A Hybrid Iterative Scheme for Mixed Equilibrium Problems, General System of Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

Abstract

We introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Using the demiclosedness principle for nonexpansive mappings, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions, and we also give some examples for mappings which satisfy conditions of the main result.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and 𝐶 a nonempty closed convex subset of 𝐻. A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive mapping if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. The fixed point set of 𝑇 is denoted by 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}.

Halpern [1] studied the following iteration formula for approximating a fixed point of 𝑇. For an arbitrary 𝑣∈𝐶, let the sequence {𝑥_𝑛} be defined by 𝑥₁∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑣+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥1,(1.1) where {𝛼_𝑛} is a sequence of real numbers in [0,1] that satisfies the following conditions: 𝐶1∶lim_𝑛→∞𝛼_𝑛=0 and 𝐶2∶∑^∞𝑛=1𝛼_𝑛=∞.

Actually, Halpern studied the special case of (1.1) in which 𝛼_𝑛=𝑛^−𝜎, 𝜎∈(0,1), and 𝑣=0 and proved that {𝑥_𝑛} converges strongly to a fixed point of 𝑇. Under a different restriction on the parameter {𝛼_𝑛}, Lions [2] improved the result of Halpern, still in Hilbert spaces. He proved strong convergence of {𝑥_𝑛} to a fixed point of 𝑇, where the real sequence {𝛼_𝑛} satisfies the following conditions: 𝐶1∶lim_𝑛→∞𝛼_𝑛=0,𝐶2∶^∞_𝑛=1𝛼_𝑛=∞,𝐶3∶lim_𝑛→∞||𝛼_𝑛+1−𝛼_𝑛||𝛼^2𝑛+1=0.(1.2) Reich [3] proved that the result of Halpern remains true when 𝐻 is uniformly smooth. It was observed that both Halpern's and Lion's conditions on the real sequence {𝛼_𝑛} excluded the canonical choice 𝛼_𝑛=1/(𝑛+1). This was overcome by Wittmann [4] who proved, still in Hilbert spaces, the strong convergence of {𝑥_𝑛} to a fixed point of 𝑇 if {𝛼_𝑛} satisfies the following conditions: 𝐶1∶lim_𝑛→∞𝛼_𝑛=0,𝐶2∶^∞_𝑛=1𝛼_𝑛=∞,𝐶3∶^∞_𝑛=1||𝛼_𝑛+1−𝛼_𝑛||<∞.(1.3)

Recall that the classical variational inequality, denoted by VI(𝐶,𝐴), is to find an 𝑥^∗∈𝐶 such that ⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶.(1.4) The variational inequality, nonconvex variational inequality, and mixed variational inequality have been widely studied in the literature; see, for example, Ceng et al. [5–14], Chang et al. [15], Noor [16–18], Peng and Yao [19–21], Plubtieng and Punpaeng [22], Yao et al. [23], Zeng and Yao [24], Zhao and He [25], and the references therein.

For solving the variational inequality problem in the finite-dimensional Euclidean space ℝ^𝑛 under the assumption that a set 𝐶⊂ℝ^𝑛 is closed and convex, a mapping 𝐴 of 𝐶 into ℝ^𝑛 is monotone and 𝑘-Lipschitz continuous, and VI(𝐶,𝐴) is nonempty, Korpelevich [26] introduced the following so-called extragradient method: 𝑦𝑥₀=𝑥∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑥_𝑛,=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑦_𝑛,(1.5) for every 𝑛=0,1,2,…, where 𝜆∈(0,1/𝑘) and 𝑃_𝐶 is the projection of ℝ^𝑛 onto 𝐶. He showed that the sequences {𝑥_𝑛} and {𝑦_𝑛} generated by this iterative process converge to the same point 𝑧∈VI(𝐶,𝐴). In 2007, Y. Yao and J. C. Yao [27] introduced a new iterative scheme for finding the common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for 𝛼-inverse-strongly monotone mappings in a real Hilbert space.

Let C be a nonempty closed convex subset of a real Hilbert space 𝐻.

Let 𝐴,𝐵∶𝐶→𝐻 be two mappings. In 2008, Ceng et al. [12] considered the following problem of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that⟨⟨𝜇𝜆𝐴𝐵𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.6) which is called a general system of variational inequalities, where 𝜆>0 and 𝜇>0 are two constants. In particular, if 𝐴=𝐵, then the problem (1.6) reduces to finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that⟨⟨𝜇𝜆𝐴𝐴𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.7) which was defined by Verma [28] and is called the new system of variational inequalities. Further, if we add up the requirement that 𝑥^∗=𝑦^∗, then the problem (1.7) reduces to the classical variational inequality VI(𝐶,𝐴). Ceng et al. [12] introduced and studied a relaxed extragradient method for finding a common element of the set of solutions of the problem (1.6) for the 𝛼- and 𝛽-inverse-strongly monotone mappings and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let 𝑥₁=𝑣∈𝐶, and {𝑥_𝑛}, {𝑦_𝑛} are given by 𝑥_𝑛+1𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜇𝐵𝑥_𝑛,=𝑎_𝑛𝑣+𝑏_𝑛𝑥_𝑛+1−𝑎_𝑛−𝑏_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1,(1.8) where 𝜆∈(0,2𝛼), 𝜇∈(0,2𝛽), and {𝑎_𝑛},{𝑏_𝑛}⊂[0,1]. Then, they proved that the sequence {𝑥_𝑛} converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the problem (1.6) under some control conditions.

Let 𝜑∶𝐶→ℝ∪{+∞} be a proper extended real-valued function and 𝐹 a bifunction from 𝐶×𝐶 to ℝ, where ℝ is the set of real numbers. Ceng and Yao [13] considered the following mixed equilibrium problem:find𝑥∈𝐶suchthat𝐹(𝑥,𝑦)+𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶.(1.9)

The set of solutions of (1.9) is denoted by MEP(𝐹,𝜑). It is easy to see that 𝑥 is a solution of the problem (1.9) impling that 𝑥∈dom𝜑={𝑥∈𝐶∣𝜑(𝑥)<+∞}.

If 𝜑=0, then the mixed equilibrium problem (1.9) becomes the following equilibrium problem:find𝑥∈𝐶suchthat𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.10) If 𝐹=0, then the mixed equilibrium problem (1.9) reduces to the convex minimization problemfind𝑥∈𝐶suchthat𝜑(𝑦)≥𝜑(𝑥),∀𝑦∈𝐶.(1.11)

The set of solutions of (1.10) is denoted by EP(𝐹).

