On Common Solutions for Fixed-Point Problems of Two Infinite Families 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 two infinite families 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 novel general iterative scheme based on the viscosity approximation method, and applicability of the results has shown difference with the results of many others 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, and 2^𝐻 denotes 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 such that𝜃∈𝐵𝑥+𝑀𝑥, (1.1) where 𝜃 is the zero vector in 𝐻. The set of solutions of the problem (1.1) is denoted by VI(𝐻,𝐵,𝑀). As special cases of the problem (1.1), we have the following. (i)If 𝑀=𝜕𝜙∶𝐻→2^𝐻, where 𝜙∶𝐻→ℝ∪{+∞} is a proper convex lower semicontinuous function such that ℝ is the set of real numbers, and 𝜕𝜙 is the subdifferential of 𝜙, then the quasivariational inclusion problem (1.1) is equivalent to find 𝑥∈𝐻 such that ⟨𝐵𝑥,𝑣−𝑥⟩+𝜙𝑦−𝜙(𝑥)≥0,∀𝑣,𝑦∈𝐻, (1.2) which is called the mixed quasivariational inequality problem (see [1]). (ii)If 𝑀=𝜕𝛿_𝐶, where 𝛿_𝐶∶𝐻→{0,+∞} is the indicator function of 𝐶, that is, 𝛿_𝐶(𝑥)=0,𝑥∈𝐶,+∞,𝑥∉𝐶, (1.3) then the quasivariational inclusion (1.1) is equivalent to find 𝑥∈𝐶 such that ⟨𝐵𝑥,𝑣−𝑥⟩≥0,∀𝑣∈𝐶, (1.4) which is called Hartman-Stampacchia variational inequality problem (see [2–4]).

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∈𝐶𝑥−𝑦. (1.5) A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.6) and the mapping 𝑓∶𝐶→𝐶 is called a contraction if there exists a constant 𝛼∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.7) 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⊂𝐻 is bounded, closed, and convex and 𝑇 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑇) is nonempty (see [5]). Recall that a mapping 𝐴∶𝐶→𝐶 is said to be (i)monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶, (1.8)(ii)𝑘-Lipschitz continuous if there exists a constant 𝑘>0 such that 𝐴𝑥−𝐴𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.9) if 𝑘=1, then 𝐴 is a nonexpansive, (iii)pseudocontractive if 𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²+(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²,∀𝑥,𝑦∈𝐶, (1.10)(iv)𝑘-strictly pseudocontractive if there exists a constant 𝑘∈[0,1) such that 𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²,∀𝑥,𝑦∈𝐶, (1.11) and it is obvious that 𝐴 is a nonexpansive if and only if 𝐴 is a 0-strictly pseudocontractive, (v)𝛼-strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (1.12)(vi)𝛼-inverse-strongly monotone (or 𝛼-cocoercive) if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶, (1.13) if 𝛼=1, then 𝐴 is called that firmly nonexpansive; it is obvious that any 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and (1/𝛼)-Lipschitz continuous, (vii)relaxed 𝛼-cocoercive if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥(−𝛼)𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶, (1.14)(viii)relaxed (𝛼,𝑟)-cocoercive if there exists two constants 𝛼,𝑟>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥(−𝛼)𝐴𝑥−𝐴𝑦²+𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (1.15) and it is obvious that any 𝑟-strongly monotonicity implies to the relaxed (𝛼,𝑟)-cocoercivity.

The existence common fixed points for a finite family of nonexpansive mappings have been considered by many authors (see [6–9] and the references therein).

In this paper, we study the mapping 𝑊_𝑛 defined by𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜇_𝑛𝑆_𝑛𝑈_𝑛,𝑛+1+1−𝜇_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜇_𝑛−1𝑆_𝑛−1𝑈_𝑛,𝑛+1−𝜇_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜇_𝑘𝑆_𝑘𝑈_𝑛,𝑘+1+1−𝜇_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜇_𝑘−1𝑆_𝑘−1𝑈_𝑛,𝑘+1−𝜇_𝑘−1⋮𝑈𝐼,_𝑛,2=𝜇₂𝑆₂𝑈_𝑛,3+1−𝜇₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜇₁𝑆₁𝑈_𝑛,2+1−𝜇₁𝐼, (1.16) 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 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.

A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping in a real Hilbert space 𝐻,min_𝑥∈Ω12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩, (1.17) where 𝐴 is a bounded linear operator on 𝐻, Ω is the fixed-point set of a nonexpansive mapping 𝑆 on 𝐻, and 𝑏 is a given point in 𝐻. Recall that 𝐴 is a strongly positive bounded linear operator on 𝐻 if there exists a constant 𝛾>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.18)

Marino and Xu [10] introduced the following iterative scheme based on the viscosity approximation method introduced by Moudafi [11]:𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑥_𝑛,∀𝑛∈ℕ, (1.19) where 𝑥₁∈𝐻, 𝐴 is a strongly positive bounded linear operator on 𝐻, 𝑓 is a contraction on 𝐻, and 𝑆 is a nonexpansive on 𝐻. They proved that under some appropriateness conditions imposed on the parameters, if 𝐹(𝑆)≠∅, then the sequence {𝑥_𝑛} generated by (1.19) converges strongly to the unique solution 𝑧=𝑃_𝐹(𝑆)(𝐼−𝐴+𝛾𝑓)𝑧 of the variational inequality 𝐴−𝛾𝑓𝑧,𝑥−𝑧≥0,∀𝑥∈𝐹(𝑆), (1.20) which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑆)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.21) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

Iiduka and Takahashi [12] introduced an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality (1.4) as in the following theorem.

Theorem IT. Let 𝐶 be a nonempty closed-convex subset of a real Hilbert space 𝐻. Let 𝐵 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻, and let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆)∩VI(𝐶,𝐵)≠∅. Suppose that 𝑥₁=𝑥∈𝐶 and {𝑥_𝑛} is the sequence defined by 𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛, (1.22) for all 𝑛∈ℕ, where {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<2𝛼 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(C2)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝜆_𝑛+1−𝜆_𝑛|<∞,
then {𝑥_𝑛} converges strongly to 𝑃_{𝐹(𝑆)∩VI(𝐶,𝐵)}𝑥.

Definition 1.1 (see [13]). 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. [13] considered the problem (1.1). To be more precise, they proved the following theorem.

Theorem ZLC. Let 𝐻 be a real Hilbert space, let 𝐵∶𝐻→𝐻 be an 𝛼-inverse-strongly monotone mapping, let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝑆∶𝐻→𝐻 be a nonexpansive mapping. Suppose that the set 𝐹(𝑆)∩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 𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑦_𝑛, (1.23) 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(𝑆)∩VI(𝐻,𝐵,𝑀)}𝑥.

