Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

Abstract

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

1. Introduction

Throughout this paper, we assume that 𝐻 is a real Hilbert space with inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 2^𝐻 denote the family of all subsets of 𝐻, and let 𝐶 be a closed-convex subset of 𝐻. Recall that a mapping 𝑇∶𝐶→𝐶 is said to be a 𝑘-strict pseudocontraction [1] if there exists 0≤𝑘<1 such that𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶, (1.1) where 𝐼 denotes the identity operator on 𝐶. When 𝑘=0, 𝑇∶𝐶→𝐶 is said to be nonexpansive [2] if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.2) And when 𝑘=1, 𝑇∶𝐶→𝐶 is said to be pseudocontraction if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1.3) Clearly, the class of 𝑘-strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of 𝑇 by 𝐹(𝑇).

A family 𝒮={𝑆(𝑠)∶0≤𝑠<∞} of mappings of 𝐶 into itself is called a nonexpansive semigroup on 𝐶 if it satisfies the following conditions: (i)𝑆(0)𝑥=𝑥 for all 𝑥∈𝐶,(ii)𝑆(𝑠+𝑡)=𝑆(𝑠)𝑆(𝑡) for all 𝑠,𝑡≥0,(iii)‖𝑆(𝑠)𝑥−𝑆(𝑠)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑠≥0,(iv)for all 𝑥∈𝐶,𝑠↦𝑆(𝑠)𝑥 is continuous.

We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮={𝑆(𝑠)∶𝑠≥0}, that is, 𝐹(𝑆)=⋂_𝑠≥0𝐹(𝑆(𝑠)). It is known that 𝐹(𝒮) is closed and convex.

Let 𝐴∶𝐻→𝐻 be a single-valued nonlinear mapping, and let 𝑀∶𝐻→2^𝐻 be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point 𝑢∈𝐻 such that𝜃∈𝐴(𝑢)+𝑀(𝑢), (1.4) where 𝜃 is the zero vector in H. The set of solutions of problem (1.4) is denoted by 𝐼(𝐴,𝑀).

Let the set-valued mapping 𝑀∶𝐻→2^𝐻 be a maximal monotone. We define the resolvent operator 𝐽_𝑀,𝜆 associate with 𝑀 and 𝜆 as follows:𝐽_𝑀,𝜆(𝑢)=(𝐼+𝜆𝑀)⁻¹(𝑢),𝑢∈𝐻, (1.5) where 𝜆 is a positive number. It is worth mentioning that the resolvent operator 𝐽_𝑀,𝜆 is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]).

Let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers, let 𝐴∶𝐶→𝐻 be a mapping, and let 𝜑∶𝐶→ℝ be a real-valued function. The generalized mixed equilibrium problem is for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.6) The set of solutions of (1.6) is denoted by GMEP(𝐹,𝜑,𝐴), that is,GMEP𝐹,𝜑,𝐴=𝑥∈𝐶∶𝐹𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.7) If 𝐴≡0, then the problem (1.6) is reduced into the mixed equilibrium problem for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.8) The set of solutions of (1.8) is denoted by MEP(𝐹,𝜑). The (generalized) mixed equilibrium problems include fixed-point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problem as special cases ([5–15]). In the last two decades, many papers have appeared in the literature on the existence of solutions of equilibrium problems; see, for example, [9] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, ([7–10, 12–20]) and references therein.

In 2006, Martinez-Yanes and Xu [21] introduced the following iterative: 𝑥₀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2⟨𝑥_𝑛−𝑥₀,𝑄,𝑧⟩_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥_𝑛−𝑥₀,𝑥⟩≤0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.9) where 𝑇 is a nonexpansive mapping in a Hilbert space 𝐻, and 𝑃_𝐶 is metric projection of 𝐻 onto a closed and convex subset 𝐶 of 𝐻. They proved that if the sequence {𝛼_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

In 2008, Zhang et al. [4] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The following iterative scheme 𝑥₀=𝑥∈𝐻 and 𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑦_𝑛, (1.10) for all 𝑛≥0. They proved the strong convergence theorem under some mind conditions.

Recently, Shehu [19] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of 𝑘-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in Hilbert spaces. Starting with an arbitrary 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖, and 𝑥₁=𝑃_𝐶1𝑥₀ define sequence {𝑥_𝑛},{𝑤_𝑛},{𝑢_𝑛},{𝑧_𝑛}, and {𝑦_𝑛,𝑖} as follows: 𝑧_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝑇^{(𝐹2,𝜑2)}_𝜆𝑛𝑧_𝑛−𝜆_𝑛𝐵𝑧_𝑛,𝑤_𝑛=𝑃_𝐶𝑢_𝑛−𝑠_𝑛𝐷𝑢_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑤_𝑛+1−𝛼_𝑛,𝑖𝑇_𝑖𝑤_𝑛𝐶,𝑛≥1,_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧≤𝑥_𝑛𝐶−𝑧,𝑛≥1,_𝑛+1=∩^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1, (1.11) where 𝑇_𝑖 is a 𝑘_𝑖-strictly pseudocontractive mapping and for some 0≤𝑘_𝑖<1, 𝐴,𝐵 is 𝛼,𝛽-inverse-strongly monotone mapping of 𝐶 into 𝐻. He proved that if the sequence {𝛼_𝑛,𝑖},{𝑟_𝑛},{𝑠_𝑛}, and {𝜆_𝑛} of parameters satisfies appropriate conditions, then {𝑥_𝑛} generated by (1.11) converges strongly to 𝑃_Ω𝑥₀.

In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu [21], Shehu [19], Zhang et al. [4], and many authors.

2. Preliminaries

Let 𝐻 be a real Hilbert space with norm ‖⋅‖ and inner product ⟨⋅,⋅⟩, and let 𝐶 be a closed-convex subset of 𝐻. When {𝑥_𝑛} is a sequence in 𝐻, 𝑥_𝑛⇀𝑥 means that {𝑥_𝑛} converges weakly to 𝑥, and 𝑥_𝑛→𝑥 means that {𝑥_𝑛} converges strongly to 𝑥. In a real Hilbert space 𝐻, we have 𝑥−𝑦²=‖𝑥‖²−𝑦²,()𝑦−2𝑥−𝑦,𝑦𝜆𝑥+1−𝜆²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦²,∀𝑥,𝑦∈𝐻, (2.1)(2.2) and 𝜆∈ℝ. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (2.3)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (2.4) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and 𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑥²,∀𝑥∈𝐻,𝑦∈𝐶. (2.5)(2.6) Recall that a mapping 𝐴 of 𝐻 into itself is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀,𝑦∈𝐻. (2.7) It is obvious that any 𝛼-inverse-strongly monotone mapping 𝐴 is (1/𝛼)-Lipschitz monotone and continuous mapping.

In order to prove our main results, we need the following Lemmas.