If 𝜑=0 and 𝐹(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶, where 𝐴 is a mapping from 𝐶 into 𝐻, then (1.9) reduces to the classical variational inequality and EP(𝐹)=VI(𝐶,𝐴).

The equilibrium problems and variational inequality problems can be applied for the problems in science and technology, economics, optimizations, and control theory. The problems of finding a solution of the systems of variational inequalities which is also a solution of the mixed equilibrium problems and the problem of finding a solution of the mixed equilibrium problems under the constraints that it is also a solution of the system of variational inequalities are very useful and important for studying problems in science and applied science.

Inspired and motivated by these facts, we introduce a new iteration process for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of a general system of variational inequalities in a real Hilbert space. Start with an arbitrary 𝑣∈𝐶, and let 𝑥₁∈𝐶, {𝑥_𝑛},{𝑢_𝑛}, and {𝑦_𝑛} be the sequences generated by𝐹𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛,=𝑎_𝑛𝑣+1−𝑎_𝑛𝑇𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1,(1.12) where 𝜆>0 and 𝜇>0 are two constants, {𝑟_𝑛}⊂(0,∞) and {𝑎_𝑛}⊂[0,1]. Using the demiclosedness principle for nonexpansive mappings, we will show that the sequence {𝑥_𝑛} converges strongly to a common element of the above three sets under some control conditions.

2. Preliminaries

In this section, we recall the well-known results and give some useful lemmas that will be used in the next section.

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and 𝐶 a nonempty closed convex subset of 𝐻.

A mapping 𝐴∶𝐶→𝐻 is called monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶.(2.1)𝐴 is called 𝛼-strongly monotone if there exists a positive real number 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(2.2) This implies that ‖𝐴𝑥−𝐴𝑦‖≥𝛼‖𝑥−𝑦‖,(2.3) that is, 𝐴 is 𝛼-expansive and, when 𝛼=1, it is expansive.

We can see easily that the following implications in monotonicity, strong monotonicity, and expansiveness hold: se⇓txrpoanngsimveonneostso.nicity⇒monotonicity(2.4)

The mapping 𝐴 is called 𝐿-Lipschitz continuous (or Lipschitzian) if there exists a constant 𝐿≥0 such that ‖𝐴𝑥−𝐴𝑦‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.5)𝐴 is called 𝛼-inverse-strongly monotone (or 𝛼-cocoercive) if there exists a positive real number 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶.(2.6) It is obvious that every 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and Lipschitz continuous. Also, if 𝐴 is 𝛼-strongly and 𝐿-Lipschitz continuous, then 𝐴 is (𝛼/𝐿²)-inverse-strongly monotone, but inverse-strongly monotone need not be strongly monotone.

A mapping 𝐴 is called relaxed c-cocoercive, if there exists a constant 𝑐>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥(−𝑐)‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶.(2.7) A mapping 𝐴 is called relaxed (c,d)-cocoercive, if there exist two constants 𝑐,𝑑>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥(−𝑐)‖𝐴𝑥−𝐴𝑦‖²+𝑑‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(2.8) For 𝑐=0, 𝐴 is 𝑑-strongly monotone. This class of mappings is more general than the class of strongly monotone mappings. As a result, we have the following implication: 𝑑-strong monotonicity ⇒ relaxed (𝑐,𝑑)-cocoercivity.

It is known that if the operator is Lipschitz continuous, then the relaxed cocoercivity is strongly monotone, but the strongly monotone does not imply the cocoercivity as shown in the following example.

Example 2.1. Let 𝐻=𝑅, 𝐶=[1,∞), and 𝐴∶𝐶→𝐻 be defined by 𝐴𝑥=𝑥², 𝑥∈𝐶. For 𝑥,𝑦∈𝐶, we have ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩==(𝑥𝑥²+−𝑦)𝑦||²𝑥−(𝑥𝑦−||₂𝑦)≥2||𝑥−𝑦||².(2.9) Hence, 𝐴 is 2-strongly monotone. If 𝐴 is 𝜇-cocoercive for some 𝜇>0, then ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩=(𝑥+𝑦)|𝑥−𝑦|²≥𝜇|𝑥²−𝑦²|². This implies 𝑥+𝑦≤1/𝜇 for all 𝑥,𝑦∈[1,∞) which is a contradiction. Hence, 𝐴 is not 𝜇-cocoercive for any 𝜇>0.

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.(2.10)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 satisfies⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²,∀𝑥,𝑦∈𝐻.(2.11) Obviously, this immediately implies that‖‖(𝑥−𝑦)−𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²≤‖𝑥−𝑦‖²−‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²,∀𝑥,𝑦∈𝐻.(2.12) Recall that 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and‖𝑥−𝑦‖⟨₂𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0,≥‖‖𝑥−𝑃_𝐶𝑥‖‖²+‖‖𝑃_𝐶𝑥−𝑦‖‖²,(2.13) for all 𝑥∈𝐻 and 𝑦∈𝐶; see Goebel and Kirk [29] for more details.

For solving the mixed equilibrium problem, let us assume 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 𝑦∈𝐶, 𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous;(A4) for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex;(A5) for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is lower 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_𝑥𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥+1𝑟⟨𝑦^𝑥−𝑧,𝑧−𝑥⟩<𝜑(𝑧);(2.14)(B2)𝐶 is a bounded set.

In the sequel we will need to use the following lemma.

Lemma 2.2 (see [30]). Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (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(𝑥)=𝑧∈𝐶∶𝐹(𝑧,𝑦)+𝜑(𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥𝜑(𝑧),∀𝑦∈𝐶(2.15) for all 𝑥∈𝐻. Then the following conclusions hold: (1)for each 𝑥∈𝐻, 𝑇_𝑟(𝑥)≠∅;(2)𝑇_𝑟 is single valued;(3)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, ‖‖𝑇_𝑟(𝑥)−𝑇_𝑟(𝑦)‖‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(2.16)(4)𝐹(𝑇_𝑟)=MEP(𝐹,𝜑);(5)MEP(𝐹,𝜑) is closed and convex.

We also need the following lemmas.

