₂⊃⋯⊃𝐶_𝑛∩𝑄_𝑛⊃⋯⊃Ω.(3.27)(2)The proof of 𝑤∈Ω is simple by the demiclosedness principle because the sequence {𝑦_𝑛} is a linear nonexpansive mapping form of the mappings 𝑊_𝑛 and 𝐽_{𝑀𝑖,𝜆𝑖}(𝐼−𝜆_𝑖𝐵_𝑖).(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, 𝑀_𝑖∶𝐻→2^𝐻 a maximal monotone mapping, and 𝐵_𝑖∶𝐻→𝐻 a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let {𝑇_𝑛∶𝐻→𝐻} be an infinite family of nonexpansive mappings. Define a mapping 𝑆_𝑛∶𝐻→𝐻 by 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻,(4.1) for all 𝑛∈ℕ, where 𝛼∈[0,1). Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑆_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁𝑖=1VI(𝐻,𝐵_𝑖,𝑀_𝑖))≠∅. For 𝑥₁=𝑢∈𝐻 chosen arbitrarily, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝜖_𝑛𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛,=𝛼_𝑛1−𝛼_𝑛‖‖‖‖𝑊_𝑛𝑥_𝑛−^𝑁_𝑖=1𝐶_𝑛+1𝛽_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐵_𝑖𝑥_𝑛‖‖‖‖²,𝑄=_𝑛+1𝑧∈𝐶_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝜖_𝑛,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐻,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(4.2) where
(C1) {𝛼_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<1,
(C2) 𝛽_𝑖∈(0,1) and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁,
(C3) ∑^𝑁𝑖=1𝛽_𝑖=1.
Then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤=𝑃_Ω(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 {𝑇_𝑛∶𝐶→𝐶} an infinite family of 𝑘-strictly pseudocontractive mappings with a fixed point such that 𝑘∈[0,1). Define a mapping 𝑆_𝑛∶𝐶→𝐶 by 𝑆_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐶,(4.3) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1). Let 𝑊_𝑛∶𝐶→𝐶 be a 𝑊-mapping generated by {𝑆_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. For 𝑥₁=𝑢∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛} is generated iteratively by 𝜖_𝑛𝑦_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛,𝐶_𝑛+1=𝛼_𝑛1−𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑥_𝑛‖‖²,𝑄=_𝑛+1𝑧∈𝐶_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝜖_𝑛,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(4.4) where {𝛼_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<1. Then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₁).

Proof. It is concluded from Theorem 3.1 immediately, by putting 𝑀_𝑖≡𝐵_𝑖≡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 {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝜎_𝑛𝑇𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛𝐷_𝑛+1𝛿_𝑛=𝜎_𝑛1−𝜎_𝑛‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²,𝑄=_𝑛+1𝑧∈𝐷_𝑛∩𝑄_𝑛∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²−𝛿_𝑛,=𝑧∈𝐷_𝑛∩𝐷𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0₁,𝑥=𝑄₁=𝐶,_𝑛+1=𝑃_{𝐷𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ,(4.5) where {𝜎_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<1. Then the sequences {x_𝑛} and {𝑦_𝑛} converge strongly to 𝑤=𝑃_𝐹(𝑇)(𝑥₁).

Proof. It is concluded from Theorem 4.2, by putting 𝛼=0. Setting 𝑇₁≡𝑇, 𝑇_𝑛≡𝐼 for all 𝑛=2,3,… and leting 𝜇_𝑛⊂(0,𝜇] for some 𝜇∈(0,1), therefore, from the definition of 𝑆_𝑛 in Theorem 4.2, we have 𝑆₁=𝑇₁=𝑇 and 𝑆_𝑛=𝐼 for all 𝑛=2,3,…. Since 𝑊_𝑛 is a 𝑊-mapping generated by {𝑆_𝑛} and {𝜇_𝑛}, therefore, by the definition of 𝑈_𝑛,𝑖 and 𝑊_𝑛 in (1.8), we have 𝑈_𝑛,𝑖=𝐼 for all 𝑖=2,3,… and 𝑊_𝑛=𝑈_𝑛,1=𝜇₁𝑆₁𝑈_𝑛,2+(1−𝜇₁)𝐼=𝜇₁𝑇+(1−𝜇₁)𝐼. Hence, by Theorem 4.2, we obtain 𝑦_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛=𝛼_𝑛𝜇₁𝑇𝑥_𝑛+1−𝜇₁𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛=𝜎_𝑛𝑇𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛,(4.6) where 𝜎_𝑛∶=𝛼_𝑛𝜇₁. Since, the same as in the proof of Theorem 3.1, we have that 𝐷_𝑛∩𝑄_𝑛 is a nonempty closed convex subset of 𝐶 for all 𝑛∈ℕ and by Theorem 4.2, we have 𝜖_𝑛=𝛼_𝑛1−𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑥_𝑛‖‖²=𝛼_𝑛1−𝛼_𝑛‖‖𝜇₁𝑇𝑥_𝑛+(1−𝜇₁)𝑥_𝑛−𝑥_𝑛‖‖²=𝛼_𝑛𝜇₁𝜇₁−𝜇₁𝛼_𝑛‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²=𝜎_𝑛𝜇₁−𝜎_𝑛‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²≤𝜎_𝑛1−𝜎_𝑛‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²=𝛿_𝑛,(4.7) for all 𝑛∈ℕ. It follows that 𝐷_𝑛⊂𝐶_𝑛 for all 𝑛∈ℕ, where 𝐶_𝑛 is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.