Lemma 2.1 (see [22]). Let 𝑉∶𝐶→𝐻 be a 𝑘-strict pseudocontraction, then
(1)the fixed-point set 𝐹(𝑉) of 𝑉 is closed convex, so that the projection 𝑃_𝐹(𝑉) is well defined;(2)define a mapping 𝑇∶𝐶→𝐻 by𝑇𝑥=𝑡𝑥+(1−𝑡)𝑉𝑥,∀𝑥∈𝐶. (2.8) If 𝑡∈[𝑘,1), then 𝑇 is a nonexpansive mapping such that 𝐹(𝑉)=𝐹(𝑇).

A family of mappings {𝑉_𝑖∶𝐶→𝐻}^∞_𝑖=1 is called a family of uniformly 𝑘-strict pseudocontractions if there exists a constant 𝑘∈[0,1) such that 𝑉_𝑖𝑥−𝑉_𝑖𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑉_𝑖)𝑥−(𝐼−𝑉_𝑖)𝑦²,∀𝑥,𝑦∈𝐶,∀𝑖≥1. (2.9) Let {𝑉_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of uniformly 𝑘-strict pseudocontractions. Let {𝑇_𝑖∶𝐶→𝐶}^∞_𝑖=1 be the sequence of nonexpansive mappings defined by (2.8), that is,𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,∀𝑥∈𝐶,∀𝑖≥1,𝑡∈[𝑘,1). (2.10)

Let {𝑇_𝑖} be a sequence of nonexpansive mappings of 𝐶 into itself defined by (2.10), and let {𝜇_𝑖} be a sequence of nonnegative numbers in [0,1]. For each 𝑛≥1, define a mapping 𝑊_𝑛 of 𝐶 into itself as follows:𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜇_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜇_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜇_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜇_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜇_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜇_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜇_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜇_𝑘−1⋮𝑈𝐼,_𝑛,2=𝜇₂𝑇₂𝑈_𝑛,3+1−𝜇₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜇₁𝑇₁𝑈_𝑛,2+1−𝜇₁𝐼. (2.11) Such a mapping 𝑊_𝑛 is nonexpansive from 𝐶 to 𝐶 and it is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜇₁,𝜇₂,…,𝜇_𝑛.

For each 𝑛,𝑘∈ℕ, let the mapping 𝑈_𝑛,𝑘 be defined by (2.11), then we can have the following crucial conclusions concerning 𝑊_𝑛. You can find them in [23]. Now, we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [23]). Let 𝐶 be a nonempty closed-convex subset of a real Hilbert space 𝐻. Let 𝑇₁,𝑇₂,… be nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛) is nonempty, and let 𝜇₁,𝜇₂,… be real numbers such that 0≤𝜇_𝑛≤𝑏<1 for every 𝑛≥1, then
(1)𝑊_𝑛 is nonexpansive and 𝐹(𝑊_𝑛)=⋂^𝑛_𝑖=1𝐹(𝑇_𝑖), ∀𝑛≥1,(2)for every 𝑥∈𝐶 and 𝑘∈ℕ, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists,(3)a mapping 𝑊∶𝐶→𝐶 defined by𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,∀𝑥∈𝐶 (2.12) is a nonexpansive mapping satisfying 𝐹(𝑊)=⋂^∞_𝑖=1𝐹(𝑇_𝑖), and it is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,… and 𝜇₁,𝜇₂,….

Lemma 2.3 (see [24]). Let 𝐶 be a nonempty closed-convex subset of a Hilbert space 𝐻, let {𝑇_𝑖∶𝐶→𝐶} be a countable family of nonexpansive mappings with ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅, and let {𝜇_𝑖} be a real sequence such that 0<𝜇_𝑖≤𝑏<1,∀𝑖≥1. If 𝐷 is any bounded subset of 𝐶, then lim_𝑛→∞sup_𝑥∈𝐷𝑊𝑥−𝑊_𝑛𝑥=0. (2.13)

Lemma 2.4 (see [25]). Each Hilbert space 𝐻 satisfies Opials condition, that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (2.14) holds for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [3]). Let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐴∶𝐻→𝐻 be a monotone mapping, then the mapping 𝑆=𝑀+𝐴∶𝐻→2^𝐻 is a maximal monotone mapping.

Remark 2.6. Lemma 2.5 implies that 𝐼(𝐴,𝑀) is closed and convex if 𝑀∶𝐻→2^𝐻 is a maximal monotone mapping and 𝐴∶𝐻→𝐻 is a monotone mapping.

Lemma 2.7 (see [4]). Let 𝑢∈𝐻 be a solution of variational inclusion (1.4) if and only if 𝑢=𝐽_𝑀,𝜆(𝑢−𝜆𝐴𝑢),∀𝜆>0, that is, 𝐼(𝐴,𝑀)=𝐹𝐽_𝑀,𝜆(𝐼−𝜆𝐴),∀𝜆>0. (2.15)

Lemma 2.8 (see [20]). Let 𝐶 be a nonempty bounded closed-convex subset of a Hilbert space 𝐻, and let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶, then for any ℎ≥0, lim_𝑡→∞sup_𝑥∈𝐶1𝑡^𝑡₀𝑆(𝑠)(ℎ)1𝑥𝑑𝑠−𝑆𝑡^𝑡₀𝑆(𝑠)𝑥𝑑𝑠=0. (2.16)

Lemma 2.9 (see [26]). Let C be a nonempty bounded closed-convex subset of H, let {𝑥_𝑛} be a sequence in C, and let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on C. If the following conditions are satisfied:
(1)𝑥_𝑛⇀𝑧,(2)limsup_𝑠→∞limsup_𝑛→∞‖𝑆(𝑠)𝑥_𝑛−𝑥_𝑛‖=0,then𝑧∈𝐹(S).
For solving the generalized mixed equilibrium problem for 𝐹∶𝐶×𝐶→ℝ, one gives the following assumptions for the bifunction F, 𝜑 and the set C:
(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,(A3)for each 𝑥,𝑦,𝑧∈𝐶,lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),(A4)for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous, (A5)for each 𝑦∈𝐶,𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous,(B1)for each 𝑥∈𝐻 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊆𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥+1𝑟𝑦_𝑥−𝑧,𝑧−𝑥<𝜑(𝑧), (2.17)(B2)C is a bounded set,
then one has the following lemma.

Lemma 2.10 (see [18]). Let C be a nonempty closed-convex subset of H. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction that satisfies (A1)–(A5), and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟^(𝐹,𝜑)(𝑥)=𝑦+1𝑧∈𝐶∶𝐹𝑧,𝑦+𝜑𝑟𝑦−𝑧,𝑧−𝑥≥𝜑(𝑧),∀𝑦∈𝐶, (2.18) for all 𝑧∈𝐻, then the following hold:(1)for each 𝑥∈𝐻,𝑇_𝑟^(𝐹,𝜑)(𝑥)≠∅,(2)𝑇_𝑟^(𝐹,𝜑) is single valued,(3)𝑇_𝑟^(𝐹,𝜑) is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦‖²≤⟨𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦,𝑥−𝑦⟩,(4)𝐹(𝑇_𝑟^(𝐹,𝜑))=𝑀𝐸𝑃(𝐹,𝜑),(5)𝑀𝐸𝑃(𝐹,𝜑) is closed and convex.