Lemma 2.3 (see [31]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property 𝑎_𝑛+1≤1−𝑡_𝑛𝑎_𝑛+𝑏_𝑛+𝑡_𝑛𝑐_𝑛,∀𝑛≥1,(2.17) where {𝑡_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} satisfy the restrictions (i)^∞_𝑛=1𝑡_𝑛=∞,(ii)^∞_𝑛=1𝑏_𝑛<∞,(iii)limsup_𝑛→∞𝑐_𝑛≤0.(2.18) Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.4 (see [32]). Let (𝐻,⟨⋅,⋅⟩) be an inner product space. Then, for all 𝑥,𝑦,𝑧∈𝐻 and 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1, one has ‖𝛼𝑥+𝛽𝑦+𝛾𝑧‖²=𝛼‖𝑥‖²+𝛽‖𝑦‖²+𝛾‖𝑧‖²−𝛼𝛽‖𝑥−𝑦‖²−𝛼𝛾‖𝑥−𝑧‖²−𝛽𝛾‖𝑦−𝑧‖².(2.19)

Lemma 2.5 (see [29] (demiclosedness principle)). Assume that 𝑇 is a nonexpansive self-mapping of a nonempty closed convex subset 𝐶 of a real Hilbert space 𝐻. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed: that is, whenever {𝑥_𝑛} is a sequence in 𝐶 converging weakly to some 𝑥∈𝐶 (for short, 𝑥_𝑛⇀𝑥∈𝐶) and the sequence {(𝐼−𝑇)𝑥_𝑛} converges strongly to some 𝑦 (for short, (𝐼−𝑇)𝑥_𝑛→𝑦), it follows that (𝐼−𝑇)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

The following lemma is an immediate consequence of an inner product.

Lemma 2.6. In a real Hilbert space 𝐻, there holds the inequality ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.(2.20)

In 2009, Kangtunyakarn and Suantai [33] introduced a new mapping called the 𝑆-mapping as follows.

Let {𝑇_𝑖}^𝑁𝑖=1 be a finite family of nonexpansive mappings of 𝐶 into itself. For each 𝑛∈ℕ, and 𝑗=1,2,…,𝑁, let 𝛼^𝑗(𝑛)=(𝛼^1𝑛,𝑗,𝛼^2𝑛,𝑗,𝛼^3𝑛,𝑗) be such that 𝛼^1𝑛,𝑗,𝛼^2𝑛,𝑗,𝛼^3𝑛,𝑗∈[0,1] with 𝛼^1𝑛,𝑗+𝛼^2𝑛,𝑗+𝛼^3𝑛,𝑗=1. Very recently, in 2009, Kangtunyakarn and Suantai [33] introduced the new mapping 𝑆_𝑛∶𝐶→𝐶 as follows: 𝑈𝑈_𝑛,0=𝐼,_𝑛,1𝑈=𝛼^1𝑛,1𝑇₁𝑈_𝑛,0+𝛼^2𝑛,1𝑈_𝑛,0+𝛼^3𝑛,1𝐼,_𝑛,2𝑈=𝛼^1𝑛,2𝑇₂𝑈_𝑛,1+𝛼^2𝑛,2𝑈_𝑛,1+𝛼^3𝑛,2𝐼,_𝑛,3𝑈=⋮𝛼^1𝑛,3𝑇₃𝑈_𝑛,2+𝛼^2𝑛,3𝑈_𝑛,2+𝛼^3𝑛,3𝐼,_{𝑛,𝑁−1}𝑆=𝛼^{1𝑛,𝑁−1}𝑇_𝑁−1𝑈_{𝑛,𝑁−2}+𝛼^{2𝑛,𝑁−1}𝑈_{𝑛,𝑁−2}+𝛼^{3𝑛,𝑁−1}𝐼,_𝑛=𝑈_𝑛,𝑁=𝛼^1𝑛,𝑁𝑇_𝑁𝑈_{𝑛,𝑁−1}+𝛼^2𝑛,𝑁𝑈_{𝑛,𝑁−1}+𝛼^3𝑛,𝑁𝐼.(2.21) The mapping 𝑆_𝑛 is called the 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝛼^1(𝑛),𝛼^2(𝑛),…,𝛼^𝑁(𝑛). Nonexpansivity of each 𝑇_𝑖 ensures the nonexpansivity of 𝑆_𝑛. They also showed the following useful fact.

Lemma 2.7 (see [33]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑖}^𝑁𝑖=1 be a finite family of nonexpansive mappings of 𝐶 into itself with ∩^𝑁𝑖=1𝐹(T_𝑖)≠∅, and let 𝛼_𝑗=(𝛼^𝑗1,𝛼^𝑗2,𝛼^𝑗3), 𝑗=1,2,…,𝑁, where 𝛼^𝑗1,𝛼^𝑗2,𝛼^𝑗3∈[0,1], 𝛼^𝑗1+𝛼^𝑗2+𝛼^𝑗3=1, 𝛼^𝑗1∈(0,1) for all 𝑗=1,2,…,𝑁−1, 𝛼^𝑁1∈(0,1] and 𝛼^𝑗2,𝛼^𝑗3∈[0,1) for all 𝑗=1,2,…,𝑁. Let 𝑆 be the 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝛼₁,𝛼₂,…,𝛼_𝑁. Then 𝐹(𝑆)=∩^𝑁𝑖=1𝐹(𝑇_𝑖).

Lemma 2.8 (see [12]). For given 𝑥^∗,𝑦^∗∈𝐶, (𝑥^∗,𝑦^∗) is a solution of the problem (1.6) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶(x−𝜇𝐵𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥),∀𝑥∈𝐶,(2.22) where 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗).

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by GVI(𝐶,𝐴,𝐵).

Next, we prove a lemma which is very useful for our consideration.

Lemma 2.9. Let 𝐺∶𝐶→𝐶 be defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥),∀𝑥∈𝐶,(2.23) where 𝜆,𝜇>0 and 𝐴,𝐵∶𝐶→𝐻 are two mappings. If 𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive mappings, then 𝐺 is nonexpansive.

Proof. For any 𝑥,𝑦∈𝐶, we have ‖𝐺(𝑥)−𝐺(𝑦)‖=‖‖−𝑃_𝐶𝑃_𝐶𝑃_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥)𝑃_𝐶(𝑦−𝜇𝐵𝑦)−𝜆𝐴𝑃_𝐶(𝑦−𝜇𝐵𝑦)‖‖²=≤‖‖‖‖𝑃(𝐼_𝐶−(𝑥𝜆−A)𝜇𝑃𝐵_𝐶𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥)−𝑃_𝐶(𝑦−𝜇𝐵𝑦)−𝜆𝐴𝑃_𝐶(𝑦−𝜇𝐵𝑦)‖‖≤‖𝑥−𝑦‖.(𝐼−𝜇𝐵)𝑥−(𝐼−𝜆𝐴)𝑃_𝐶(𝐼−𝜇𝐵)𝑦‖‖(2.24) This shows that 𝐺 is a nonexpansive mapping.

