A General Iterative Scheme Based on Regularization for Solving Equilibrium and Constrained Convex Minimization Problems

Abstract

The present paper is divided into two parts. First, we introduce implicit and explicit iterative schemes based on the regularization for solving equilibrium and constrained convex minimization problems. We establish results on the strong convergence of the sequences generated by the proposed schemes to a common solution of minimization and equilibrium problem. Such a point is also a solution of a variational inequality. In the second part, as applications, we apply the algorithm to solve split feasibility problem and equilibrium problem.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝜙 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. Consider the equilibrium problem (EP) which is to find 𝑧∈𝐶 such that 𝜙𝑧,𝑦≥0,∀𝑦∈𝐶. (1)

We denoted the set of solutions of EP by EP(𝜙). Given a mapping 𝑇∶𝐶→𝐻, let 𝜙(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶; then 𝑧∈EP(𝜙) if and only if ⟨𝑇𝑧,𝑦−𝑧⟩≥0 for all 𝑦∈𝐶; that is, 𝑧 is a solution of the variational inequality. Numerous problems in physics, optimizations and economics reduce to find a solution of (1). Some methods have been proposed to solve the equilibrium problem; see for instance see [1–8] and the references therein.

Some composite iterative algorithms were proposed by many authors for finding the common solution of equilibrium problem and fixed point problem. Next, we list some main results as follows.

With some appropriate assumptions, Ceng et al. [9] established the following iterative scheme: 𝑥₁∈𝐻 and 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛,∀𝑛∈ℕ. (2) Under certain conditions, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge weakly to an element of EP(𝜙)∩𝐹(𝑆).

For finding an element of EP(𝜙)∩𝐹(𝑆), S. Takahashi and W. Takahashi [10] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: 𝑥₁∈𝐻 and 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛ℎ𝑥_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛,∀𝑛∈ℕ. (3) Under suitable conditions, some strong convergence theorems are obtained.

In 2009, Liu [11] introduced two iterative schemes by the general iterative method for finding an element of EP(𝜙)∩𝐹(𝑆), where 𝑆∶𝐶→𝐻 is a 𝑘-strictly pseudocontraction nonself mapping in the setting of a real Hilbert space. Let {𝑥_𝑛} be a sequence generated by 𝜙𝑢_𝑛+1,𝑦𝜆_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,𝑥_𝑛=𝛼_𝑛𝛾ℎ𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑦_𝑛,∀𝑛∈ℕ, (4) and 𝑥₁∈𝐻 arbitrarily, 𝜙𝑢_𝑛+1,𝑦𝜆_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾ℎ𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑦_𝑛,∀𝑛∈ℕ, (5) where 𝐵 is a strongly positive bounded linear operator on 𝐻. Under some assumptions, the strong convergence theorems are obtained.

In 2012, based on the concept of the shrinking projection method, Reich and Sabach [12] consider the following algorithm for finding the common solution of finite equilibrium problems in a reflexive Banach space 𝑥₀𝑄∈𝑋,_𝑖0𝑦=𝑋,𝑖=1,2,…,𝑁,_𝑖𝑛=Res^𝑓_{𝜆𝑖𝑛𝑔𝑖}𝑥_𝑛+𝑒^𝑖_𝑛,𝑄^𝑖_𝑛+1=𝑧∈𝑄^𝑖_𝑛∶∇𝑓𝑥_𝑛+𝑒^𝑖_𝑛−∇𝑓𝑦^𝑖_𝑛,𝑧−𝑦^𝑖_𝑛,𝑄≤0_𝑛+1∶=∩^𝑁_𝑖=1𝑄^𝑖_𝑛+1,𝑥_𝑛+1=proj_𝑄^𝑓_𝑛+1𝑥₀,𝑛=0,1,2,…. (6) Under some consumption, the sequence {𝑥_𝑛}_𝑛∈ℕ converges strongly to Proj^𝑓_𝐸(𝑥₀).