3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed-convex subset of a real Hilbert Space 𝐻. Let 𝐹₁,𝐹₂ be bifunctions of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5), and let 𝜑₁,𝜑₂∶𝐶→ℝ∪{+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let 𝐴,𝐵,𝐸₁,𝐸₂ be 𝛼,𝛽,𝜂₁,𝜂₂-inverse-strongly monotone mappings of 𝐶 into 𝐻, respectively, and let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be maximal monotone mappings. Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐶, and let {𝑡_𝑛} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of uniformly 𝑘-strict pseudocontractions, let {𝑇_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,∀𝑥∈𝐶,∀𝑖≥1,𝑡∈[𝑘,1), and let 𝑊_𝑛 be the 𝑊-mapping defined by (2.11) and 𝑊 a mapping defined by (2.12) with 𝐹(𝑊)≠∅. Suppose that Θ∶=𝐹(S)∩𝐹(𝑊)∩𝐺𝑀𝐸𝑃(𝐹₁,𝜑₁,𝐴)∩𝐺𝑀𝐸𝑃(𝐹₂,𝜑₂,𝐵)∩𝐼(𝐸₁,𝑀₁)∩𝐼(𝐸₂,𝑀₂)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖,𝑥₁=𝑃_𝐶1𝑥₀, and 𝑡_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑢_𝑛=𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛,𝑣_𝑛=𝐽_𝑀1,𝜆1𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛,𝑤_𝑛=𝐽_𝑀2,𝜆2𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝐶𝑑𝑠,_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,𝑧_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀, (3.1) for every 𝑛≥0, where {𝛼_𝑛,𝑖}^∞_𝑛=1⊂(0,1), {𝑟_𝑛},{𝑞_𝑛}⊂(0,∞),𝜆₁∈(0,2𝜂₁), and 𝜆₂∈(0,2𝜂₂) satisfy the following conditions:(i)0<𝑎≤𝑟_𝑛≤𝑏<2𝛼,(ii)0<𝑐≤𝑞_𝑛≤𝑑<2𝛽,(iii)lim_𝑛→∞𝛼_𝑛,𝑖=0, (iv)0<𝑒≤𝜆₁≤𝑓<2𝜂₁,(v)0<𝑔≤𝜆₂≤𝑗<2𝜂₂,
then {𝑥_𝑛} converges strongly to 𝑃_Θ𝑥₀.