3. Main Results

In this section, we prove strong convergence theorems of the iterative scheme (1.12) to a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of a general system of variational inequality in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹 be a function from 𝐶×𝐶 to ℝ satisfying (A1)–(A5) and 𝜑∶𝐶→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let 𝜆,𝜇>0, and let 𝐴,𝐵∶𝐶→𝐻 be such that 𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive mappings. Let 𝑇 be a nonexpansive self-mapping of 𝐶 such that Ω=𝐹(𝑇)∩GVI(𝐶,𝐴,𝐵)∩MEP(𝐹,𝜑)≠∅. Assume that either (B1) or (B2) holds and that 𝑣 is an arbitrary point in 𝐶. Let the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} be defined by (1.12), {𝑟_𝑛}⊂(0,∞) and {𝑎_𝑛}⊂[0,1] such that (C1)lim_𝑛→∞𝑎_𝑛=0, lim_𝑛→∞(𝑎_𝑛+1/𝑎_𝑛)=1, and ∑^∞𝑛=1𝑎_𝑛=∞, (C2)liminf_𝑛→∞𝑟_𝑛>0 and ∑^∞𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞.If lim_𝑛→∞‖𝐴𝑦_𝑛−𝐴𝑦^∗‖=0 and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑥^∗‖=0 for all 𝑥^∗∈Ω and 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗), then {𝑥_𝑛} converges strongly to 𝑥=𝑃_Ω𝑣 and (𝑥,𝑦) is a solution of the problem (1.6), where 𝑦=𝑃_𝐶(𝑥−𝜇𝐵𝑥).