Peng et al. [14] introduced an iterative schemeΦ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐻,_𝑛=𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑦_𝑛, (1.24) for all 𝑛∈ℕ, where 𝑥₁∈𝐻, 𝐵 is an 𝛼-cocoercive mapping on H, 𝑓 is a contraction on 𝐻, 𝑆 is a nonexpansive on 𝐻, 𝑀 is a maximal monotone mapping of 𝐻 into 2^𝐻, and Φ is a bifunction from 𝐻×𝐻 into ℝ.

We note that their iteration is well defined if we let 𝐶=𝐻, and the appropriateness of the control conditions 𝛼_𝑛 and 𝜆 of their iteration should be {𝛼_𝑛}⊂(0,1) and 𝜆∈(0,2𝛼) (see Theorem  3.1 in [14]). They proved that under some appropriateness imposed on the other parameters, if Ω=𝐹(𝑆)∩VI(𝐻,𝐵,𝑀)∩EP(Φ)≠∅, then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} generated by (1.24) converge strongly to 𝑧=𝑃_Ω𝑓(𝑧) of the variational inequality 𝑧−𝑓(𝑧),𝑥−𝑧≥0,∀𝑥∈Ω, (1.25) where EP(Φ) is the set of solutions of equilibrium problem defined by EP(Φ)=𝑥∈𝐻∶Φ𝑥,𝑦≥0,∀𝑦∈𝐻. (1.26)

Moreover, Plubtieng and Sriprad [15] introduced an iterative schemeΦ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐻,_𝑛=𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆_𝑛𝑦_𝑛, (1.27) for all 𝑛∈ℕ, where 𝑥₁∈𝐻, 𝐴 is a strongly bounded linear operator on 𝐻, 𝐵 is an 𝛼-cocoercive mapping on H, 𝑓 is a contraction on 𝐻, 𝑆_𝑛 is a nonexpansive on 𝐻, 𝑀 is a maximal monotone mapping of 𝐻 into 2^𝐻, and Φ is a bifunction from 𝐻×𝐻 into ℝ.

We note that the appropriateness of the control conditions 𝛼_𝑛 and 𝜆 of their iteration should be {𝛼_𝑛}⊂(0,1) and 𝜆∈(0,2𝛼) (see Theorem 3.2 in [15]). They proved that under some appropriateness imposed on the other parameters, if Ω=⋂^∞_𝑛=1𝐹(𝑆_𝑛)∩VI(𝐻,𝐵,𝑀)∩EP(Φ)≠∅, then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} generated by (1.27) converge strongly to 𝑧=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑧.

On the other hand, Li and Wu [16] introduced an iterative scheme for finding a common element of the set of fixed points of a 𝑘-strictly pseudocontractive mapping with a fixed point and the set of solutions of relaxed cocoercive quasivariational inclusions as follows:𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝜇𝑆_𝑘𝑥_𝑛+1−𝜇𝑦_𝑛, (1.28) for all 𝑛∈ℕ, where 𝑥₁∈𝐻, 𝐴 is a strongly positive bounded linear operator on 𝐻, 𝑓 is a contraction on 𝐻, 𝑆_𝑘 is a mapping on 𝐻 defined by 𝑆_𝑘𝑥=𝑘𝑥+(1−𝑘)𝑆𝑥 for all 𝑥∈𝐻, such that 𝑆 is a 𝑘-strictly pseudocontractive mapping on 𝐻 with a fixed point, 𝐵 is relaxed cocoercive and Lipschitz continuous mappings on 𝐻, and 𝑀 is a maximal monotone mapping of 𝐻 into 2^𝐻.

They proved that under the missing condition of 𝜇, which should be 0<𝜇<1 (see Theorem  2.1 in [16]) and some appropriateness imposed on the other parameters, if Ω=𝐹(𝑆)∩VI(𝐻,𝐵,𝑀)≠∅, then the sequence {𝑥_𝑛} generated by (1.28) converges strongly to 𝑧=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑧.

Very recently, Tianchai and Wangkeeree [17] introduced an implicit iterative scheme for finding a common element of the set of common fixed points of an infinite family of a 𝑘_𝑛-strictly pseudocontractive mapping and the set of solutions of the system of generalized relaxed cocoercive quasivariational inclusions as follows:𝑧_𝑛=𝐽_𝑀2,𝜆2𝑥_𝑛−𝜆₂𝐵₂+𝐶₂𝑥_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆1𝑧_𝑛−𝜆₁𝐵₁+𝐶₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑊_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛𝑦_𝑛, (1.29) for all 𝑛∈ℕ, where 𝑥₁∈𝐻, 𝐴 is a strongly positive bounded linear operator on 𝐻, 𝑓 is a contraction on 𝐻, 𝑊_𝑛 is a 𝑊-mapping on 𝐻 generated by {𝑆_𝑛} and {𝜇_𝑛} such that 𝑆_𝑛𝑥=𝛿_𝑛𝑥+(1−𝛿_𝑛)𝑇_𝑛𝑥 for all 𝑥∈𝐻, 𝑇_𝑛 is a 𝑘_𝑛-strictly pseudocontractive mapping on 𝐻 with a fixed point, 𝑀_𝑖 is a maximal monotone mapping of 𝐻 into 2^𝐻, and 𝐵_𝑖, 𝐶_𝑖 are two mappings of relaxed cocoercive and Lipschitz continuous mappings on 𝐻 for each 𝑖=1,2.

They proved that under some appropriateness imposed on the parameters, if Ω=⋂^∞_𝑛=1𝐹(𝑇_𝑛)∩𝐹(𝐷)≠∅ such that the mapping 𝐷∶𝐻→𝐻 defined by 𝐷𝑥=𝐽_𝑀1,𝜆1𝐼−𝜆₁𝐵₁+𝐶₁𝐽_𝑀2,𝜆2𝐼−𝜆₂𝐵₂+𝐶₂𝑥,∀𝑥∈𝐻, (1.30) then the sequence {𝑥_𝑛} generated by (1.29) converges strongly to 𝑧=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑧.

In this paper, we introduce a novel general iterative scheme (1.32) below by the viscosity approximation method to find a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. Firstly, we introduce a mapping 𝑊_𝑛, where 𝑊_𝑛 is a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} for solving a common fixed point for two infinite families of strictly pseudocontractive mappings by iteration such that the mapping 𝑅_𝑛∶𝐻→𝐻 defined by𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (1.31) for all 𝑛∈ℕ, where {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} are two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point, respectively, and {𝜇_𝑛}⊂(0,𝜇] for some 𝜇∈(0,1). It follows that a linear general iterative scheme of the mappings 𝑊_𝑛 and 𝐽_{𝑀𝑖,𝜆𝑖}(𝐼−𝜆_𝑖𝐶_𝑖) is obtained as follows:𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (1.32) for all 𝑛∈ℕ, where 𝑥₁=𝑢∈𝐻, 𝑀_𝑖∶𝐻→2^𝐻 is a maximal monotone mapping, 𝐶_𝑖∶𝐻→𝐻 is a cocoercive mapping for each 𝑖=1,2,…,𝑁, 𝑓∶𝐻→𝐻 is a contraction mapping, and 𝐴,𝐵∶𝐻→𝐻 are two mappings of the strongly positive linear bounded self-adjoint operator mappings.