Proof. First, we show that 𝐼−𝜆₁𝐸₁ and 𝐼−𝜆₂𝐸₂ are nonexpansive. Indeed, for all 𝑥,𝑦∈𝐶 and 𝜆₁∈(0,2𝜂₁), we obtain 𝐼−𝜆₁𝐸₁𝑥−𝐼−𝜆₁𝐸₁𝑦²=𝑥−𝑦−𝜆₁𝐸₁𝑥−𝐸₁𝑦²=𝑥−𝑦²−2𝜆₁𝑥−𝑦,𝐸₁𝑥−𝐸₁𝑦+𝜆²₁𝐸₁𝑥−𝐸₁𝑦²≤𝑥−𝑦²−2𝜆₁𝜂₁𝐸₁𝑥−𝐸₁𝑦²+𝜆²₁𝐸₁𝑥−𝐸₁𝑦²≤𝑥−𝑦²+𝜆₁𝜆₁−2𝜂₁𝐸₁𝑥−𝐸₁𝑦²≤𝑥−𝑦², (3.2) which implies that the mapping 𝐼−𝜆₁𝐸₁ is nonexpansive, so is 𝐼−𝜆₂𝐸₂. Let 𝑝∈Θ. We observe that 𝑤_𝑛−𝑝²=𝐽_𝑀2,𝜆2𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝐽_𝑀2,𝜆2𝑝−𝜆₂𝐸₂𝑝²≤𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝑝−𝜆₂𝐸₂𝑝²≤𝑣_𝑛−𝑝²=𝐽_𝑀1,𝜆1𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝐽_𝑀1,𝜆1𝑝−𝜆₁𝐸₁𝑝²≤𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝑝−𝜆₁𝐸₁𝑝²=𝑢_𝑛−𝑝². (3.3) Since both 𝐼−𝑟_𝑛𝐴 and 𝐼−𝑞_𝑛𝐵 are nonexpansive for each 𝑛≥1, let 𝑝∈Θ, then 𝑝=𝑇^𝐹1,𝜑1_𝑟𝑛(𝑝−𝑟_𝑛𝐴𝑝) and 𝑝=𝑇^𝐹2,𝜑2_𝑞𝑛(𝑝−𝑞_𝑛𝐵𝑝); by conditions (i) and (ii), we have 𝑢_𝑛−𝑝²=𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝐼−𝑞_𝑛𝐵𝑡_𝑛−𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝐼−𝑞_𝑛𝐵𝑝²≤𝐼−𝑞_𝑛𝐵𝑡_𝑛−𝐼−𝑞_𝑛𝐵𝑝²≤𝑡_𝑛−𝑝²+𝑞_𝑛𝑞_𝑛−2𝛽𝐵𝑡_𝑛−𝐵𝑝²≤𝑡_𝑛−𝑝²,𝑡_𝑛−𝑝²=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝐼−𝑟_𝑛𝐴𝑝²≤𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑝²≤𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝². (3.4) Therefore, we get 𝑢_𝑛−𝑝≤𝑥_𝑛−𝑝. (3.5) Next, we will divide the proof into five steps.
Step 1. We show that {𝑥_𝑛} is well defined. Let 𝑛=1, then 𝐶_1,𝑖=𝐶 is closed and convex for each 𝑖≥1. Suppose that 𝐶_𝑛,𝑖 is closed convex for some 𝑛>1, then, from the definition of 𝐶_𝑛+1,𝑖, we know that 𝐶_𝑛+1,𝑖 is closed convex for the same 𝑛≥1. Hence, 𝐶_𝑛,𝑖 is closed convex for 𝑛≥1 and for each 𝑖≥1. This implies that 𝐶_𝑛 is closed convex for 𝑛≥1. Furthermore, we show that Θ⊂𝐶_𝑛. For 𝑛=1,Θ⊂𝐶=𝐶_1,𝑖. For 𝑛≥2, let 𝑝∈Θ, then 𝑦_𝑛,𝑖−𝑝²=𝛼_𝑛,𝑖𝑥₀−𝑝+1−𝛼_𝑛,𝑖(1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠−𝑝)²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²=𝑤_𝑛−𝑝²+𝛼_𝑛,𝑖𝑥₀−𝑝²−𝑤_𝑛−𝑝²≤𝑥_𝑛−𝑝²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝑝, (3.6) which shows that 𝑝∈𝐶_𝑛,𝑖,∀𝑛≥2,∀𝑖≥1. Thus, Θ⊂𝐶_𝑛,𝑖,∀𝑛≥1,∀𝑖≥1. Hence, it follows that ∅≠Θ⊂𝐶_𝑛,∀𝑛≥1. This implies that {𝑥_𝑛} is well defined.Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑦_𝑛,𝑖−𝑥_𝑛‖=0, for 𝑖≥1. Since 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛,∀𝑛≥1, we have 𝑥_𝑛−𝑥₀≤𝑥_𝑛+1−𝑥₀,∀𝑛≥1. (3.7) Also, as Θ⊂𝐶_𝑛 by (2.1), it follows that 𝑥_𝑛−𝑥₀≤𝑧−𝑥₀,𝑧∈Θ,∀𝑛≥1. (3.8) Form (3.7) and (3.8), we have that lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists. Hence, {𝑥_𝑛} is bounded and so are {𝑦_𝑛,𝑖},∀𝑖≥1, {𝑤_𝑛}, {𝑣_𝑛}, {𝑢_𝑛}, {𝑡_𝑛}, {𝐴𝑥_𝑛}, {𝐵𝑡_𝑛}, {𝐸₁𝑢_𝑛}, {𝐸₂𝑣_𝑛}, {𝑊_𝑛𝑤_𝑛}, and {(1/𝑡_𝑛)∫^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠}. For 𝑚>𝑛≥1, we have that 𝑥_𝑚=𝑃_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛. By (2.5), we obtain 𝑥_𝑚−𝑥_𝑛²≤𝑥_𝑛−𝑥₀²−𝑥_𝑚−𝑥₀². (3.9) Letting 𝑚,𝑛→∞ and taking the limit in (3.9), we have ‖𝑥_𝑚−𝑥_𝑛‖→0, which shows that {𝑥_𝑛} is Cauchy. In particular, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.10) Since {𝑥_𝑛} is Cauchy, we assume that 𝑥_𝑛→𝑧∈𝐶. Since 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, then 𝑦_𝑛,𝑖−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥_𝑛+1⟶0, (3.11) and it follows that 𝑦_𝑛,𝑖−𝑥_𝑛≤𝑦_𝑛,𝑖−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛. (3.12) Therefore, lim_𝑛→∞𝑦_𝑛,𝑖−𝑥_𝑛=0,∀𝑖≥1. (3.13)Step 3. We claim that the following statements hold: (1)lim_𝑛→∞‖𝑢_𝑛−𝑡_𝑛‖=0,(2)lim_𝑛→∞‖𝑡_𝑛−𝑥_𝑛‖=0,(3)lim_𝑛→∞‖𝑤_𝑛−𝑣_𝑛‖=0,(4)lim_𝑛→∞‖𝑣_𝑛−𝑢_𝑛‖=0.
For 𝑝∈Θ, from (3.4), and (3.6), we obtain𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑡_𝑛−𝑝²+𝑞_𝑛𝑞_𝑛−2𝛽𝐵𝑡_𝑛−𝐵𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛,𝑖𝑞_𝑛𝑞_𝑛−2𝛽𝐵𝑡_𝑛−𝐵𝑝². (3.14) Since 0<𝑐≤𝑞_𝑛≤𝑑<2𝛽, we have 1−𝛼_𝑛,𝑖𝑞_𝑛2𝛽−𝑞_𝑛𝐵𝑡_𝑛−𝐵𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝. (3.15) Hence, by condition (iii) and (3.13), we have lim_𝑛→∞𝐵𝑡_𝑛−𝐵𝑝=0. (3.16) From (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑝². (3.17) On the other hand, 𝑢_𝑛−𝑝²=𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛−𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝑝−𝑞_𝑛𝐵𝑝²≤𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛−𝑝−𝑞_𝑛𝐵𝑝,𝑢_𝑛=1−𝑝2𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛−𝑝−𝑞_𝑛𝐵𝑝²+𝑢_𝑛−𝑝²−𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛−𝑝−𝑞_𝑛𝐵𝑝−𝑢_𝑛−𝑝²≤12𝑡_𝑛−𝑝²+𝑢_𝑛−𝑝²−𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛−𝑝−𝑞_𝑛𝐵𝑝−𝑢_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+𝑢_𝑛−𝑝²−𝑢_𝑛−𝑡_𝑛²+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝑠²_𝑛𝐵𝑡_𝑛−𝐵𝑝², (3.18) and