Proof. Let 𝑥^∗∈Ω and {𝑇_𝑟𝑛} a sequence of mappings defined as in Lemma 2.2. It follows from Lemma 2.8 that 𝑥^∗=𝑃_𝐶𝑃_𝐶𝑥^∗−𝜇𝐵𝑥^∗−𝜆𝐴𝑃_𝐶𝑥^∗−𝜇𝐵𝑥^∗.(3.1) Put 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗) and 𝑡_𝑛=𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛), then 𝑥^∗=𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗) and 𝑥_𝑛+1=𝑎_𝑛𝑣+1−𝑎_𝑛𝑇𝑡_𝑛.(3.2) Since 𝐼−𝜆𝐴, 𝐼−𝜇𝐵, 𝑇_𝑟𝑛, and 𝑃_𝐶 are nonexpansive mappings, we have ‖‖𝑡_𝑛−𝑥^∗‖‖²=‖‖𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤‖‖𝑦_𝑛−𝑦^∗‖‖²=‖‖𝑃_𝐶(𝑢_𝑛−𝜇𝐵𝑢_𝑛)−𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗)‖‖²≤‖‖𝑢_𝑛−𝑥^∗‖‖²=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²,(3.3) which implies that ‖‖𝑥_𝑛+1−𝑥^∗‖‖=≤𝑎‖‖_𝑛𝑎_𝑛𝑣+1−𝑎_𝑛𝑇𝑡_𝑛−𝑥^∗‖‖≤𝑎_𝑛‖𝑣−𝑥^∗‖+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖≤ma‖x𝑣−‖𝑥𝑣^∗−‖𝑥+_∗1−𝑎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖‖,‖‖𝑥₁−𝑥^∗‖‖.(3.4) Thus, {𝑥_𝑛} is bounded. Consequently, the sequences {𝑢_𝑛}, {𝑦_𝑛}, {𝑡_𝑛}, {𝐴𝑦_𝑛}, {𝐵𝑢_𝑛}, and {𝑇𝑡_𝑛} are also bounded. Also, observe that ‖‖𝑡_𝑛+1−𝑡_𝑛‖‖≤=‖‖‖‖𝑃𝑦_𝐶_𝑛+1𝑦_𝑛+1−𝜆𝐴𝑦_𝑛+1−𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖=‖‖𝑃_𝐶−𝑦_𝑛‖‖≤‖‖𝑢_𝑛+1𝑢_𝑛+1−𝜇𝐵𝑢_𝑛+1−𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛‖‖−𝑢_𝑛‖‖.(3.5) On the other hand, from 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛∈dom𝜑 and 𝑢_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1∈dom𝜑, we have 𝐹𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛𝐹𝑢⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,(3.6)_𝑛+1,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐶.(3.7) Putting 𝑦=𝑢_𝑛+1 in (3.6) and 𝑦=𝑢_𝑛 in (3.7), we have 𝐹𝑢_𝑛,𝑢_𝑛+1+𝜑𝑢_𝑛+1−𝜑𝑢_𝑛+1𝑟_𝑛𝑢_𝑛+1𝐹𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,_𝑛+1,𝑢_𝑛+𝜑𝑢_𝑛−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.8) From the monotonicity of 𝐹, we obtain that 𝑢^𝑛+1−𝑢_𝑛,𝑢^𝑛−𝑥_𝑛𝑟_𝑛−𝑢^𝑛+1−𝑥_𝑛+1𝑟_𝑛+1≥0,(3.9) and hence 𝑢^𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.10) Without loss of generality, let us assume that there exists a real number 𝑐 such that 𝑟_𝑛>𝑐>0 for all 𝑛∈ℕ. Then, we have ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+1−𝑟^𝑛𝑟_𝑛+1𝑢^𝑛+1≤‖‖𝑢_𝑛+1−𝑥_𝑛+1−𝑢_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||1−𝑟^𝑛𝑟_𝑛+1||||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖,(3.11) and hence ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1𝑟_𝑛+1≤‖‖𝑥_𝑛+1||𝑟_𝑛+1−𝑟_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+1𝑐||𝑟^𝑛+1−𝑟_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖.(3.12) It follows from (3.5) and (3.12) that ‖‖𝑡_𝑛+1−𝑡_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1𝑐||𝑟^𝑛+1−𝑟_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖.(3.13) Next, we show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. From (3.13), we have ‖‖𝑥_𝑛+2−𝑥_𝑛+1‖‖==‖‖‖‖𝑎𝑎_𝑛+1𝑣_𝑛+1+1−𝑎_𝑛+1𝑇𝑡_𝑛+1−𝑎_𝑛𝑣−1−𝑎_𝑛𝑇𝑡_𝑛‖‖≤||𝑎_𝑛+1−𝑎_n𝑣+𝑎_𝑛−𝑎_𝑛+1𝑇𝑡_𝑛+1+1−𝑎_𝑛𝑇𝑡_𝑛+1−𝑇𝑡_𝑛‖‖≤||𝑎_𝑛+1−𝑎_𝑛||‖𝑣‖+‖‖𝑇𝑡_𝑛+1‖‖+1−𝑎_𝑛‖‖𝑡_𝑛+1−𝑡_𝑛‖‖+1−𝑎_𝑛||‖𝑣‖+‖‖𝑇𝑡_𝑛+1‖‖+1−𝑎_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝑐||𝑟^𝑛+1≤1−𝑎_𝑛−𝑟_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1𝑐||𝑟^𝑛+1−𝑟_𝑛||𝑀+2𝑀||𝑎_𝑛+1−𝑎_𝑛||,(3.14) where 𝑀=max{‖𝑣‖,sup_𝑛‖𝑇𝑡_𝑛+1‖,sup_𝑛‖𝑢_𝑛+1−𝑥_𝑛+1‖}. By the assumptions on {𝑎_𝑛} and {𝑟_𝑛} and Lemma 2.3, we conclude that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.15) From (C1), (C2), (3.5), (3.12), and (3.15), we also have ‖𝑢_𝑛+1−𝑢_𝑛‖→0, ‖𝑡_𝑛+1−𝑡_𝑛‖→0, and ‖𝑦_𝑛+1−𝑦_𝑛‖→0, as 𝑛→∞.
Since 𝑥_𝑛+1−𝑥_𝑛=𝑎_𝑛𝑣−𝑥_𝑛+1−𝑎_𝑛𝑇𝑡_𝑛−𝑥_𝑛,(3.16) we have that ‖‖𝑇𝑡_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.17) Next, we prove that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. From Lemma 2.2(3), we have ‖‖𝑢_𝑛−𝑥^∗‖‖²==⟨‖‖𝑢𝑇_𝑟𝑛𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖‖²≤𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗,𝑥_𝑛−𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗⟩=12‖‖𝑢^𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.18) Hence, ‖‖𝑢_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.19) From Lemma 2.4, (3.3), and (3.19), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑢_𝑛−𝑥^∗‖‖²≤≤𝑎𝑎^𝑛_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝑎_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.20) It follows that 1−𝑎_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.21) From conditions (C1) and (3.15), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.22) By (3.17) and (3.22), we have ‖‖𝑇𝑡_𝑛−𝑢_𝑛‖‖≤‖‖𝑇𝑡_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑢_𝑛‖‖⟶0,as𝑛⟶∞.