As special cases of the iterative scheme (1.32), we have the following. (i)If 𝜖_𝑛=0 for all 𝑛∈ℕ, then (1.32) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆i}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.33)(ii)If 𝐵≡𝐼, then (1.32) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝜖_𝑛−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.34)(iii)If 𝜖_𝑛=0 for all 𝑛∈ℕ, then (1.34) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.35)(iv)If 𝛽_𝑛=0 for all 𝑛∈ℕ, then (1.34) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝜖_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.36)(v)If 𝜖_𝑛=0 for all 𝑛∈ℕ, then (1.36) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.37)(vi)If 𝛾=1 and 𝐴≡𝐼, then (1.37) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛∈ℕ. (1.38)(vii)If 𝑀_𝑖≡𝐶_𝑖≡0 for each 𝑖=1,2,…,𝑁 and ∑^𝑁_𝑖=1𝜌_𝑖=1, then (1.32) is reduced to the iterative scheme 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛∈ℕ. (1.39)

Furthermore, if 𝑆_𝑛≡𝑇_𝑛 for all 𝑛∈ℕ, then the mapping 𝑅_𝑛∶𝐻→𝐻 in (1.31) is reduced to𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (1.40) for all 𝑛∈ℕ. It follows that the iterative scheme (1.32) is reduced to find 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.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings; it follows that if the mapping 𝑅_𝑛 is defined as (1.31) and 𝑘₁=𝑘₂=0, then the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for two infinite families of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces, and if the mapping 𝑅_𝑛 is defined as (1.40) and 𝑘₁=𝑘₂=0, then the iterative scheme (1.32) 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.

We suggest and analyze the iterative scheme (1.32) above under some appropriateness conditions imposed on the parameters, the strong convergence theorem for the above two sets is obtained, and applicability of the results has shown difference with the results of many others existing in the current literature.

2. Preliminaries

We collect the following lemmas which are used in the proof for the main results in the next section.

Lemma 2.1. Let 𝐶 be a nonempty closed-convex subset of a Hilbert space 𝐻 then the following inequalities hold:
(1)⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦⟩≥0,∀𝑥∈𝐻,𝑦∈𝐶,(2)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.

Lemma 2.2 (see [10]). Let 𝐻 be a Hilbert space, let 𝑓∶𝐻→𝐻 be a contraction with coefficient 0<𝛼<1, and let 𝐴∶𝐻→𝐻 be a strongly positive linear bounded operator with coefficient 𝛾>0, then
(1)if 0<𝛾<𝛾/𝛼, then 𝑥−𝑦,𝐴−𝛾𝑓𝑥−𝐴−𝛾𝑓𝑦≥𝛾−𝛾𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐻, (2.1)(2)if 0<𝜌≤‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.3 (see [18]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜂_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥1, (2.2) where {𝜂_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that
(1)lim_𝑛→∞𝜂_𝑛=0 and ∑^∞_𝑛=1𝜂_𝑛=∞,(2)limsup_𝑛→∞(𝛿_𝑛/𝜂_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞,then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.4 (see [9]). Let 𝐶 be a nonempty closed-convex subset of a Hilbert space 𝐻, define mapping 𝑊_𝑛 as (1.16), 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)=⋂^𝑛_𝑖=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 2.5 (see [13]). The resolvent operator 𝐽_𝑀,𝜆 associated with 𝑀 is single-valued and nonexpansive for all 𝜆>0.

Lemma 2.6 (see [13]). 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)=𝐹𝐽_𝑀,𝜆(𝐼−𝜆𝐵),∀𝜆>0. (2.4)

Lemma 2.7 (see [19]). 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 2.8 (see [2]). 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 2.9 (see [20]). 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 a nonexpansive such that 𝐹(𝑆)=𝐹(𝑇).

3. Main Results

Lemma 3.1. Let 𝐶 be a nonempty closed-convex subset of a real Hilbert space 𝐻, and let 𝑆,𝑇∶𝐶→𝐶 be two mappings of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point, respectively. Suppose that 𝐹(𝑆)∩𝐹(𝑇)≠∅ and define a mapping 𝑅∶𝐶→𝐶 by 𝑅𝑥=𝛼𝑥+(1−𝛼)(𝛼𝑆𝑥+(1−𝛼)𝑇𝑥),∀𝑥∈𝐶, (3.1) where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}, then 𝑅 is well defined, nonexpansive, and 𝐹(𝑅)=𝐹(𝑆)∩𝐹(𝑇).

Proof. Define the mappings 𝑆₁,𝑇₁∶𝐶→𝐶 as follows: 𝑆₁𝑥=𝛼𝑥+(1−𝛼)𝑆𝑥,𝑇₁𝑥=𝛼𝑥+(1−𝛼)𝑇𝑥, (3.2) for all 𝑥∈𝐶. By Lemma 2.9, we have 𝑆₁ and 𝑇₁ as nonexpansive such that 𝐹(𝑆₁)=𝐹(𝑆) and 𝐹(𝑇₁)=𝐹(𝑇). Therefore, for all 𝑥∈𝐶, we have (())()()𝑅𝑥=𝛼𝑥+1−𝛼)(𝛼𝑆𝑥+1−𝛼𝑇𝑥=𝛼𝑥+𝛼1−𝛼𝑆𝑥+1−𝛼²𝑇𝑥=𝛼²𝑥+𝛼(1−𝛼)𝑆𝑥+(1−𝛼)𝛼𝑥+(1−𝛼)²(())+(())𝑇𝑥=𝛼𝛼𝑥+1−𝛼𝑆𝑥1−𝛼)(𝛼𝑥+1−𝛼𝑇𝑥=𝛼𝑆₁𝑥+(1−𝛼)𝑇₁𝑥. (3.3) It follows from Lemma 2.7 that 𝑅 is well defined, nonexpansive, and 𝐹(𝑅)=𝐹(𝑆₁)∩𝐹(𝑇₁)=𝐹(𝑆)∩𝐹(𝑇).