hence, 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑢_𝑛−𝑡_𝑛²+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛𝐵𝑡_𝑛−𝐵𝑝. (3.19) Putting (3.19) into (3.17), for 𝑖≥1, we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝑢_𝑛−𝑡_𝑛²+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛𝐵𝑡_𝑛−𝐵𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛,𝑖𝑢_𝑛−𝑡_𝑛²+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛𝐵𝑡_𝑛−𝐵𝑝. (3.20) It follows that 1−𝛼_𝑛,𝑖𝑢_𝑛−𝑡_𝑛²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛𝐵𝑡_𝑛−𝐵𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝+2𝑞_𝑛𝑡_𝑛−𝑢_𝑛𝐵𝑡_𝑛−𝐵𝑝. (3.21) Therefore, from condition (iii), (3.13), and (3.16), we have lim_𝑛→∞𝑢_𝑛−𝑡_𝑛=0. (3.22) Furthermore, from (3.4), and (3.6), we get 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑡_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛,𝑖𝑟_𝑛𝑟_𝑛−2𝛼𝐴𝑥_𝑛−𝐴𝑝². (3.23) Since 0<𝑎≤𝑟_𝑛≤𝑏<2𝛼, we have 1−𝛼_𝑛,𝑖𝑟_𝑛2𝛼−𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝. (3.24) Then, by condition (iii) and (3.13), we obtain that lim_𝑛→∞𝐴𝑥_𝑛−𝐴𝑝=0. (3.25) From (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑡_𝑛−𝑝². (3.26) On the other hand, we note that 𝑡_𝑛−𝑝²≤𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑝−𝑟_𝑛𝐴𝑝²≤𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑝−𝑟_𝑛𝐴𝑝,𝑡_𝑛=1−𝑝2𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑝−𝑟_𝑛𝐴𝑝²+𝑡_𝑛−𝑝²−𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑝−𝑟_𝑛𝐴𝑝−𝑡_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+𝑡_𝑛−𝑝²−𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑝−𝑟_𝑛𝐴𝑝−𝑡_𝑛−𝑝²=12𝑥_𝑛−𝑝²+𝑡_𝑛−𝑝²−𝑡_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑡_𝑛,𝐴𝑥_𝑛−𝐴𝑝−𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝², (3.27) and hence, 𝑡_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑡_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑡_𝑛𝐴𝑥_𝑛−𝐴𝑝. (3.28) Putting (3.28) into (3.26), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝑡_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑡_𝑛𝐴𝑥_𝑛−𝐴𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛,𝑖𝑡_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑥_𝑛−𝑡_𝑛𝐴𝑥_𝑛−𝐴𝑝. (3.29) It follows that 1−𝛼_𝑛,𝑖𝑥_𝑛−𝑡_𝑛²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝+2𝑟_𝑛𝑥_𝑛−𝑧_𝑛𝐴𝑥_𝑛−𝐴𝑝. (3.30) Therefore, by condition (iii), (3.13), and (3.25), we have lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=0. (3.31) Condition (iii) implies that 𝑦_𝑛,𝑖−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠²=𝛼_𝑛,𝑖𝑥₀−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠⟶0. (3.32) It follows that 𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠≤𝑥_𝑛−𝑦_𝑛,𝑖+𝑦_𝑛,𝑖+1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠⟶0. (3.33) From (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²=𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝐽_𝑀2,𝜆2(𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛)−𝐽_𝑀2,𝜆2(𝑝−𝜆₂𝐸₂𝑝)²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖(𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛)−(𝑝−𝜆₂𝐸₂𝑝)²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑣_𝑛−𝑝²+𝜆₂𝜆₂−2𝜂₂𝐸₂𝑣_𝑛−𝐸₂𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛,𝑖𝜆₂𝜆₂−2𝜂₂𝐸₂𝑣_𝑛−𝐸₂𝑝². (3.34) Since 0<𝑔≤𝜆₂≤𝑗<2𝜂₂, we have 1−𝛼_𝑛,𝑖𝜆₂2𝜂₂−𝜆₂𝐸₂𝑣_𝑛−𝐸₂𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝. (3.35) Then, by condition (iii) and (3.13), we obtain that lim_𝑛→∞𝐸₂𝑣_𝑛−𝐸₂𝑝=0. (3.36) From (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝². (3.37) On the other hand, we note that 𝑤_𝑛−𝑝²≤𝐽_𝑀2,𝜆2𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝐽_𝑀2,𝜆2𝑝−𝜆₂𝐸₂𝑝²≤⟨𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝑝−𝜆₂𝐸₂𝑝,𝑤_𝑛=1−𝑝⟩2𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝑝−𝜆₂𝐸₂𝑝²+𝑤_𝑛−𝑝²−𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛−𝑝−𝜆₂𝐸₂𝑝−𝑤_𝑛−𝑝²≤12𝑣_𝑛−𝑝²+𝑤_𝑛−𝑝²−(𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛)−(𝑝−𝜆₂𝐸₂𝑝)−(𝑤_𝑛−𝑝)²≤12𝑥_𝑛−𝑝²+𝑤_𝑛−𝑝²−𝑤_𝑛−𝑣_𝑛²+2𝜆₂⟨𝑣_𝑛−𝑤_𝑛,𝐸₂𝑣_𝑛−𝐸₂𝑝⟩−𝜆²₂𝐸₂𝑣_𝑛−𝐸₂𝑝², (3.38) and hence, 𝑤_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑤_𝑛−𝑣_𝑛²+2𝜆₂𝑣_𝑛−𝑤_𝑛𝐸₂𝑣_𝑛−𝐸₂𝑝. (3.39) Putting (3.39) into (3.37), 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝑤_𝑛−𝑣_𝑛²+2𝜆₂𝑣_𝑛−𝑤_𝑛𝐸₂𝑣_𝑛−𝐸₂𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛,𝑖𝑤_𝑛−𝑣_𝑛²+2𝜆₂𝑣_𝑛−𝑤_𝑛𝐸₂𝑣_𝑛−𝐸₂𝑝, (3.40) this implies that 