(3.23) Next, we prove that ‖𝑇𝑡_𝑛−𝑡_𝑛‖→0 as 𝑛→∞. From (2.11) and nonexpansiveness of 𝐼−𝜇𝐵, we get ‖‖𝑦_𝑛−𝑦^∗‖‖²=‖‖𝑃_𝐶(𝑢_𝑛−𝜇𝐵𝑢_𝑛)−𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗)‖‖²≤=1𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝑥^∗−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑦^∗2‖‖(𝑢^𝑛−𝜇𝐵𝑢_𝑛)−(𝑥^∗−𝜇𝐵𝑥^∗)‖‖²+‖‖𝑦_𝑛−𝑦^∗‖‖²≤1−‖‖(𝑢_𝑛−𝜇𝐵𝑢_𝑛)−(𝑥^∗−𝜇𝐵𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²2‖‖𝑢^𝑛−𝑥^∗‖‖²+‖‖𝑦_𝑛−𝑦^∗‖‖²−‖‖(𝑢_𝑛−𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²+2𝜇⟨𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗⟩−𝜇²‖𝐵𝑢_𝑛−𝐵𝑥^∗‖².(3.24) By (3.3), we obtain ‖‖𝑦_𝑛−𝑦^∗‖‖²≤‖‖𝑢_𝑛−𝑥^∗‖‖²−‖‖(𝑢_𝑛−𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²+2𝜇𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗−𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖(𝑢_𝑛−𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²+2𝜇𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗−𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖².(3.25) Hence, ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖²≤×𝑎_𝑛‖‖‖𝑣𝑥−_𝑛𝑥^∗‖²+1−𝑎_𝑛−𝑥^∗‖‖²−‖‖(𝑢_𝑛−𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²≤𝑎_𝑛+2𝜇𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗−𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−1−𝑎_𝑛‖‖(𝑢_𝑛−𝑥^∗)−(𝑦_𝑛−𝑦^∗)‖‖²+1−𝑎_𝑛2𝜇‖‖𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗‖‖‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖,(3.26) which implies that 1−𝑎_𝑛‖‖(𝑢_𝑛−𝑥^∗)−𝑦_𝑛−𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²++‖‖1𝑥−_𝑛𝑎_𝑛2𝜇‖‖𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗‖‖‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.27) By this together with (C1), (3.15), and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑥^∗‖=0, we obtain ‖‖𝑢_𝑛−𝑥^∗−𝑦_𝑛−𝑦^∗‖‖⟶0as𝑛⟶∞.(3.28) From Lemma 2.6 and (2.12), it follows that ‖‖(𝑦_𝑛−𝑡_𝑛)+(𝑥^∗−𝑦^∗)‖‖²=‖‖(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)−𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐶𝑦^∗−𝜆𝐴𝑦^∗+𝜆(𝐴𝑦_𝑛−𝐴𝑦^∗)‖‖²≤‖‖(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)−𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐶𝑦^∗−𝜆𝐴𝑦^∗‖‖²≤+‖‖(2𝑦𝜆_𝑛⟨𝐴𝑦_𝑛−𝐴𝑦^∗,𝑦_𝑛−𝑡_𝑛+𝑥^∗−𝑦^∗⟩−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²−‖‖𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤+‖‖(2𝑦𝜆_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖‖‖𝑦_𝑛−𝑡_𝑛+𝑥^∗−𝑦^∗‖‖−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²−‖‖𝑇𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑇𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤+‖‖2𝜆𝑦_𝑛‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖‖‖𝑦_𝑛−𝑡_𝑛+𝑥^∗−𝑦^∗‖‖×‖‖𝑦−_𝑛𝜆𝐴𝑦_𝑛−𝑦^∗−𝜆𝐴𝑦^∗−𝑇𝑡_𝑛−𝑥^∗‖‖+2𝜆‖‖𝐴𝑦−_𝑛𝜆𝐴𝑦_𝑛−𝑦^∗−𝜆𝐴𝑦^∗‖‖+‖‖𝑇𝑡_𝑛−𝑥^∗‖‖=‖‖𝑢_𝑛−𝐴𝑦^∗‖‖‖‖𝑦_𝑛−𝑡_𝑛+𝑥^∗−𝑦^∗‖‖×‖‖−𝑦𝑇_𝑛𝑡_𝑛+𝑥^∗−𝑦^∗−𝑢_𝑛−𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝐴𝑦^∗‖‖+2𝜆‖‖𝐴𝑦−_𝑛𝜆𝐴𝑦_𝑛−𝑦^∗−𝜆𝐴𝑦^∗‖‖+‖‖𝑇𝑡_𝑛−𝑥^∗‖‖−𝐴𝑦^∗‖‖‖‖𝑦_𝑛−𝑡_𝑛+𝑥^∗−𝑦^∗‖‖.(3.29) By this together with (3.23), (3.28), and lim_𝑛→∞‖𝐴𝑦_𝑛−𝐴𝑦^∗‖=0, we obtain ‖(𝑦_𝑛−𝑡_𝑛)+(𝑥^∗−𝑦^∗)‖→0 as 𝑛→∞. This together with (3.17), (3.22), and (3.28), we obtain that ‖‖𝑇𝑡_𝑛−𝑡_𝑛‖‖≤+‖‖‖‖𝑇𝑡_𝑛𝑦−_𝑛𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑦_𝑛−𝑥^∗−𝑦^∗‖‖−𝑡_𝑛+𝑥^∗−𝑦^∗‖‖⟶0,as𝑛⟶∞.(3.30) Next, we show that limsup_𝑛→∞⟨𝑣−𝑥,𝑥_𝑛−𝑥⟩≤0,(3.31) where 𝑥=𝑃_Ω𝑣.
Indeed, since {𝑡_𝑛} and {𝑇𝑡_𝑛} are two bounded sequences in 𝐶, we can choose a subsequence {𝑡_𝑛𝑖} of {𝑡_𝑛} such that 𝑡_𝑛𝑖⇀𝑧∈𝐶 and limsup_𝑛→∞𝑣−𝑥,𝑇𝑡_𝑛−𝑥=lim_𝑖→∞𝑣−𝑥,𝑇𝑡_𝑛𝑖−𝑥.(3.32) Since lim_𝑛→∞‖𝑇𝑡_𝑛−𝑡_𝑛‖=0, we obtain that 𝑇𝑡_𝑛𝑖⇀𝑧 as 𝑖→∞.
Next, we show that 𝑧∈Ω.
(a) We first show 𝑧∈𝐹(𝑇).
Since 𝑡_𝑛𝑖⇀𝑧 and ‖𝑇𝑡_𝑛−𝑡_𝑛‖→0, we obtain by Lemma 2.5 that 𝑧∈𝐹(𝑇).
(b) Now, we show that 𝑧∈GVI(𝐶,𝐴,𝐵).
From (3.30) and (3.17), we have ‖‖𝑡_𝑛−𝑥_𝑛‖‖≤‖‖𝑇𝑡_𝑛−𝑡_𝑛‖‖+‖‖𝑇𝑡_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.33) Furthermore, by Lemma 2.9, we have that 𝐺∶𝐶→𝐶 is nonexpansive. Then, we have ‖‖𝑡_𝑛−𝐺𝑡_𝑛‖‖==‖‖‖‖𝑃𝑃_𝐶𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝐺𝑡_𝑛‖‖=‖‖𝐺𝑢𝑃_𝑛𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝜆𝐴𝑃𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝐺𝑡_𝑛‖‖≤‖‖𝑢_𝑛−𝐺𝑡_𝑛‖‖≤‖‖𝑢_n−𝑡_𝑛‖‖−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑡_𝑛‖‖⟶0as𝑛⟶∞.(3.34) Again by Lemma 2.5, we have 𝑧∈GVI(𝐶,𝐴,𝐵).
(c) We show that 𝑧∈MEP(𝐹,𝜑). Since 𝑡_𝑛𝑖⇀𝑧 and ‖𝑥_𝑛−𝑡_𝑛‖→0, we obtain that 𝑥_𝑛𝑖⇀𝑧. From ‖𝑢_𝑛−𝑥_𝑛‖→0, we also obtain that 𝑢_𝑛𝑖⇀𝑧. By using the same argument as that in the proof of [30, Theorem  3.1, page 1825], we can show that 𝑧∈MEP(𝐹,𝜑). Therefore, there holds 𝑧∈Ω.
On the other hand, it follows from (2.13), (3.17), and 𝑇𝑡_𝑛𝑖⇀𝑧 as 𝑖→∞ that limsup_𝑛→∞𝑣−𝑥,x_𝑛−𝑥==lim𝑣−sup_𝑛→∞𝑣−𝑥,𝑇𝑡_𝑛−𝑥=lim_𝑖→∞𝑣−𝑥,𝑇𝑡_𝑛𝑖−𝑥𝑥,𝑧−𝑥≤0.(3.35) Hence, we have ‖‖𝑥_𝑛+1−𝑥‖‖²==𝑎_𝑛𝑎_𝑛𝑣+1−𝑎_𝑛𝑇𝑡_𝑛−𝑥,𝑥_𝑛+1−𝑥≤𝑎_𝑛𝑣−𝑥,𝑥_𝑛+1−𝑥+1−𝑎_𝑛𝑇𝑡_𝑛−𝑥,𝑥_𝑛+1−𝑥≤𝑎_𝑛𝑣−𝑥,𝑥_𝑛+1−𝑥+121−𝑎^𝑛‖‖𝑡_𝑛−𝑥‖‖²+‖‖𝑥_𝑛+1−𝑥‖‖²𝑣−𝑥,𝑥_𝑛+1−𝑥+121−𝑎^𝑛‖‖𝑥_𝑛−𝑥‖‖²+‖‖𝑥_𝑛+1−𝑥‖‖²,(3.36) which implies that ‖‖𝑥_𝑛+1−𝑥‖‖²≤1−𝑎_𝑛‖‖𝑥_𝑛−𝑥‖‖²+2𝑎_𝑛⟨𝑣−𝑥,𝑥_𝑛+1−𝑥⟩.(3.37) By this together with (C1) and (3.35), we have by Lemma 2.3 that {𝑥_𝑛} converges strongly to 𝑥. This completes the proof.