Theorem 3.2. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (3.4) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (3.5) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (3.6)

Proof. From ‖𝐵‖=𝛽∈(0,1], 𝜖_𝑛≤𝛼_𝑛 for all 𝑛∈ℕ, (C1) and (C2), we have 𝛼_𝑛→0, 𝜖_𝑛→0 as 𝑛→∞ and limsup_𝑛→∞𝛽_𝑛<1. Thus, we may assume without loss of generality that 𝛼_𝑛<(1−𝜖_𝑛−𝛽_𝑛‖𝐵‖)‖𝐴‖⁻¹ for all 𝑛∈ℕ. For any 𝑥,𝑦∈𝐻 and for each 𝑖=1,2,…,𝑁, by the 𝜉_𝑖-cocoercivity of 𝐶_𝑖, we have 𝐼−𝜆_𝑖𝐶_𝑖𝑥−𝐼−𝜆_𝑖𝐶_𝑖𝑦²=𝑥−𝑦−𝜆_𝑖𝐶_𝑖𝑥−𝐶_𝑖𝑦²=𝑥−𝑦²−2𝜆_𝑖𝑥−𝑦,𝐶_𝑖𝑥−𝐶_𝑖𝑦+𝜆²_𝑖𝐶_𝑖𝑥−𝐶_𝑖𝑦²≤𝑥−𝑦²−2𝜉_𝑖−𝜆_𝑖𝜆_𝑖𝐶_𝑖𝑥−𝐶_𝑖𝑦²≤𝑥−𝑦², (3.7) which implies that 𝐼−𝜆_𝑖𝐶_𝑖 is a nonexpansive. Since 𝐴 and 𝐵 are two mappings of the linear bounded self-adjoint operators, we have ‖𝐴‖‖𝑥‖},‖𝐵‖‖𝑥‖}.=sup{|⟨𝐴𝑥,𝑥⟩|∶𝑥∈𝐻,=1=sup{|⟨𝐵𝑥,𝑥⟩|∶𝑥∈𝐻,=1 (3.8) Observe that 1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑥,𝑥=1−𝜖_𝑛⟨𝑥,𝑥⟩−𝛽_𝑛⟨𝐵𝑥,𝑥⟩−𝛼_𝑛⟨𝐴𝑥,𝑥⟩≥1−𝜖_𝑛−𝛽_𝑛‖𝐵‖−𝛼_𝑛‖𝐴‖>0. (3.9) Therefore, we obtain that (1−𝜖_𝑛)𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴 is positive. Thus, by the strong positivity of 𝐴 and 𝐵, we get 1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴=sup1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑥,𝑥∶𝑥∈𝐻,‖𝑥‖=1=sup1−𝜖_𝑛⟨𝑥,𝑥⟩−𝛽_𝑛⟨𝐵𝑥,𝑥⟩−𝛼_𝑛⟨𝐴𝑥,𝑥⟩∶𝑥∈𝐻,‖𝑥‖=1≤1−𝜖_𝑛−𝛽_𝑛𝛽−𝛼_𝑛𝛿≤1−𝛽_𝑛𝛽−𝛼_𝑛𝛿. (3.10)
Define the sequences of mappings {𝑃_𝑛∶𝐻→𝐻} and {𝑄_𝑛∶𝐻→𝐻} as follows: 𝑃_𝑛𝑥=𝛼_𝑛𝛾𝑓(𝑥)+𝛽_𝑛𝐵𝑥+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑄_𝑛𝑄𝑥,∀𝑥∈𝐻,_𝑛𝑥=𝛾_𝑛𝑊_𝑛𝑥+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑥,∀𝑥∈𝐻, (3.11) for all 𝑛∈ℕ. Firstly, we prove that 𝑃_𝑛 has a unique fixed point in 𝐻. Note that for all 𝑥,𝑦∈𝐻, by (3.11), (C3), the nonexpansiveness of 𝑊_𝑛,𝐽_{𝑀𝑖,𝜆𝑖}, and 𝐼−𝜆_𝑖𝐶_𝑖, we have 𝑄_𝑛𝑥−𝑄_𝑛𝑦≤𝛾_𝑛𝑊_𝑛𝑥−𝑊_𝑛𝑦+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑥−𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑦≤𝛾_𝑛𝑥−𝑦+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐼−𝜆_𝑖𝐶_𝑖𝑥−𝐼−𝜆_𝑖𝐶_𝑖𝑦≤𝛾_𝑛𝑥−𝑦+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖=.𝑥−𝑦𝑥−𝑦 (3.12) Therefore, 𝑄_𝑛 is a nonexpansive. It follows from (3.10), (3.11), (3.12), the contraction of 𝑓, and the linearity of 𝐴 and 𝐵 that 𝑃_𝑛𝑥−𝑃_𝑛𝑦≤𝛼_𝑛𝛾𝑓(𝑥)−𝑓𝑦+𝛽_𝑛‖𝐵+‖𝑥−𝑦1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑄_𝑛𝑥−𝑄_𝑛𝑦≤𝛼_𝑛𝛾𝛿𝑥−𝑦+𝛽_𝑛𝛽𝑥−𝑦+1−𝛽_𝑛𝛽−𝛼_𝑛𝛿=𝑥−𝑦1−𝛿−𝛾𝛿𝛼_𝑛𝑥−𝑦. (3.13) Hence, 𝑃_𝑛 is a contraction with coefficient 1−(𝛿−𝛾𝛿)𝛼_𝑛∈(0,1). Therefore, Banach contraction principle guarantees that 𝑃_𝑛 has a unique fixed point in 𝐻, and so the iteration (3.5) is well defined.
Next, we prove that {𝑥_𝑛} is bounded. Pick 𝑝∈Ω. Therefore, by Lemma 2.6, we have 𝑝=𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑝, (3.14) for each 𝑖=1,2,…,𝑁. By (3.14), the nonexpansiveness of 𝐽_{𝑀𝑖,𝜆𝑖}, and 𝐼−𝜆_𝑖𝐶_𝑖, we have 𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−𝑝=𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−𝐽_{𝑀𝑖,𝜆𝑖}𝑝−𝜆_𝑖𝐶_𝑖𝑝≤𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−𝑝−𝜆_𝑖𝐶_𝑖𝑝≤𝑥_𝑛−𝑝. (3.15)
Let 𝑡_𝑛=∑^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}(𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛). By (3.14), (C3), the nonexpansiveness of 𝐽_{𝑀𝑖,𝜆𝑖}, and 𝐼−𝜆_𝑖𝐶_𝑖, we have 𝑡_𝑛−𝑝=^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−^𝑁_𝑖=1𝜌_𝑖𝑝=^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑝−𝜆_𝑖𝐶_𝑖𝑝≤^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−𝐽_{𝑀𝑖,𝜆𝑖}𝑝−𝜆_𝑖𝐶_𝑖𝑝≤^𝑁_𝑖=1𝜌_𝑖𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛−𝑝−𝜆_𝑖𝐶_𝑖𝑝≤^𝑁_𝑖=1𝜌_𝑖𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (3.16) Since 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)(𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥), where 𝛼∈[𝑘,1)⧵{0}, {𝑆_𝑛} and {𝑇_𝑛} are two infinite families of 𝑘₁ and 𝑘₂-strict pseudocontractions with a fixed point, respectively, such that 𝑘=max{𝑘₁,𝑘₂}; therefore, by Lemma 3.1, we have that 𝑅_𝑛 is a nonexpansive and 𝐹(𝑅_𝑛)=𝐹(𝑆_𝑛)∩𝐹(𝑇_𝑛) for all 𝑛∈ℕ. It follows from Lemma 2.4(1) that we get 𝐹(𝑊_𝑛)=⋂^𝑛_𝑖=1𝐹(𝑅_𝑖)=(⋂^𝑛_𝑖=1𝐹(𝑆_𝑖))∩(⋂^𝑛_𝑖=1𝐹(𝑇_𝑖)), which implies that 𝑊_𝑛𝑝=𝑝. Hence, by (3.16) and the nonexpansiveness of 𝑊_𝑛, we have 𝑦_𝑛−𝑝=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛𝑡_𝑛=𝛾−𝑝_𝑛𝑊_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑡_𝑛−𝑝≤𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑝+1−𝛾_𝑛𝑡_𝑛−𝑝≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (3.17) By (3.10), (3.17), the contraction of 𝑓, and the linearity of 𝐴 and 𝐵, we have 𝑥_𝑛+1−𝑝=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛=𝛼−𝑝_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝐵𝑥_𝑛+−𝑝1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑝−𝜖_𝑛𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛‖𝐵‖𝑥_𝑛+−𝑝1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑝+𝜖_𝑛𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝛽_𝑛𝛽𝑥_𝑛+−𝑝1−𝛽_𝑛𝛽−𝛼_𝑛𝛿𝑥_𝑛−𝑝+𝛼_𝑛𝑝≤1−𝛿−𝛾𝛿𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑝+𝑝𝑥𝛾𝑓−𝐴𝑝≤max_𝑛,𝑝+𝑝−𝑝𝛾𝑓−𝐴𝑝.