The gradient-projection algorithm is a classical power method for solving constrained convex optimization problems and has been studied by many authors (see [13–26] and the reference therein). The method has recently been applied to solve split feasibility problems which find applications in image reconstructions and the intensity modulated radiation therapy (see [27–34]).

Consider the problem of minimizing 𝑓 over the constraint set 𝐶 (assuming 𝐶 is a nonempty closed and convex subset of a real Hilbert space 𝐻). The main results we all know about the gradient projection are that if 𝑓∶𝐻→ℝ is a convex and continuously Fréchet differentiable functional, the gradient-projection algorithm generates a sequence {𝑥_𝑛}^∞_𝑛=0 determined by the gradient of 𝑓 and the metric projection onto 𝐶. Under the condition that 𝑓 has a Lipschitz continuous and strongly monotone gradient, the sequence {𝑥_𝑛}^∞_𝑛=0 can be strongly convergent to a minimizer of 𝑓 in 𝐶. If the gradient of 𝑓 is only assumed to be inverse strongly monotone, then {𝑥_𝑛}^∞_𝑛=0 can only be weakly convergent if 𝐻 is infinite-dimensional.

Recently, Xu [35] gave an operator-oriented approach as an alternative to the gradient-projection method and to the relaxed gradient-projection algorithm, namely, an averaged mapping approach. He also presented two modifications of gradient-projection algorithms which are shown to have strong convergence.

On the other hand, regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems [36]. The disadvantage is the weak convergence of the method RGPA for the regularization problem under some conditions.

The purpose of the paper is to study the iterative method for finding the common solution of an equilibrium problem and a constrained convex minimization problem. Based on the Viscosity method [18], we combine the RGPA and averaged mapping approaches to propose implicit and explicit composite iterative methods for finding the common element of the set of solutions of an equilibrium problem and the solution set of a constrained convex minimization problem and also to prove some strong convergence theorems.

2. Preliminaries

Throughout the paper, we assume that 𝐻 is a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively, and 𝐶 is a nonempty closed convex subset of 𝐻. The set of fixed points of a mapping 𝑇 is denoted by Fix(𝑇); that is, Fix(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. The fact that the sequence {𝑥_𝑛} converges strongly to 𝑥 is denoted by 𝑥_𝑛→𝑥. The following definition and results are needed in the subsequent sections.

Recall that a mapping 𝑉∶𝐻→𝐻 is said to be 𝐿-Lipschitzian if 𝑉𝑥−𝑉𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐻, (7) where 𝐿>0 is a constant. In particular, if 𝐿∈[0,1), then 𝑉 is called a contraction on 𝐻; if 𝐿=1, then 𝑉 is called a nonexpansive mapping on 𝐻. 𝑉 is called firmly nonexpansive if 2𝑉−𝐼 is nonexpansive, or equivalently, ⟨𝑥−𝑦,𝑉𝑥−𝑉𝑦⟩≥‖𝑉𝑥−𝑉𝑦‖²,for all 𝑥,𝑦∈𝐻. Alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 𝑇=(1/2)(𝐼+𝑆), where 𝑆∶𝐻→𝐻 is nonexpansive.

In 1978, Baillon et al. [37] defined the concept of averaged mapping which is used very frequently now.

Definition 1 (see [37]). A mapping 𝑇∶𝐻→𝐻 is said to be an averaged mapping if it can be written as the average of the identity 𝐼 and a nonexpansive mapping; that is, 𝑇=(1−𝛼)𝐼+𝛼𝑆, (8) where 𝛼 is a number in (0,1) and 𝑆∶𝐻→𝐻 is nonexpansive. More precisely, when (8) holds, we say that 𝑇 is 𝛼-averaged. Clearly, a firmly nonexpansive mapping (in particular, projection) is a (1/2)-averaged map.

Proposition 2 (see [28, 38]). For given operators 𝑆,𝑇,𝑉∶𝐻→𝐻 one has the following.

(i) If 𝑇=(1−𝛼)𝑆+𝛼𝑉 for some 𝛼∈(0,1) and if 𝑆 is averaged and 𝑉 is nonexpansive, then 𝑇 is averaged.