The following examples provide mappings 𝐴 and 𝐵 which satisfy those conditions in Theorem 3.1.

Example 3.2. Let 𝐴,𝐵∶𝐶→𝐻 be 𝛼-inverse-strongly monotone and 𝛽-inverse-strongly monotone, respectively. If 𝜆∈(0,2𝛼) and 𝜇∈(0,2𝛽), then we have(1)𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive,(2)‖𝐴𝑦_𝑛−𝐴𝑦^∗‖→0 and ‖𝐵𝑢_𝑛−𝐵𝑥^∗‖→0 as 𝑛→∞ for all 𝑥^∗∈Ω and 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗), where {𝑦_𝑛} and {𝑢_𝑛} are sequences defined as in Theorem 3.1.

Proof. (1) For any 𝑥,𝑦∈𝐶, we have ‖(𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦‖²=‖(𝑥−𝑦)−𝜆(𝐴𝑥−𝐴𝑦)‖²=≤‖‖𝑥𝑥−−𝑦𝑦‖‖²₂+𝜆²‖𝐴𝑥−𝐴𝑦‖²−2𝜆⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩+𝜆²‖𝐴𝑥−𝐴𝑦‖²−2𝜆𝛼‖𝐴𝑥−𝐴𝑦‖²=‖𝑥−𝑦‖²+𝜆(𝜆−2𝛼)‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²,(3.38) hence, 𝐼−𝜆𝐴 is nonexpansive. Similarly, we can show that 𝐼−𝜇𝐵 is nonexpansive.
(2) Let {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} be the sequences defined as in Theorem 3.1. From (3.3), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²=𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤≤𝑎𝑎_𝑛𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²‖‖𝑥_𝑛+1‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝑎_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²,−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖(𝑢_𝑛−𝜇𝐵𝑢_𝑛)−(𝑥^∗−𝜇𝐵𝑥^∗)‖‖²≤≤𝑎𝑎_𝑛𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑢_𝑛−𝑥^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝑎_𝑛𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖².(3.39) Therefore, we have −−11−−𝑎𝑎_𝑛𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖,𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.40) From (3.15) and (C1), we obtain ‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖⟶0,‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖⟶0as𝑛⟶∞.(3.41)

Example 3.3. Let 𝐴∶𝐶→𝐻 be an 𝐿_𝐴-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping and 𝐵∶𝐶→𝐻 an 𝐿_𝐵-Lipschitzian and relaxed (𝑐^,𝑑^)-cocoercive mapping. If 0<𝜆<(2(𝑑−𝑐𝐿^2𝐴))/𝐿^2𝐴 and 0<𝜇<(2(𝑑^−𝑐^𝐿^2𝐵))/𝐿^2𝐵, then we have(1)𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive,(2)‖𝐴𝑦_𝑛−𝐴𝑦^∗‖→0 and ‖𝐵𝑢_𝑛−𝐵𝑥^∗‖→0 as 𝑛→∞ for all 𝑥^∗∈Ω and 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗), where {𝑦_𝑛} and {𝑢_𝑛} are sequences defined as in Theorem 3.1.

Proof. (1) For any 𝑥,𝑦∈𝐶, we have ‖(𝐼−𝜇𝐵)𝑥−(𝐼−𝜇𝐵)𝑦‖²=‖(𝑥−𝑦)−𝜇(𝐵𝑥−𝐵𝑦)‖²≤=‖‖𝑥𝑥−−𝑦𝑦‖‖²₂+𝜇²‖𝐵𝑥−𝐵𝑦‖²−2𝜇⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩+𝜇²‖𝐵𝑥−𝐵𝑦‖²=−‖𝑥2𝜇−𝑦−‖𝑐^₂‖𝐵𝑥−𝐵𝑦‖²+𝑑^‖𝑥−𝑦‖²+𝜇²‖𝐵𝑥−𝐵𝑦‖²+2𝜇𝑐^‖𝐵𝑥−𝐵𝑦‖²−2𝜇𝑑^‖𝑥−𝑦‖²≤‖𝑥−𝑦‖²+𝜇²𝐿^2𝐵‖𝑥−𝑦‖²+2𝜇𝑐^𝐿^2𝐵‖𝑥−𝑦‖²−2𝜇𝑑^‖𝑥−𝑦‖²=1+2𝜇𝑐^𝐿^2𝐵−2𝜇𝑑^+𝜇²𝐿^2𝐵‖𝑥−𝑦‖²≤‖𝑥−𝑦‖²,(3.42) hence, 𝐼−𝜇𝐵 is nonexpansive. Similarly, we can show that 𝐼−𝜆𝐴 is nonexpansive.
(2) Let {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} be the sequences defined as in Theorem 3.1. From (3.3), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖(𝑢_𝑛−𝜇𝐵𝑢_𝑛)−(𝑥^∗−𝜇𝐵𝑥^∗)‖‖²≤×𝑎_𝑛‖‖‖𝑣𝑢−_𝑛𝑥^∗‖²+1−𝑎_𝑛≤𝑎_𝑛−𝑥^∗‖‖²−2𝜇⟨𝑢_𝑛−𝑥^∗,𝐵𝑢_𝑛−𝐵𝑥^∗⟩+𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²×‖‖‖𝑣𝑢−_𝑛𝑥^∗‖²+1−𝑎_𝑛≤𝑎_𝑛−𝑥^∗‖‖²+2𝜇𝑐^‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²−2𝜇𝑑^‖‖𝑢_𝑛−𝑥^∗‖‖²+𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜇𝑐^+𝜇²−2𝜇𝑑^𝐿^2𝐵‖‖𝐵𝑢^𝑛−𝐵𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝑎_𝑛2𝜇𝑐^+𝜇²−2𝜇𝑑^𝐿^2𝐵‖‖𝐵𝑢^𝑛‖‖𝑥_𝑛+1−𝐵𝑥^∗‖‖²,−𝑥^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖²=𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑃_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑃_𝐶(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−(𝑦^∗−𝜆𝐴𝑦^∗)‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+1−𝑎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜆𝑐+𝜆²−2𝜆𝑑𝐿^2𝐴‖‖𝐴𝑦^𝑛−𝐴𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝑎_𝑛2𝜆𝑐+𝜆²−2𝜆𝑑𝐿^2𝐴‖‖𝐴𝑦^𝑛−𝐴𝑦^∗‖‖².(3.43) Therefore, we have −1−𝑎_𝑛2𝜇𝑐^+𝜇²−2𝜇𝑑^𝐿^2𝐵‖‖𝐵𝑢^𝑛−𝐵𝑥^∗‖‖²−1≤−𝑎𝑎_𝑛‖_𝑛𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖,2𝜆𝑐+𝜆²−2𝜆𝑑𝐿^2𝐴‖‖𝐴𝑦^𝑛−𝐴𝑦^∗‖‖²≤𝑎_𝑛‖𝑣−𝑥^∗‖²+‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.44) From (3.15) and (C1), we obtain ‖‖𝐴𝑦_𝑛−𝐴𝑦^∗‖‖⟶0,‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖⟶0as𝑛⟶∞.(3.45)

By using the same proof as in Examples 3.2 and 3.3, we can obtain the following example.

Example 3.4. Let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻 and 𝐵 be a L-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping of 𝐶 into 𝐻. If 𝜆∈(0,2𝛼) and 0<𝜇<(2(𝑑−𝑐𝐿²))/𝐿², then we have(1)𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive,(2)‖𝐴𝑦_𝑛−𝐴𝑦^∗‖→0 and ‖𝐵𝑢_𝑛−𝐵𝑥^∗‖→0 as 𝑛→∞ for all 𝑥^∗∈Ω and 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗), where {𝑦_𝑛} and {𝑢_𝑛} are sequences defined as in Theorem 3.1.