𝛿−𝛾𝛿 (3.18) It follows from induction that 𝑥_𝑛+1−𝑝≤max𝑥₁,𝑝+𝑝−𝑝𝛾𝑓−𝐴𝑝𝛿−𝛾𝛿, (3.19) for all 𝑛∈ℕ. Hence, {𝑥_𝑛} is bounded, and so are {𝑦_𝑛}, {𝑊_𝑛𝑥_𝑛}, {𝑡_𝑛}, {𝑓(𝑥_𝑛)}, {𝐴𝑦_𝑛}, {𝐵𝑥_𝑛}, and {𝐵𝑦_𝑛}.
Next, we prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. By (C3), the nonexpansiveness of 𝐽_{𝑀𝑖,𝜆𝑖}, and 𝐼−𝜆_𝑖𝐶_𝑖, we have 𝑡_𝑛+1−𝑡_𝑛=^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛+1−𝜆_𝑖𝐶_𝑖𝑥_𝑛+1−^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛≤^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛+1−𝜆_𝑖𝐶_𝑖𝑥_𝑛+1−𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛≤^𝑁_𝑖=1𝜌_𝑖𝑥_𝑛+1−𝜆_𝑖𝐶_𝑖𝑥_𝑛+1−𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛≤^𝑁_𝑖=1𝜌_𝑖𝑥_𝑛+1−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛. (3.20) By the nonexpansiveness of 𝑅_𝑖 and 𝑈_𝑛,𝑖, we have 𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛=𝑈_𝑛+1,1𝑥_𝑛−𝑈_𝑛,1𝑥_𝑛=𝜇₁𝑅₁𝑈_𝑛+1,2𝑥_𝑛+1−𝜇₁𝑥_𝑛−𝜇₁𝑅₁𝑈_𝑛,2𝑥_𝑛+1−𝜇₁𝑥_𝑛≤𝜇₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛,2𝑥_𝑛=𝜇₁𝜇₂𝑅₂𝑈_𝑛+1,3𝑥_𝑛+1−𝜇₂𝑥_𝑛−𝜇₂𝑅₂𝑈_𝑛,3𝑥_𝑛+1−𝜇₂𝑥_𝑛≤𝜇₁𝜇₂𝑈_𝑛+1,3𝑥_𝑛−𝑈_𝑛,3𝑥_𝑛⋮≤^𝑛_𝑖=1𝜇_𝑖𝑈_{𝑛+1,𝑛+1}𝑥_𝑛−𝑈_𝑛,𝑛+1𝑥_𝑛≤𝑀^𝑛_𝑖=1𝜇_𝑖, (3.21) for some constant 𝑀 such that 𝑀≥‖𝑈_{𝑛+1,𝑛+1}𝑥_𝑛−𝑈_𝑛,𝑛+1𝑥_𝑛‖≥0. Therefore, from (3.21), by the nonexpansiveness of 𝑊_𝑛+1, we have 𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛𝑥_𝑛≤𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛+1𝑥_𝑛+𝑊_𝑛+1𝑥_𝑛−𝑊_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀^𝑛_𝑖=1𝜇_𝑖. (3.22) Since 𝑦_𝑛+1−𝑦_𝑛=𝛾_𝑛+1𝑊_𝑛+1𝑥_𝑛+1+1−𝛾_𝑛+1𝑡_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛𝑡_𝑛=𝛾_𝑛+1𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑡_𝑛+1−𝛾_𝑛+1𝑡_𝑛+1−𝑡_𝑛, (3.23) combining (3.20), (3.22), and (3.23), we have 𝑦_𝑛+1−𝑦_𝑛≤𝛾_𝑛+1𝑊_𝑛+1𝑥_𝑛+1−𝑊_𝑛𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑡_𝑛+1−𝛾_𝑛+1𝑡_𝑛+1−𝑡_𝑛≤𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑀^𝑛_𝑖=1𝜇_𝑖+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑡_𝑛+1−𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀^𝑛_𝑖=1𝜇_𝑖+𝛾_𝑛+1−𝛾_𝑛𝑊_𝑛𝑥_𝑛−𝑡_𝑛. (3.24) By the linearity of 𝐴 and 𝐵, we have 𝑥_𝑛+2−𝑥_𝑛+1=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+𝛽_𝑛+1𝐵𝑥_𝑛+1+1−𝜖_𝑛+1𝐼−𝛽_𝑛+1𝐵−𝛼_𝑛+1𝐴𝑦_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛=1−𝜖_𝑛+1𝐼−𝛽_𝑛+1𝐵−𝛼_𝑛+1𝐴𝑦_𝑛+1−𝑦_𝑛+𝛽_𝑛−𝛽_𝑛+1𝐵𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐴𝑦_𝑛+𝜖_𝑛−𝜖_𝑛+1𝑦_𝑛+𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛾𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛+1𝐵𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝐵𝑥_𝑛. (3.25) Therefore, by (3.10), (3.24), (3.25), and the contraction of 𝑓, we have 𝑥_𝑛+2−𝑥_𝑛+1≤1−𝜖_𝑛+1𝐼−𝛽_𝑛+1𝐵−𝛼_𝑛+1𝐴𝑦_𝑛+1−𝑦_𝑛+𝛽_𝑛−𝛽_𝑛+1𝐵𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐴𝑦_𝑛+𝜖_𝑛−𝜖_𝑛+1𝑦_𝑛+𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛾𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛+1‖𝐵‖𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝐵𝑥_𝑛≤1−𝛽_𝑛+1𝛽−𝛼_𝑛+1𝛿𝑦_𝑛+1−𝑦_𝑛+𝛽_𝑛−𝛽_𝑛+1𝐵𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐴𝑦_𝑛+𝜖_𝑛−𝜖_𝑛+1𝑦_𝑛+𝛼_𝑛+1𝛾𝛿𝑥_𝑛+1−𝑥_𝑛+𝛾𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛+1𝛽𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝐵𝑥_𝑛≤1−𝜂_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛, (3.26) where 𝜂_𝑛∶=(𝛿−𝛾𝛿)𝛼_𝑛+1∈(0,1) and 𝛿_𝑛∶=𝑀^𝑛_𝑖=1𝜇_𝑖+𝑁𝛾_𝑛−𝛾_𝑛+1+𝜖_𝑛−𝜖_𝑛+1+𝛽_𝑛−𝛽_𝑛+1+𝛼_𝑛−𝛼_𝑛+1, (3.27) such that 𝑁=maxsup_𝑛≥1𝑊_𝑛𝑥_𝑛−𝑡_𝑛,sup_𝑛≥1𝐵𝑦_𝑛+𝐵𝑥_𝑛,sup_𝑛≥1𝑦_𝑛,sup_𝑛≥1𝐴𝑦_𝑛+𝛾𝑓𝑥_𝑛. (3.28) By (C1), (C3), (C4), and (C5), we can find that lim_𝑛→∞𝜂_𝑛=0, ∑^∞_𝑛=1𝜂_𝑛=∞, and ∑^∞_𝑛=1𝛿_𝑛<∞; therefore, by (3.26) and Lemma 2.3, we obtain 𝑥_𝑛+1−𝑥_𝑛⟶0as𝑛⟶∞. (3.29)