(ii) 𝑇 is firmly nonexpansive if and only if the complement 𝐼−𝑇 is firmly nonexpansive.

(iii) If 𝑇=(1−𝛼)𝑆+𝛼𝑉 for some 𝛼∈(0,1) and if S is firmly nonexpansive and 𝑉 is nonexpansive, then 𝑇 is averaged.

Recall that the metric (or nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦=∶𝑑(𝑥,𝐶). (9)

In 1984, Goebel and Reich [39] discussed the properties of the nearest point projection.

Lemma 3 (see [39]). For given 𝑥∈𝐻 one has the following:(i)𝑧=𝑃_𝐶𝑥 if and only if ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶; (10)(ii)𝑧=𝑃_𝐶𝑥 if and only if ‖𝑥−𝑧‖²≤𝑥−𝑦²−𝑦−𝑧²,∀𝑦∈𝐶; (11)(iii)𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (12)
Consequently, 𝑃_𝐶 is nonexpansive and monotone.

Lemma 4. The following inequality holds in a Hilbert space 𝑋𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝑋. (13)

Lemma 5 (see [40]). In a Hilbert space 𝐻, one has 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦²∀𝑥,𝑦∈𝐻,𝜆∈[0,1]. (14)

Lemma 6 (Demiclosedness Principle [40]). Let 𝐻 be a Hilbert space, 𝐾 a closed convex subset of 𝐻, and 𝑇∶𝐾→𝐾 a nonexpansive mapping with Fix(𝑇)≠∅; if {𝑥_𝑛} is a sequence in 𝐾 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦; in particular if 𝑦=0 then 𝑥∈Fix(𝑇).

Definition 7. A nonlinear operator 𝑇 whose domain 𝐷(𝑇)⊆𝐻 and range 𝑅(𝑇)⊆𝐻 is said to be(i) monotone if ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥0,∀𝑥,𝑦∈𝐷(𝑇); (15)(ii)𝛽-strongly monotone if there exists 𝛽>0 such that ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇); (16)(iii)𝜈-inverse strongly monotone (for short, 𝜈-ism) if there exists 𝜈>0 such that 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (17)

Proposition 8 (see [28]). Let 𝑇∶𝐻→𝐻 be an operator from 𝐻 to itself.(i)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is (1/2)-ism.(ii)If 𝑇 is 𝜈-ism, then for 𝛾>0, 𝛾𝑇 is (𝜈/𝛾)-ism.(iii)𝑇 is averaged if and only if the complement 𝐼−𝑇 is 𝜈-ism for some 𝜈>1/2. Indeed, for 𝛼∈(0,1), 𝑇 is 𝛼-averaged if and only if 𝐼−𝑇 is (1/2𝛼)-ism.

Lemma 9 (see [18]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (18) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that(i) ∑^∞_𝑛=1𝛾_𝑛=∞; (ii) limsup_𝑛→∞𝛿_𝑛≤0𝑜𝑟∑^∞_𝑛=1𝛾_𝑛|𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

In order to solve the equilibrium problem for a bifunction 𝜙∶𝐶×𝐶→ℝ, let us assume that 𝜙 satisfies the following conditions:(A1)𝜙(𝑥,𝑥)=0, for all 𝑥∈𝐶;(A2)𝜙 is monotone; that is, 𝜙(𝑥,𝑦)+𝜙(𝑦,𝑥)≤0,for all 𝑥,𝑦∈𝐶;(A3) For all 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝜙𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝜙𝑥,𝑦; (19)(A4) for each fixed 𝑥∈𝐶, the function 𝑦↦𝜙(𝑥,𝑦) is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 10 (see [28]). Let 𝜙 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that 𝜙+1𝑧,𝑦𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶. (20) Further, if 𝑇_𝑟𝑥={𝑧∈𝐶∶𝜙(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶}, then the following holds:(1)𝑇_𝑟 is single-valued; (2)𝑇_𝑟 is firmly nonexpansive; that is, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩,∀𝑥,𝑦∈𝐻; (21)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝜙); (4)𝐸𝑃(𝜙) is closed and convex.