1−𝛼_𝑛,𝑖𝑤_𝑛−𝑣_𝑛²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²+2𝜆₂𝑤_𝑛−𝑣_𝑛𝐸₂𝑣_𝑛−𝐸₂𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝+2𝜆₂𝑤_𝑛−𝑣_𝑛𝐸₂𝑣_𝑛−𝐸₂𝑝. (3.41) Therefore, by condition (iii), (3.13), and (3.36), we have lim_𝑛→∞𝑤_𝑛−𝑣_𝑛=0. (3.42) Furthermore, from (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑣_𝑛−𝑝²=𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝐽_𝑀1,𝜆1(𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛)−𝐽_𝑀1,𝜆1(𝑝−𝜆₁𝐸₁𝑝)²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖(𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛)−(𝑝−𝜆₁𝐸₁𝑝)²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑢_𝑛−𝑝²+𝜆₁𝜆₁−2𝜂₁𝐸₁𝑢_𝑛−𝐸₁𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛,𝑖𝜆₁𝜆₁−2𝜂₁𝐸₁𝑢_𝑛−𝐸₁𝑝². (3.43) Since 0<𝑒≤𝜆₁≤𝑓<2𝜂₁, we have 1−𝛼_𝑛,𝑖𝜆₁2𝜂₁−𝜆₁𝐸₁𝑢_𝑛−𝐸₁𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝. (3.44) Then, by condition (iii) and (3.13), we obtain that lim_𝑛→∞𝐸₁𝑢_𝑛−𝐸₁𝑝=0. (3.45) From (3.6), we have 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑤_𝑛−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑣_𝑛−𝑝². (3.46) On the other hand, we note that 𝑣_𝑛−𝑝²≤𝐽_𝑀1,𝜆1(𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛)−𝐽_𝑀1,𝜆1(𝑝−𝜆₁𝐸₁𝑝)²≤𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝑝−𝜆₁𝐸₁𝑝,𝑣_𝑛=1−𝑝2𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝑝−𝜆₁𝐸₁𝑝²+𝑣_𝑛−𝑝²−𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝑝−𝜆₁𝐸₁𝑝−𝑣_𝑛−𝑝²≤12𝑢_𝑛−𝑝²+𝑣_𝑛−𝑝²−𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛−𝑝−𝜆₁𝐸₁𝑝−𝑣_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+𝑣_𝑛−𝑝²−𝑣_𝑛−𝑢_𝑛²+2𝜆₁𝑢_𝑛−𝑣_𝑛,𝐸₁𝑢_𝑛−𝐸₁𝑝−𝜆²₁𝐸₁𝑢_𝑛−𝐸₁𝑝², (3.47) and hence, 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑣_𝑛−𝑢_𝑛²+2𝜆₁𝑢_𝑛−𝑣_𝑛𝐸₁𝑢_𝑛−𝐸₁𝑝. (3.48) Putting (3.48) into (3.46), 𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+1−𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝑣_𝑛−𝑢_𝑛²+2𝜆₁𝑢_𝑛−𝑣_𝑛𝐸₁𝑢_𝑛−𝐸₁𝑝≤𝛼_𝑛,i𝑥₀−𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛,𝑖𝑣_𝑛−𝑢_𝑛²+2𝜆₁𝑢_𝑛−𝑣_𝑛𝐸₁𝑢_𝑛−𝐸₁𝑝, (3.49) this implies that 1−𝛼_𝑛,𝑖𝑣_𝑛−𝑢_𝑛²≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛,𝑖−𝑝²+2𝜆₁𝑣_𝑛−𝑢_𝑛𝐸₁𝑢_𝑛−𝐸₁𝑝≤𝛼_𝑛,𝑖𝑥₀−𝑝²+𝑥_𝑛−𝑦_𝑛,𝑖𝑥_𝑛−𝑝+𝑦_𝑛,𝑖−𝑝+2𝜆₁𝑣_𝑛−𝑢_𝑛𝐸₁𝑢_𝑛−𝐸₁𝑝. (3.50) Therefore, by condition (iii), (3.13), and (3.45), we have lim_𝑛→∞𝑣_𝑛−𝑢_𝑛=0. (3.51)Step 4. We show that 𝑧∈Θ∶=𝐹(𝒮)∩𝐹(𝑊)∩GMEP(𝐹₁,𝜑₁,𝐴)∩GMEP(𝐹₂,𝜑₂,𝐵)∩𝐼(𝐸₁,𝑀₁)∩𝐼(𝐸₂,𝑀₂). Since {𝑤_𝑛𝑖} is bounded, there exists a subsequence {𝑤_𝑛𝑖𝑗} of {𝑤_𝑛𝑖} which converges weakly to 𝑧∈𝐶. Without loss of generality, we can assume that 𝑤_𝑛𝑖⇀𝑧.
(1) First, we prove that 𝑧∈𝐹(𝒮). From (3.22), (3.31), (3.33), (3.42), and (3.51), we getlim_𝑛→∞𝑤_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠=0. (3.52) Since {𝑊_𝑛𝑤_𝑛} is bounded and from Lemma 2.8 for all s≥0, we have lim_𝑛→∞1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛(𝑠)1𝑑𝑠−𝑆𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠=0, (3.53) and since 𝑤_𝑛−𝑆(𝑠)𝑤_𝑛≤𝑤_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛(𝑠)1𝑑𝑠−𝑆𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛+𝑆(𝑠)1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠−𝑆(𝑠)𝑤_𝑛≤2𝑤_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛(𝑠)1𝑑𝑠−𝑆𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝑑𝑠. (3.54) It follows from (3.52) and (3.53) that lim_n→∞𝑤_𝑛−𝑆(𝑠)𝑤_𝑛=0. (3.55) Indeed, from Lemma 2.9 and (3.55), we get 𝑧∈𝐹(𝒮), that is, 𝑧=𝑆(𝑠)𝑧,∀𝑠≥0.
(2) Next, we show that 𝑧∈𝐹(𝑊)=⋂^∞_𝑛=1𝐹(𝑊_𝑛), where 𝐹(𝑊_𝑛)=⋂^𝑛_𝑖=1𝐹(𝑇_𝑖),∀𝑛≥1, and 𝐹(𝑊_𝑛+1)⊂𝐹(𝑊_𝑛). Assume that 𝑧∉𝐹(𝑊), then there exists a positive integer 𝑚 such that 𝑧∉𝐹(𝑇_𝑚), and so 𝑧∉⋂^𝑚_𝑖=1𝐹(𝑇_𝑖). Hence, for any 𝑛≥𝑚,𝑧∉⋂^𝑛_𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑊_𝑛), that is, 𝑧≠𝑊_𝑛𝑧. This together with 𝑧=𝑆(𝑠)𝑧,∀𝑠≥0 shows that 𝑧=𝑆(𝑠)𝑧≠𝑆(𝑠)𝑊_𝑛𝑧,∀𝑠≥0; therefore, we have 𝑧≠(1/𝑡_𝑛)∫^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑧𝑑𝑠,∀𝑛≥𝑚. It follows from the Opial's condition and (3.52) that liminf_𝑖→∞𝑤_𝑛𝑖−𝑧<liminf_𝑖→∞𝑤_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑆(𝑠)𝑊_𝑛𝑖𝑧𝑑𝑠≤liminf_𝑖→∞𝑤_𝑛𝑖−1𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑆(𝑠)𝑊_𝑛𝑖𝑤_𝑛𝑖+1𝑑𝑠𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑆(𝑠)𝑊_𝑛𝑖𝑤_𝑛𝑖1𝑑𝑠−𝑡_𝑛𝑖^𝑡𝑛𝑖₀𝑆(𝑠)𝑊_𝑛𝑖𝑧𝑑𝑠≤liminf_𝑖→∞𝑤_𝑛𝑖−𝑧, (3.56) which is a contradiction. Thus, we get 𝑧∈𝐹(𝑊).