Next, we prove that ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. By the linearity of 𝐵, we have 𝑥_𝑛+1−𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛+𝛽_𝑛𝐵𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛𝐵𝑥_𝑛+1−𝑦_𝑛−𝜖_𝑛𝑦_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛+𝛽_𝑛‖𝐵‖𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛‖𝐵‖𝑥_𝑛+1−𝑦_𝑛+𝜖_𝑛𝑦_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛+𝑦_𝑛+𝛽_𝑛𝛽𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛𝛽𝑥_𝑛+1−𝑦_𝑛. (3.30) It follows that 1−𝛽_𝑛𝛽𝑥_𝑛+1−𝑦_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑦_𝑛+𝑦_𝑛+𝛽_𝑛𝛽𝑥_𝑛−𝑥_𝑛+1. (3.31) Hence, by (C1), (C2), (3.29), and (3.31), we have 𝑥_𝑛+1−𝑦_𝑛⟶0as𝑛⟶∞. (3.32) Since 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛, (3.33) therefore, by (3.29) and (3.32), we obtain 𝑥_𝑛−𝑦_𝑛⟶0as𝑛⟶∞. (3.34)
For all 𝑥,𝑦∈𝐻, by Lemma 2.2(2), the nonexpansiveness of 𝑃_Ω, the contraction of 𝑓, and the linearity of 𝐴, we have 𝑃_Ω𝐼−𝐴+𝛾𝑓𝑥−𝑃_Ω𝑦≤𝑦𝑓(𝑥)𝑦+‖+𝐼−𝐴+𝛾𝑓𝐼−𝐴+𝛾𝑓𝑥−𝐼−𝐴+𝛾𝑓≤𝛾−𝑓𝐼−𝐴‖𝑥−𝑦≤𝛾𝛿𝑥−𝑦1−𝛿=𝑥−𝑦1−𝛿−𝛾𝛿𝑥−𝑦. (3.35) Therefore, 𝑃_Ω(𝐼−𝐴+𝛾𝑓) is a contraction with coefficient 1−(𝛿−𝛾𝛿)∈(0,1); Banach contraction principle guarantees that 𝑃_Ω(𝐼−𝐴+𝛾𝑓) has a unique fixed point, say 𝑤∈𝐻, that is, 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤. Hence, by Lemma 2.1(1), we obtain 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (3.36)
Next, we claim that limsup_𝑛→∞𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛−𝑤≤0. (3.37) To show this inequality, we choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛−𝑤=lim_𝑖→∞𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛𝑖−𝑤. (3.38) Since {𝑥_𝑛𝑖} is bounded, there exists a subsequence {𝑥_𝑛𝑖𝑗} of {𝑥_𝑛𝑖} which converges weakly to 𝑤. Without loss of generality, we can assume that 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞.
Next, we prove that 𝑤∈Ω. Define the sequence of mappings {𝑄_𝑛∶𝐻→𝐻} and the mapping 𝑄∶𝐻→𝐻 by 𝑄_𝑛𝑥=𝛾_𝑛𝑊_𝑛𝑥+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑥,∀𝑥∈𝐻,𝑄𝑥=lim_𝑛→∞𝑄_𝑛𝑥, (3.39) for all 𝑛∈ℕ. Therefore, by (C2) and Lemma 2.4(3), we have 𝑄𝑥=𝑎𝑊𝑥+(1−𝑎)^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖𝑥,∀𝑥∈𝐻, (3.40) where 0<𝑎=lim_𝑛→∞𝛾_𝑛<1. From (C3), Lemma 2.4(3), we have that 𝑊 and ∑^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}(𝐼−𝜆_𝑖𝐶_𝑖) are nonexpansive. Therefore, by (C3), Lemmas 2.4(3), 2.6, 2.7, and 3.1, we have 𝐹(𝑄)=𝐹(𝑊)∩𝐹^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖=^∞_𝑖=1𝐹𝑅_𝑖∩^𝑁_𝑖=1𝐹𝐽_{𝑀𝑖,𝜆𝑖}𝐼−𝜆_𝑖𝐶_𝑖=^∞_𝑖=1𝐹𝑆_𝑖∩^∞_𝑖=1𝐹𝑇_𝑖∩^𝑁_𝑖=1VI𝐻,𝐶_𝑖,𝑀_𝑖, (3.41) that is, 𝐹(𝑄)=Ω. From (3.34), we have ‖𝑦_𝑛𝑖−𝑥_𝑛𝑖‖→0as𝑖→∞. Thus, from (3.5) and (3.39), we get ‖𝑄𝑥_𝑛𝑖−𝑥_𝑛𝑖‖→0as𝑖→∞. It follows from 𝑥_𝑛𝑖⇀𝑤 and by Lemma 2.8 that 𝑤∈𝐹(𝑄), that is, 𝑤∈Ω. Therefore, from (3.36) and (3.38), we obtain limsup_𝑛→∞𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛−𝑤=lim_𝑖→∞𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛𝑖=−𝑤𝛾𝑓−𝐴𝑤,𝑤−𝑤≤0. (3.42)
Next, we prove that 𝑥_𝑛→𝑤 as 𝑛→∞. Since 𝑤∈Ω, the same as in (3.17), we have 𝑦_𝑛−𝑤≤𝑥_𝑛−𝑤. (3.43) Therefore, by (3.10), (3.43), Lemma 2.1(2), the contraction of 𝑓, and the linearity of 𝐴 and 𝐵, we have 𝑥_𝑛+1−𝑤²=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑤²=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑤+𝛽_𝑛𝐵𝑥_𝑛+−𝑤1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑤−𝜖_𝑛𝑤²≤𝛽_𝑛𝐵𝑥_𝑛−𝑤+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑤²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑤−𝜖_𝑛𝑤,𝑥_𝑛+1≤𝛽−𝑤_𝑛‖𝐵‖𝑥_𝑛−𝑤+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛−𝑤²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑤),𝑥_𝑛+1−𝑤+2𝛼_𝑛𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛+1−𝑤−2𝜖_𝑛𝑤,𝑥_𝑛+1≤𝛽−𝑤_𝑛𝛽𝑥_𝑛−𝑤+1−𝛽_𝑛𝛽−𝛼_𝑛𝛿𝑥_𝑛−𝑤²+2𝛼_𝑛𝛾𝛿𝑥_𝑛−𝑤𝑥_𝑛+1−𝑤+2𝛼_𝑛𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛+1−𝑤−2𝜖_𝑛𝑤,𝑥_𝑛+1≤−𝑤1−𝛼_𝑛𝛿²𝑥_𝑛−𝑤²+𝛼_𝑛𝛾𝛿𝑥_𝑛−𝑤²+𝑥_𝑛+1−𝑤²+2𝛼_𝑛𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛+1−𝑤−2𝜖_𝑛𝑤,𝑥_𝑛+1−𝑤. (3.44) If follows that 𝑥_𝑛+1−𝑤²≤1−2𝛼_𝑛𝛿+𝛼_𝑛𝛾𝛿1−𝛼_𝑛𝑥𝛾𝛿^𝑛−𝑤²+𝛿^′_𝑛=21−𝛿−𝛾𝛿𝛼_𝑛1−𝛼_𝑛𝑥𝛾𝛿^𝑛−𝑤²+𝛿^′_𝑛≤1−𝜂^′_𝑛𝑥_𝑛−𝑤²+𝛿^′_𝑛, (3.45) where 𝜂^′_𝑛∶=(𝛿−𝛾𝛿)𝛼_𝑛/(1−𝛼_𝑛𝛾𝛿)∈(0,1) and 𝛿^′_𝑛1∶=1−𝛼_𝑛𝛼𝛾𝛿^2𝑛𝛿²𝑥_𝑛−𝑤²+2𝛼_𝑛𝛾𝑓(𝑤)−𝐴𝑤,𝑥_𝑛+1−𝑤−2𝜖_𝑛𝑤,𝑥_𝑛+1−𝑤. (3.46) By (3.29), (3.42), (C1), and (C3), we can found that lim_𝑛→∞𝜂^′_𝑛=0, ∑^∞_𝑛=1𝜂^′_𝑛=∞, and limsup_𝑛→∞(𝛿^′_𝑛/𝜂^′_𝑛)≤0. Therefore, by Lemma 2.3, we obtain that {𝑥_𝑛} converges strongly to 𝑤, and so is {𝑦_𝑛}. This completes the proof.