3. Main Results

We now look at the constrained convex minimization problem: min_𝑥∈𝐶𝑓(𝑥), (22) where 𝐶 is a closed and convex subset of a Hilbert space 𝐻 and 𝑓∶𝐶→ℝ is a real-valued convex function. If 𝑓 is Frchet differentiable, then the gradient-projection algorithm (GPA) generates a sequence {𝑥_𝑛}^∞_𝑛=0 according to the recursive formula 𝑥_𝑛+1=Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛,𝑛≥0, (23) or more generally, 𝑥_𝑛+1=Proj_𝐶𝐼−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (24) where, in both (23) and (24), the initial guess 𝑥₀ is taken from 𝐶 arbitrarily and the parameters 𝛾 or 𝛾_𝑛 are positive real numbers.

As a matter of fact, it is known that, if ∇𝑓 fails to be strongly monotone and is only 1/𝐿-ism; namely, there is constant 𝐿>0 such that (𝑥)𝑦≥1∇𝑓−∇𝑓,𝑥−𝑦𝐿∇𝑓(𝑥)−∇𝑓(𝑦)²,𝑥,𝑦∈𝐶, (25) under some assumption for 𝛾 or 𝛾_𝑛, then algorithms (23) and (24) can still converge in the weak topology.

Now, consider the regularized minimization problem min_𝑥∈𝐶𝑓_𝛼(𝑥)(𝑥)+𝛼∶=𝑓2‖𝑥‖², (26) where 𝛼>0 is the regularization parameter, and again 𝑓 is convex with 1/𝐿-ism continuous gradient ∇𝑓.

It is obvious that there exists a unique point 𝑥_𝛼∈𝐶 such that 𝑥_𝛼 is the unique fixed point of the mapping 𝑉_𝛼∶=Proj_𝐶𝐼−𝛾∇𝑓_𝛼=Proj_𝐶𝐼−𝛾∇𝑓+𝛼𝐼. (27)

We can prove that {𝑥_𝛼}⇀𝑥^∗, where 𝑥^∗ is a solution of the constrained convex minimization problem.

Throughout the rest of this paper, assume that the minimization problem (22) is consistent and let 𝑈 denote its solution set; we always assume that ℎ is a contraction of 𝐶 into 𝐻 with coefficient 𝜌∈(0,1); let {𝐺_𝛽𝑛} be a sequence of mappings defined as Lemma 3 and define a mapping 𝑇_𝑛∶𝐶→𝐶 by 𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛=𝜆_𝑛𝐼+1−𝜆_𝑛𝑇_𝑛𝜆_𝑛∶=2−𝛾𝐿+𝛼_𝑛4 (28) Consider the following mapping 𝑄_𝑛 on 𝐻 defined by 𝑄_𝑛𝑥=𝜃_𝑛ℎ(𝑥)+1−𝜃_𝑛𝑇_𝑛𝐺_𝛽𝑛𝑥,𝑥∈𝐻,𝑛∈𝑁, (29) where 𝜃_𝑛∈(0,1); then by Lemmas 3 and 10𝑄_𝑛𝑥−𝑄_𝑛𝑦≤𝜃_𝑛𝜌𝑥−𝑦+1−𝜃_𝑛𝑥−𝑦=1−1−𝜌𝜃_𝑛. (30)

Since 0<1−(1−𝜌)𝜃_𝑛<1, it follows that 𝑄_𝑛 is a contraction. Therefore, by the Banach contraction principle, 𝑄_𝑛 has a unique fixed point 𝑥^ℎ_𝑛∈𝐻 such that 𝑥^ℎ_𝑛=𝜃_𝑛ℎ𝑥^ℎ_𝑛+1−𝜃_𝑛𝑇_𝑛𝐺_𝛽𝑛𝑥^ℎ_𝑛. (31)