Let 𝒜 be the class of all 𝛼-inverse-strongly monotone mappings from 𝐶 into 𝐻, ℬ the class of all 𝛽-inverse-strongly monotone mappings from 𝐶 into 𝐻, 𝒞 the class of all 𝐿-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mappings from 𝐶 into 𝐻, and 𝒟 the class of all 𝐿₁-Lipschitzian and relaxed (𝑐^,𝑑^)-cocoercive mappings from 𝐶 into 𝐻.

Theorem 3.5. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹 be a function from 𝐶×𝐶 to ℝ satisfying (A1)–(A5) and 𝜑∶𝐶→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let {𝑇_𝑖}^𝑁𝑖=1 be a finite family of nonexpansive self-mappings of 𝐶 such that Ω=∩^𝑁𝑖=1𝐹(𝑇_𝑖)∩GVI(𝐶,𝐴,𝐵)∩MEP(𝐹,𝜑)≠∅. Let 𝛼_𝑗=(𝛼^𝑗1,𝛼^j₂,𝛼^𝑗3), 𝑗=1,2,…,𝑁, where 𝛼^𝑗1,𝛼^𝑗2,𝛼^𝑗3∈[0,1], 𝛼^𝑗1+𝛼^𝑗2+𝛼^𝑗3=1, 𝛼^𝑗1∈(0,1) for all 𝑗=1,2,…,𝑁−1, 𝛼^𝑁1∈(0,1] and 𝛼^𝑗2,𝛼^𝑗3∈[0,1) for all 𝑗=1,2,…,𝑁. Let 𝑆 be the 𝑆-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝛼₁,𝛼₂,…,𝛼_𝑁. Assume that either (B1) or (B2) holds and that 𝑣 is an arbitrary point in 𝐶. Let 𝑥₁∈𝐶 and {𝑥_𝑛},{𝑢_𝑛},{𝑦_𝑛} the sequences generated by 𝐹𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛,=𝑎_𝑛𝑣+1−𝑎_𝑛𝑆𝑃_C𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1.(3.46) If one of the following conditions is satisfied: (1)𝐴∈𝒜, 𝐵∈ℬ, 𝜆∈(0,2𝛼), and 𝜇∈(0,2𝛽),(2)𝐴∈𝒞, 𝐵∈𝒟, 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿², and 0<𝜇<(2(𝑑^−𝑐^𝐿²¹))/𝐿²¹,(3)𝐴∈𝒜, 𝐵∈𝒞, 𝜆∈(0,2𝛼), and 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿²,and the sequences {𝑟_𝑛} and {𝑎_𝑛} are as in Theorem 3.1, then {𝑥_𝑛} converges strongly to 𝑥=𝑃_Ω𝑣 and (𝑥,𝑦) is a solution of the problem (1.6), where 𝑦=𝑃_𝐶(𝑥−𝜇𝐵𝑥).

Proof. By Lemma 2.7, we obtain that 𝑆 is nonexpansive and 𝐹(𝑆)=∩^𝑁𝑖=1𝐹(𝑇_𝑖). Hence, the result is obtained directly from Theorem 3.1 and Examples 3.2–3.4.

From Theorem 3.1 and Examples 3.2–3.4, we obtain the following result.

Corollary 3.6. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹 be a function from 𝐶×𝐶 to ℝ satisfying (A1)–(A5) and 𝜑∶𝐶→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let 𝑇 be a nonexpansive self-mapping of 𝐶 such that Ω=𝐹(𝑇)∩GVI(𝐶,𝐴,𝐵)∩MEP(𝐹,𝜑)≠∅. Assume that either (B1) or (B2) holds and that 𝑣 is an arbitrary point in 𝐶. If one of the following conditions is satisfied: (1)𝐴∈𝒜, 𝐵∈ℬ, 𝜆∈(0,2𝛼), and 𝜇∈(0,2𝛽),(2)𝐴∈𝒞, 𝐵∈𝒟, 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿², and 0<𝜇<(2(𝑑^−𝑐^𝐿²¹))/𝐿²¹,(3)𝐴∈𝒜, 𝐵∈𝒞, 𝜆∈(0,2𝛼), and 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿²,and the sequences {𝑟_𝑛}, {𝑎_𝑛} are as in Theorem 3.1, then the sequence {𝑥_𝑛} generated by (1.12) converges strongly to 𝑥=𝑃_Ω𝑣 and (x,𝑦) is a solution of the problem (1.6), where 𝑦=𝑃_𝐶(𝑥−𝜇𝐵𝑥).

Let 𝜑=0 in Theorem 3.1. From Theorem 3.1 and Examples 3.2–3.4, we obtain the following result.

Corollary 3.7. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹 be a function from 𝐶×𝐶 to ℝ satisfying (A1)–(A5). Let 𝑇 be a nonexpansive self-mapping of 𝐶 such that Ω=𝐹(𝑇)∩GVI(𝐶,𝐴,𝐵)∩EP(𝐹)≠∅. Let 𝑥₁,𝑣∈𝐶 and {𝑥_𝑛},{𝑢_𝑛},{𝑦_𝑛} be the sequences generated by 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛,=𝑎_𝑛𝑣+1−𝑎_𝑛𝑇𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1.(3.47) If one of the following conditions is satisfied: (1)𝐴∈𝒜, 𝐵∈ℬ, 𝜆∈(0,2𝛼), and 𝜇∈(0,2𝛽),(2)𝐴∈𝒞, 𝐵∈𝒟, 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿², and 0<𝜇<(2(𝑑^−𝑐^𝐿²¹))/𝐿²¹,(3)𝐴∈𝒜, 𝐵∈𝒞, 𝜆∈(0,2𝛼), and 0<𝜆<(2(𝑑−𝑐𝐿²))/𝐿²,and the sequences {𝑟_𝑛}, {𝑎_𝑛} are as in Theorem 3.1, then {𝑥_𝑛} converges strongly to 𝑥=𝑃_Ω𝑣 and (𝑥,𝑦) is a solution of the problem (1.6), where 𝑦=𝑃_𝐶(𝑥−𝜇𝐵𝑥).

Remark 3.8. In Theorem 3.5, if 𝐹≡0, then the sequence {𝑥_𝑛} generated by (3.46) converges strongly to a solution of the minimization problem which is also a solution of a system of variational inequalities.

Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and thank the National Research University Project under Thailand’s Office of the Higher Education Commission, the Centre of Excellence in Mathematics, the Thailand Research Fund and the Thaksin university for financial support.