Remark 3.3. The iteration (3.5) is the difference with many others as follows.
(1)Two mappings 𝐴 and 𝐵 of the strongly positive linear bounded self-adjoint operator mappings are used in the iteration of {𝑥_𝑛}, which used only one mapping 𝐴 by many others. (2)Three parameters 𝛼_𝑛,𝛽_𝑛, and 𝜖_𝑛 are used in the iteration of {𝑥_𝑛}, which used only two parameters 𝛼_𝑛 and 𝛽_𝑛 by many others. (3)The parameter 𝛽_𝑛 can be chosen to be 𝛽_𝑛=0 for all 𝑛∈ℕ, because the condition liminf_𝑛→∞𝛽_𝑛>0 of Suzuki's Lemma (see [21]) is ignored in the control conditions of the iteration, which is used by many others. (4)A solving of a common fixed point for two infinite families of strictly pseudocontractive mappings by iteration is obtained by the mapping 𝑊_𝑛, where 𝑊_𝑛 is a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that 𝑅_𝑛 is defined as in Theorem 3.2.

4. Applications

Theorem 4.1. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.1) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿.For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (4.2) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), {𝛽_𝑛}⊂[0,1), 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.3)

Proof. It is concluded from Theorem 3.2 immediately, by putting 𝜖_𝑛=0 for all 𝑛∈ℕ.

Theorem 4.2. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶H→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴∶𝐻→𝐻 be a strongly positive linear bounded self-adjoint operator mapping with coefficient 𝛿∈(0,1] such that 𝛿≤‖𝐴‖≤1, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.4) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝜖_𝑛−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛, (4.5) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.6)

Proof. It is concluded from Theorem 3.2 immediately, by putting 𝐵≡𝐼.

Theorem 4.3. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴∶𝐻→𝐻 be a strongly positive linear bounded self-adjoint operator mapping with coefficient 𝛿∈(0,1] such that 𝛿≤‖𝐴‖≤1, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.7) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛, (4.8) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), {𝛽_𝑛}⊂[0,1), 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.9)

Proof. It is concluded from Theorem 4.2 immediately, by putting 𝜖_𝑛=0 for all 𝑛∈ℕ.

Theorem 4.4. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴∶𝐻→𝐻 be a strongly positive linear bounded self-adjoint operator mapping with coefficient 𝛿∈(0,1] such that 𝛿≤‖𝐴‖≤1, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.10) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝜖_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛, (4.11) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), {𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.12)

Proof. It is concluded from Theorem 4.2 immediately, by putting 𝛽_𝑛=0 for all 𝑛∈ℕ.

Theorem 4.5. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴∶𝐻→𝐻 be a strongly positive linear bounded self-adjoint operator mapping with coefficient 𝛿∈(0,1] such that 𝛿≤‖𝐴‖≤1, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.13) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾∑_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑦_𝑛, (4.14) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.15)

Proof. It is concluded from Theorem 4.4 immediately, by putting 𝜖_𝑛=0 for all 𝑛∈ℕ.

Theorem 4.6. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1), and let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.16) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛, (4.17) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1), 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω𝑓(𝑤) is a unique solution of the variational inequality 𝐼−𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.18)

Proof. It is concluded from Theorem 4.5 immediately, by putting 𝛾=𝛿=1 and 𝐴≡𝐼.