For simplicity, we will write 𝑥_𝑛 for 𝑥^ℎ_𝑛 provided that no confusion occurs. Next, we prove the convergence of {𝑥_𝑛} while we claim the existence of the 𝑞∈𝑈⋂EP(𝜙) which solves the variational inequality (𝐼−ℎ)𝑞,𝑝−𝑞≥0,∀𝑝∈𝑈𝐸𝑃𝜙. (32)

3.1. Convergence of the Implicit Scheme

Proposition 11. If 0<𝛾<2/𝐿, 0<𝛼<2/𝛾−𝐿, ∇𝑓 is (1/𝐿)-ism, for all 𝑥∈𝐶, Proj_𝐶𝐼−𝛾∇𝑓_𝛼=1−𝜇_𝛼𝐼+𝜇_𝛼𝑇_𝛼,Proj_𝐶𝐼−𝛾∇𝑓=1−𝜇𝐼+𝜇𝑇, (33) where 𝜇_𝛼=(2+𝛾(𝐿+𝛼))/4, 𝜇=(2+𝛾L)/4, then 𝑇_𝛼𝑥−𝑇𝑥≤𝛼𝑀(𝑥), (34) where 𝑀(𝑥)=𝛾(5‖𝑥‖+‖𝑇𝑥‖). (35)

Proof. One here Proj_𝐶𝐼−𝛾∇𝑓_𝛼𝑥−Proj_𝐶𝐼−𝛾∇𝑓_𝛼𝑥=𝜇−𝜇_𝛼𝑥+𝜇_𝛼𝑇_𝛼≤𝑥−𝜇𝑇𝑥𝐼−𝛾∇𝑓_𝛼𝑥−𝐼−𝛾∇𝑓𝑥=𝛾∇𝑓_𝛼(𝑥)−∇𝑓(𝑥)=𝛼𝛾‖𝑥‖, (36) then 𝜇_𝛼𝑇_𝛼𝑥−𝜇𝑇𝑥≤𝜇−𝜇_𝛼𝑥‖‖𝑥‖,𝑇‖+𝛼𝛾_𝛼≤(5‖𝑥‖+‖𝑥−𝑇𝑥𝛼𝛾𝑇𝑥‖)()(𝑥),2+𝛾𝐿+𝛼≤𝛼𝑀 (37) where, 𝑀(𝑥)=𝛾(5‖𝑥‖+‖𝑇𝑥‖).

Theorem 12. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and ℎ∶𝐶→𝐻 a contraction with 𝜌∈(0,1), 𝑈∩𝐸𝑃(𝜙)≠∅, and 𝜙 a bifunction from 𝐶×𝐶 into ℝ satisfying (𝐴1), (𝐴2), (𝐴3), and (𝐴4). Let {𝑥_𝑛} be sequence generated by 𝜙𝑢_𝑛+1,𝑦𝛽_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛=𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛, (38) where, 𝑃_𝑐[𝐼−𝛾∇𝑓_𝛼𝑛]=𝜆_𝑛𝐼+(1−𝜆_𝑛)𝑇_𝑛, 0<𝛾<2/𝐿 and 𝜆_𝑛=(2−𝛾(𝐿+𝛼_𝑛))/4.(i){𝛽_𝑛}⊂(0,+∞)liminf_𝑛→∞𝛽_𝑛>0;(ii){𝜃_𝑛}⊂(0,1), lim_𝑛→∞𝜃_𝑛=0;(iii)𝛼_𝑛=𝑜(𝜃_𝑛).
Then, {𝑥_𝑛} converges strongly to a point 𝑞∈𝑈∩EP(𝜙) which solves the variational inequality (32).