(3) Now, we prove that 𝑧∈GMEP(𝐹₁,𝜑,𝐴). Since 𝑡_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛),𝑛≥1, we have for any 𝑦∈𝐶 that 𝐹₁𝑡_𝑛,𝑦+𝜑₁𝑦−𝜑₁𝑡_𝑛+⟨𝐴𝑥_𝑛,𝑦−𝑡_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈C. (3.57) From (A2), we also have 𝜑₁𝑦−𝜑₁𝑡_𝑛+⟨𝐴𝑥_𝑛,𝑦−𝑡_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛⟩≥𝐹₁𝑦,𝑡_𝑛,∀𝑦∈C. (3.58) For 𝑡 with 0<𝑡≤1 and 𝑦∈C, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑧. Since 𝑦∈𝐶 and 𝑧∈𝐶, we have 𝑦_𝑡∈𝐶. Then, we have ⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑦_𝑡⟩≥⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑦_𝑡⟩−𝜑₁𝑦_𝑡+𝜑₁𝑡_𝑛𝑖−⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑥_𝑛𝑖⟩−𝑦_𝑡−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+𝐹₁𝑦_𝑡,𝑡_𝑛𝑖=⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑦_𝑡−𝐴𝑡_𝑛𝑖⟩+⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑡_𝑛𝑖−𝐴𝑥_𝑛𝑖⟩−𝜑₁𝑦_𝑡+𝜑₁𝑡_𝑛𝑖−𝑦_𝑡−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖+𝐹₁𝑦_𝑡,𝑡_𝑛𝑖. (3.59) Since ‖𝑡_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have ‖𝐴𝑡_𝑛𝑖−𝐴𝑥_𝑛𝑖‖→0. Furthermore, from the inverse-strongly monotonicity of 𝐴, we have ⟨𝑦_𝑡−𝑡_𝑛𝑖,𝐴𝑦_𝑡−𝐴𝑡_𝑛𝑖⟩≥0. So, from (A4), (A5), and the weak lower semicontinuity of 𝜑₁,(𝑡_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑡_𝑛𝑖⇀𝑧, we have at the limit ⟨𝑦_𝑡−𝑧,𝐴𝑦_𝑡⟩≥−𝜑₁𝑦_𝑡+𝜑₁(𝑧)+𝐹₁𝑦_𝑡,𝑧, (3.60) as 𝑖→∞. From (A1), (A4), and (3.60), we also get 0=𝐹₁𝑦_𝑡,𝑦_𝑡+𝜑₁𝑦_𝑡−𝜑₁𝑦_𝑡≤𝑡𝐹₁𝑦_𝑡,𝑦+(1−𝑡)𝐹₁𝑦_𝑡,𝑧+𝑡𝜑₁𝑦−(1−𝑡)𝜑₁(𝑧)−𝜑𝑦_𝑡=𝑡𝐹₁𝑦_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑦_𝑡+(1−𝑡)𝐹₁𝑦_𝑡,𝑧+𝜑₁(𝑧)−𝜑₁𝑦_𝑡≤𝑡𝐹₁𝑦_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑦_𝑡+(1−𝑡)𝑦_𝑡−𝑧,𝐴𝑦_𝑡=𝑡𝐹₁𝑦_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑦_𝑡+(1−𝑡)𝑡⟨𝑦−𝑧,𝐴𝑦_𝑡⟩, (3.61) and hence, 0≤𝐹₁𝑦_𝑡,𝑦+𝜑₁𝑦−𝜑₁𝑦_𝑡+(1−𝑡)⟨𝑦−𝑧,𝐴𝑦_𝑡⟩. (3.62) Letting 𝑡→0, we have, for each 𝑦∈𝐶, 𝐹₁𝑧,𝑦+𝜑₁𝑦−𝜑₁(𝑧)+𝑦−𝑧,𝐴𝑧≥0. (3.63) This implies that 𝑧∈GMEP(𝐹₁,𝜑,𝐴). By following the same arguments, we can show that 𝑧∈GMEP(𝐹₂,𝜑,𝐵).
(4) At last, we show that 𝑧∈𝐼(𝐸₂,𝑀₂). Infact, since 𝐸₂ is 𝜂₂-inverse-strongly monotone, this implies that 𝐸₂ is(1/𝜂₂)-Lipschitz continuous monotone mapping and domain of 𝐸₂ equal to 𝐻. It follows from Lemma 2.5 that 𝑀₂+𝐸₂ is a maximal monotone. Let (𝑦,𝑔)∈𝐺(𝑀₂+𝐸₂), that is, 𝑔−𝐸₂𝑦∈𝑀₂(𝑦). Since 𝑤_𝑛𝑖=𝐽_𝑀2,𝜆2(𝑣_𝑛𝑖−𝜆₂𝐸₂𝑣_𝑛𝑖), we have 𝑣_𝑛𝑖−𝜆₂𝐸₂𝑣_𝑛𝑖∈(𝐼+𝜆₂𝑀₂)(𝑤_𝑛𝑖), that is, 1𝜆₂𝑣_𝑛𝑖−𝑤_𝑛𝑖−𝜆₂𝐸₂𝑣_𝑛𝑖∈𝑀₂𝑤_𝑛𝑖. (3.64) Since 𝑀₂+𝐸₂ is a maximal monotone, we have 𝑦−𝑤_𝑛𝑖1,𝑔−𝐵𝑦−𝜆₂𝑣_𝑛𝑖−𝑤_𝑛𝑖−𝜆₂𝐸₂𝑣_𝑛𝑖≥0, (3.65) and so ⟨𝑦−𝑤_𝑛𝑖,𝑔⟩≥𝑦−𝑤_𝑛𝑖,𝐸₂1𝑦+𝜆₂𝑣_𝑛𝑖−𝑤_𝑛𝑖−𝜆₂𝐸₂𝑣_𝑛𝑖=𝑦−𝑤_𝑛𝑖,𝐸₂𝑦−𝐸₂𝑤_𝑛𝑖+𝐸₂𝑤_𝑛𝑖−𝐸₂𝑣_𝑛𝑖+1𝜆₂𝑣_𝑛𝑖−𝑤_𝑛𝑖≥0+𝑦−𝑤_𝑛𝑖,𝐸₂𝑤_𝑛𝑖−𝐸₂𝑣_𝑛𝑖+𝑦−𝑤_𝑛𝑖,1𝜆₂𝑣_𝑛𝑖−𝑤_𝑛𝑖. (3.66) It follows from ‖𝑣_𝑛−𝑤_𝑛‖→0, ‖𝐸₂𝑣_𝑛−𝐸₂𝑤_𝑛‖→0, and 𝑤_𝑛𝑖⇀𝑧 that lim_𝑖→∞⟨𝑦−𝑤_𝑛𝑖,𝑔⟩=⟨𝑦−𝑤,𝑔⟩≥0. (3.67) It follows from the maximal monotonicity of 𝑀₂+𝐸₂ that 0∈(𝑀₂+𝐸₂)(𝑧), that is, 𝑧∈𝐼(𝐸₂,𝑀₂). By following the same arguments, we can show that 𝑧∈𝐼(𝐸₁,𝑀₁). Hence, by (1)–(4), we have 𝑧∈Θ.Step 5. Noting that 𝑥_𝑛=𝑃_𝐶𝑛𝑥₀, by (2.5), we have ⟨𝑥₀−𝑥_𝑛,𝑦−𝑥_𝑛⟩≤0,∀𝑦∈𝐶_𝑛. (3.68) Since Θ⊂C_𝑛 and by the continuity of inner product, we obtain from the above inequality that ⟨𝑥₀−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶. (3.69) By (2.5) again, we conclude that z=𝑃_Θ𝑥₀. This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let F₁,F₂ be bifunctions of 𝐶×𝐶 into real numbers ℝ satisfying (A1)-(A5), and let 𝜑₁,𝜑₂∶C→ℝ∪{+∞} be proper lower semicontinuous and convex functions with assumption (B1) or (B2). Let A,B,E₁,E₂ be 𝛼,𝛽,𝜂₁,𝜂₂-inverse-strongly monotone mappings of C into H, respectively. Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on C, and let {t_n} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of uniformly k-strict pseudocontractions, let {𝑇_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of nonexpansive mappings defined by T_ix=tx+(1−t)V_ix,∀x∈C,∀i≥1,t∈[k,1), and let W_n be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ∶=𝐹(𝒮)∩𝐹(𝑊)∩𝐺𝑀𝐸𝑃(𝐹₁,𝜑₁,𝐴)∩𝐺𝑀𝐸𝑃(𝐹₂,𝜑₂,𝐵)∩𝑉𝐼(𝐶,𝐸₁)∩𝑉𝐼(𝐶,𝐸₂)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖,𝑥₁=𝑃_𝐶1𝑥₀, and 𝑡_𝑛=𝑇^{(𝐹1,𝜑1)}_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑢_𝑛=𝑇^{(𝐹2,𝜑2)}_𝑞𝑛𝑡_𝑛−𝑞_𝑛𝐵𝑡_𝑛,𝑣_𝑛=𝑃_𝐶𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛,𝑤_𝑛=𝑃_𝐶𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝐶𝑑𝑠,_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,𝑧_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀, (3.70) for every n≥0, where {𝛼_𝑛,𝑖}^∞_𝑛=1⊂(0,1), {𝑟_𝑛},{𝑞_𝑛}⊂(0,∞),𝜆₁∈(0,2𝜂₁), and 𝜆₂∈(0,2𝜂₂) satisfy the following conditions:(i)0<𝑎≤𝑟_𝑛≤𝑏<2𝛼, (ii)0<𝑐≤𝑞_𝑛≤𝑑<2𝛽, (iii)lim_𝑛→∞𝛼_𝑛,𝑖=0, (iv)0<𝑒≤𝜆₁≤𝑓<2𝜂₁, (v)0<g≤𝜆₂≤j<2𝜂₂,
then {𝑥_𝑛} converges strongly to 𝑃_Θ𝑥₀.