Theorem 4.7. Let 𝐻 be a real Hilbert space. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of 𝑘₁ and 𝑘₂-strictly pseudocontractive mappings with a fixed point such that 𝑘₁,𝑘₂∈[0,1), respectively. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.19) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1)⧵{0} such that 𝑘=max{𝑘₁,𝑘₂}. Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))≠∅ and 0 <𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (4.20) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1) and {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.21)

Proof. It is concluded from Theorem 3.2 immediately, by putting 𝑀_𝑖≡𝐶_𝑖≡0 for each 𝑖=1,2,…,𝑁.

Theorem 4.8. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). 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 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.22) for all 𝑛∈ℕ, where 𝛼∈[𝑘,1). Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (4.23) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1) and {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.24)

Proof. It is concluded from Theorem 3.2 immediately, by putting 𝑆_𝑛≡𝑇_𝑛 for all 𝑛∈ℕ, and note that 𝛼∈[𝑘,1) by Lemma 2.9.

Theorem 4.9. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑆_𝑛∶𝐻→𝐻} and {𝑇_𝑛∶𝐻→𝐻} be two infinite families of nonexpansive mappings. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝛼𝑆_𝑛𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.25) for all 𝑛∈ℕ, where 𝛼∈(0,1). Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (4.26) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1) and {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.27)

Proof. It is concluded from Theorem 3.2 immediately, by putting 𝑘₁=𝑘₂=0.

Theorem 4.10. Let 𝐻 be a real Hilbert space, let 𝑀_𝑖∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐶_𝑖∶𝐻→𝐻 be a 𝜉_𝑖-cocoercive mapping for each 𝑖=1,2,…,𝑁. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let {𝑇_𝑛∶𝐻→𝐻} be an infinite family of nonexpansive mappings. Define a mapping 𝑅_𝑛∶𝐻→𝐻 by 𝑅_𝑛𝑥=𝛼𝑥+(1−𝛼)𝑇_𝑛𝑥,∀𝑥∈𝐻, (4.28) for all 𝑛∈ℕ, where 𝛼∈[0,1). Let 𝑊_𝑛∶𝐻→𝐻 be a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛} such that {𝜇_𝑛}⊂(0,𝜇], for some 𝜇∈(0,1). Assume that Ω∶=(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^𝑁_𝑖=1VI(𝐻,𝐶_𝑖,𝑀_𝑖))≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝑦_𝑛, (4.29) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝛾_𝑛}⊂(0,1) and {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛, 𝜌_𝑖∈(0,1), and 𝜆_𝑖∈(0,2𝜉_𝑖] for each 𝑖=1,2,…,𝑁 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝛾_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞ and ∑^𝑁_𝑖=1𝜌_𝑖=1,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞,(C5)∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|<∞ and ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈Ω where 𝑤=𝑃_Ω(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈Ω. (4.30)

Proof. It is concluded from Theorem 4.8 immediately, by putting 𝑘=0.

Theorem 4.11. Let 𝐻 be a real Hilbert space. Let 𝐴,𝐵∶𝐻→𝐻 be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients 𝛿,𝛽∈(0,1] such that 𝛿≤‖𝐴‖≤1 and ‖𝐵‖=𝛽, respectively, and let 𝑓∶𝐻→𝐻 be a contraction mapping with coefficient 𝛿∈(0,1). Let 𝑇∶𝐻→𝐻 be a nonexpansive mapping. Assume that 𝐹(𝑇)≠∅ and 0<𝛾<𝛿/𝛿. For 𝑥₁=𝑢∈𝐻, suppose that {𝑥_𝑛} is generated iteratively by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝐵𝑥_𝑛+1−𝜖_𝑛𝐼−𝛽_𝑛𝐵−𝛼_𝑛𝐴𝜎_𝑛𝑇𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛, (4.31) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝜎_𝑛}⊂(0,1) and {𝛽_𝑛},{𝜖_𝑛}⊂[0,1) such that 𝜖_𝑛≤𝛼_𝑛 satisfying the following conditions:
(C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞(𝜖_𝑛/𝛼_𝑛)=0,(C2)0<lim_𝑛→∞𝜎_𝑛<1 and limsup_𝑛→∞𝛽_𝑛<1,(C3)∑^∞_𝑛=1𝛼_𝑛=∞,(C4)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,(C5)∑^∞_𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞ and ∑^∞_𝑛=1|𝜎_𝑛+1−𝜎_𝑛|<∞,
then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤∈𝐹(𝑇) where 𝑤=𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑤 is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑤,𝑦−𝑤≥0,∀𝑦∈𝐹(𝑇). (4.32)

Proof. From Theorem 4.10, putting 𝛼=0 and 𝑀_𝑖≡𝐶_𝑖≡0 for all 𝑖=1,2,…,𝑁. Setting 𝑇₁≡𝑇, 𝑇_𝑛≡𝐼 for all 𝑛=2,3,…, and let 𝜇_𝑛⊂(0,𝜇] for some 𝜇∈(0,1) such that ∑^∞_𝑛=1∏^𝑛_𝑖=1𝜇_𝑖<∞. Therefore, from the definition of 𝑅_𝑛 in Theorem 4.10, we have 𝑅₁=𝑇₁=𝑇 and 𝑅_𝑛=𝐼 for all 𝑛=2,3,…. Since 𝑊_𝑛 is a 𝑊-mapping generated by {𝑅_𝑛} and {𝜇_𝑛}, therefore by the definition of 𝑈_𝑛,𝑖 and 𝑊_𝑛 in (1.16), we have 𝑈_𝑛,𝑖=𝐼 for all 𝑖=2,3,… and 𝑊_𝑛=𝑈_𝑛,1=𝜇₁𝑅₁𝑈_𝑛,2+(1−𝜇₁)𝐼=𝜇₁𝑇+(1−𝜇₁)𝐼. Hence, by Theorem 4.10, we obtain 𝑦_𝑛=𝛾_𝑛𝑊_𝑛𝑥_𝑛+1−𝛾_𝑛^N_𝑖=1𝜌_𝑖𝐽_{𝑀𝑖,𝜆𝑖}𝑥_𝑛−𝜆_𝑖𝐶_𝑖𝑥_𝑛=𝛾_𝑛𝜇₁𝑇𝑥_𝑛+1−𝜇₁𝑥_𝑛+1−𝛾_𝑛^𝑁_𝑖=1𝜌_𝑖𝑥_𝑛=𝛾_𝑛𝜇₁𝑇𝑥_𝑛+1−𝜇₁𝑥_𝑛+1−𝛾_𝑛𝑥_𝑛=𝜎_𝑛𝑇𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛, (4.33) where 𝜎_𝑛∶=𝛾_𝑛𝜇₁. This completes the proof.

Acknowledgment

The author would like to thank the Faculty of Science, Maejo University for its financial support.