Proof. Pick any 𝑝∈𝑈∩EP(𝜙), 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛, 𝑝=𝐺_𝛽𝑛𝑝; then we have 𝑢_𝑛−𝑝≤𝑥_𝑛−𝑝, (39) (noting 𝑇𝑝=𝑝) 𝑥_𝑛=𝜃−𝑝_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛=𝜃−𝑝_𝑛ℎ𝑥_𝑛−ℎ𝑝+𝜃_𝑛ℎ𝑝+−𝑝1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑝≤𝜃_𝑛𝜌𝑥_𝑛−𝑝+𝜃_𝑛ℎ𝑝+−𝑝1−𝜃_𝑛𝑢_𝑛−𝑝+𝑇_𝑛𝑝𝑝≤𝜃−𝑇1−1−𝜌_𝑛𝑥_𝑛−𝑝+𝜃_𝑛ℎ𝑝−𝑝+𝛼_𝑛𝑀₁; (40) hence, 𝑥_𝑛≤1−𝑝ℎ𝑝1−𝜌−𝑝+𝑀₁. (41) So, {𝑥_𝑛} is bounded. Next, we claim that ‖𝑥_𝑛−𝑢_𝑛‖→0.
Take 𝑝∈𝑈∩EP(𝜙); by Lemma 3, we have 𝑢_𝑛−𝑝²=𝑇_𝛽𝑛𝑥_𝑛−𝑇_𝛽𝑛𝑝²≤𝑥_𝑛−𝑝,𝑢_𝑛=1−𝑝2𝑥_𝑛−𝑝²+𝑢_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛². (42) It follows that 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²,𝑥_𝑛−𝑝²=𝜃_𝑛ℎ(𝑥_𝑛)−𝑝+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑝²≤1−𝜃_𝑛²𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑝+𝑇_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛ℎ𝑥_𝑛−𝑝,𝑥_𝑛≤−𝑝1−𝜃_𝑛²×𝑢_𝑛−𝑝²+2𝑢_𝑛×𝑇−𝑝_𝑛𝑝−𝑇𝑝+𝑇_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛ℎ𝑥_𝑛−𝑝,𝑥_𝑛≤−𝑝1−𝜃_𝑛²×𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑢_𝑛−𝑝𝑇_𝑛+𝑇𝑝−𝑇𝑝_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛𝜌𝑥_𝑛−𝑝²+2𝜃_𝑛ℎ𝑝−𝑝,𝑥_𝑛−𝑝. (43) So, ‖𝑥_𝑛−𝑢_𝑛‖→0. Next, we show that ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0, 𝑥_𝑛−𝑇_𝑛𝑥_𝑛=𝑥_𝑛−𝑇_𝑛𝑢_𝑛+𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑥_𝑛≤𝜃_𝑛ℎ𝑥_𝑛−𝑇_𝑛𝑢_𝑛+𝑢_𝑛−𝑥_𝑛𝑢⟶0._𝑛−𝑇_𝑛𝑢_𝑛=𝑢_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑢_𝑛⟶0. (44) Observe that 𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛=𝜆_𝑛𝑢_𝑛+1−𝜆_𝑛𝑇_𝑛𝑢_𝑛−𝑢_𝑛=1−𝜆_𝑛𝑇_𝑛𝑢_𝑛−𝑢_𝑛≤𝑇_𝑛𝑢_𝑛−𝑢_𝑛. (45) Hence, 𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑢_𝑛≤𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛+𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛≤𝛾𝛼_𝑛𝑢_𝑛+𝑇_𝑛𝑢_𝑛−𝑢_𝑛⟶0. (46) So, lim_𝑛→∞𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑢_𝑛=0. (47)