Proof. From Theorem 3.1, put 𝑀=𝜕𝛿_𝐶, then 𝐽_𝑀,𝜆1=𝑃_𝐶 and 𝐽_𝑀,𝜆2=𝑃_𝐶. So we have 𝑣_𝑛=𝑃_𝐶(𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛) and 𝑤_𝑛=𝑃_𝐶(𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛). The conclusion of Corollary 3.2 can be obtained from Theorem 3.1 immediately.

4. Applications

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions: minℎ₁(𝑥)minℎ₂(𝑥)𝑥∈C, (4.1) where ℎ(𝑥) is a convex and lower semicontinuous functional, and define C as a closed-convex subset of a real Hilbert space 𝐻. We denote the set of solutions of (4.1) by 𝑀(ℎ₁) and 𝑀(ℎ₂). Let 𝐹_𝑖∶𝐶×𝐶→ℝ be a bifunction defined by 𝐹_𝑖(𝑥,𝑦)=ℎ_𝑖(𝑦)−ℎ_𝑖(𝑥). We consider the equilibrium problem, and it is obvious that 𝐸𝑃(𝐹_𝑖)=𝑀(ℎ_𝑖),𝑖=1,2. Therefore, from Theorem 3.1, we obtain the following Corollaries.

Corollary 4.1. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let ℎ₁,ℎ₂∶C→ℝ∪{+∞} be proper lower semicontinuous and convex functions. Let E₁,E₂ be 𝜂₁,𝜂₂-inverse-strongly monotone mappings of C into H, respectively, and let M₁,M₂∶H→2^H be maximal monotone mappings. Let 𝒮={S(s)∶0≤s<∞} be a nonexpansive semigroup on C, and let {t_n} be a positive real divergent sequence. Let {V_i∶C→C}^∞_i=1 be a countable family of uniformly k-strict pseudocontractions, let {T_i∶C→C}^∞_i=1 be a countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,∀𝑥∈𝐶,∀𝑖≥1,𝑡∈[𝑘,1), and let W_𝑛 be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ∶=F(𝒮)∩F(W)∩M(ℎ₁)∩M(ℎ₂)∩𝐼(E₁,M₁)∩𝐼(E₂,M₂)≠∅. Let {x_n} be a sequence generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖,𝑥₁=𝑃_𝐶1𝑥₀, and ℎ₁(𝑡)−ℎ₁𝑡_𝑛+1𝑟_𝑛⟨𝑡−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛ℎ⟩≥0,∀𝑡∈𝐶,₂(𝑢)−ℎ₂𝑢_𝑛+1𝑞_𝑛⟨𝑢−𝑢_𝑛,𝑢_𝑛−𝑡_𝑛𝑣⟩≥0,∀𝑢∈𝐶,_𝑛=𝐽_𝑀1,𝜆1𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛,𝑤_𝑛=𝐽_𝑀2,𝜆2𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝐶𝑑𝑠,_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2𝑥_𝑛−𝑥₀,𝐶,𝑧_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀, (4.2) for every n≥0, where {𝛼_n,i}^∞_n=1⊂(0,1), {r_n},{q_n}⊂(0,∞),𝜆₁∈(0,2𝜂₁), and 𝜆₂∈(0,2𝜂₂) satisfy the following conditions:(i)liminf_𝑛→∞𝑟_𝑛>0, (ii)liminf_𝑛→∞𝑞_𝑛>0, (iii)lim_𝑛→∞𝛼_𝑛,𝑖=0, (iv)0<e≤𝜆₁≤f<2𝜂₁, (v)0<g≤𝜆₂≤j<2𝜂₂, then {𝑥_𝑛} converges strongly to 𝑃_Θ𝑥₀.

Proof. From Theorem 3.1, put 𝐹₁(𝑡_𝑛,𝑡)=ℎ₁(𝑡)−ℎ₁(𝑡_𝑛),𝐹₂(𝑢_𝑛,𝑢)=ℎ₂(𝑢)−ℎ₂(𝑢_𝑛), and 𝐴,𝐵,𝜑₁,𝜑₂≡0. The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.

Corollary 4.2. Let C be a nonempty closed-convex subset of a real Hilbert Space H. Let ℎ₁,ℎ₂∶C→ℝ∪{+∞} be proper lower semicontinuous and convex functions. Let E₁,E₂ be 𝜂₁,𝜂₂-inverse-strongly monotone mappings of C into H, respectively, and let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be maximal monotone mappings. Let 𝒮={𝑆(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on C, and let {t_n} be a positive real divergent sequence. Let {𝑉_𝑖∶𝐶→𝐶}^∞_𝑖=1 be a countable family of uniformly k-strict pseudocontractions, let {T_i∶C→C}^∞_i=1 be a countable family of nonexpansive mappings defined by 𝑇_𝑖𝑥=𝑡𝑥+(1−𝑡)𝑉_𝑖𝑥,∀𝑥∈𝐶,∀𝑖≥1,𝑡∈[𝑘,1), and let 𝑊_𝑛 be the W-mapping defined by (2.11) and W a mapping defined by (2.12) with F(W)≠∅. Suppose that Θ∶=F(𝒮)∩F(W)∩M(ℎ₁)∩M(ℎ₂)∩𝑉𝐼(C,E₁)∩𝑉𝐼(C,E₂)≠∅. Let {x_n} be a sequence generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖,𝑥₁=𝑃_𝐶1𝑥₀, and ℎ₁(𝑡)−ℎ₁𝑡_𝑛+1𝑟_𝑛⟨𝑡−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛ℎ⟩≥0,∀𝑡∈𝐶,₂(𝑢)−ℎ₂𝑢_𝑛+1𝑞_𝑛⟨𝑢−𝑢_𝑛,𝑢_𝑛−𝑡_𝑛𝑣⟩≥0,∀𝑢∈𝐶,_𝑛=𝑃_𝐶𝑢_𝑛−𝜆₁𝐸₁𝑢_𝑛,𝑤_𝑛=𝑃_𝐶𝑣_𝑛−𝜆₂𝐸₂𝑣_𝑛,𝑦_𝑛,𝑖=𝛼_𝑛,𝑖𝑥₀+1−𝛼_𝑛,𝑖1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑤_𝑛𝐶𝑑𝑠,_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝑦_𝑛,𝑖−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛,𝑖𝑥₀²+2⟨𝑥_𝑛−𝑥₀,𝐶,𝑧⟩_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀, (4.3) for every n≥0, where {𝛼_n,i}^∞_n=1⊂(0,1), {r_n},{q_n}⊂(0,∞),𝜆₁∈(0,2𝜂₁), and 𝜆₂∈(0,2𝜂₂) satisfy the following conditions:
(i)liminf_𝑛→∞𝑟_𝑛>0, (ii)liminf_𝑛→∞𝑞_𝑛>0, (iii)lim_𝑛→∞𝛼_𝑛,𝑖=0, (iv)0<𝑒≤𝜆₁≤𝑓<2𝜂₁, (v)0<𝑔≤𝜆₂≤𝑗<2𝜂₂, then {𝑥_𝑛} converges strongly to 𝑃_Θ𝑥₀.

Acknowledgments

The authors thank the referees for their appreciation, valuable comments, and suggestions. They would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support. Furthermore, they would like to thank the Faculty of science (KMUTT) and the National Research Council of Thailand. This work was completed with the support of the National Research Council of Thailand (NRCT 2010-2011).