Since {𝑢_𝑛} is bounded, there exists {𝑢_𝑛𝑖} such that {𝑢_𝑛𝑖}⇀𝑞. Since 𝐶 is closed and convex, 𝐶 is weakly closed. So, we have 𝑞∈𝐶. Let us show that 𝑞∈𝑈. Assume that 𝑞∈𝑈. Since 𝑢_𝑛𝑖⇀𝑞 and 𝑞≠𝑇𝑞, it follows from the Opial’s condition that liminf_𝑛→∞𝑢_𝑛𝑖−𝑞<liminf_𝑛→∞𝑢_𝑛𝑖−𝑇𝑞≤liminf_𝑛→∞𝑢_𝑛𝑖−𝑇𝑢_𝑛𝑖+𝑇𝑢_𝑛𝑖−𝑇𝑞≤liminf_𝑛→∞𝑢_𝑛𝑖−𝑞. (48) This is a contradiction. So, we get 𝑞∈𝑈.
Next, we show that 𝑞∈EP(𝜙). Since 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛, for any 𝑦∈𝐶, we obtain 𝜙𝑢_𝑛+1,𝑦𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0. (49) From (𝐴2), we have 1𝛽_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝜙𝑦,𝑢_𝑛. (50) Replacing 𝑛 with 𝑛_𝑖, we have 𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝛽_𝑛𝑖≥𝜙𝑦,𝑢_𝑛𝑖. (51) Since (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝛽_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑞, it follows from (𝐴₄) that 0≥𝜙(𝑦,𝑞), for all 𝑦∈𝐶. Let 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑞 for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑡∈𝐶 and hence 𝜙(𝑧_𝑡,𝑞)≤0. Thus, from (𝐴1) and (𝐴4) we have 0=𝜙𝑧_𝑡,𝑧_𝑡≤𝑡𝜙𝑧_𝑡,𝑦+(1−𝑡)𝜙𝑧_𝑡,𝑞≤𝑡𝜙𝑧_𝑡,𝑦 (52) and hence 0≤𝜙(𝑧_𝑡,𝑦). From (𝐴₃), we have 0≤𝜙(𝑞,𝑦) for all 𝑦∈𝐶 and hence 𝑞∈EP(𝜙). Therefore, 𝑞∈𝑈∩EP(𝜙).
On the other hand, 𝑥_𝑛−𝑞=𝜃_𝑛ℎ𝑥_𝑛−𝑞+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛,𝑥−𝑞_𝑛−𝑞²=𝜃_𝑛ℎ𝑥_𝑛−𝑞,𝑥_𝑛−𝑞+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑞,𝑥_𝑛−𝑞=𝜃_𝑛⟨ℎ𝑥_𝑛−ℎ𝑞,𝑥_𝑛+−𝑞⟩1−𝜃_𝑛⟨𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑞,𝑥_𝑛+−𝑞⟩1−𝜃_𝑛⟨𝑇_𝑛𝑞−𝑇𝑞,𝑥_𝑛−𝑞⟩+𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛−𝑞⟩≤𝜃_𝑛𝜌𝑥_𝑛−𝑞²+1−𝜃_𝑛𝑥_𝑛−𝑞²+𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛−𝑞⟩+1−𝜃_𝑛𝛼_𝑛𝑀₂𝑥_𝑛−𝑞; (53)(54) then 𝑥_𝑛𝑗→𝑞 if 𝑥_𝑛𝑗⇀𝑞.
Next, we prove that 𝑞 solves the VI (problem): (𝐼−ℎ)𝑥_𝑛1=−𝜃_𝑛𝐼−𝑇_𝑛𝑄_𝛽𝑛𝑥_𝑛+𝐼−𝑇_𝑛𝑄_𝛽𝑛𝑥_𝑛. (55) Note that ⟨(𝐼−ℎ)𝑞,𝑞−𝑝⟩=lim_𝑗→∞⟨(𝐼−ℎ)𝑥_𝑛𝑗,𝑥_𝑛𝑗−𝑝⟩=lim_𝑗→∞−1𝜃_𝑛𝑗𝐼−𝑇_𝑛𝑄_𝛽𝑛𝑗𝑥_𝑛𝑗−𝐼−𝑇_𝑛𝑗𝑄_𝛽𝑛𝑗𝑝,𝑥_𝑛𝑗−1−𝑝𝜃_𝑛𝑗𝐼−𝑇_𝑛𝑗𝑄_𝛽𝑛𝑗𝑝,𝑥_𝑛𝑗+−𝑝𝐼−𝑇_𝑛𝑗𝑄_𝛽𝑛𝑗𝑥_𝑛𝑗,𝑥_𝑛𝑗−𝑝≤0. (56)

3.2. Convergence of the Explicit Scheme

Theorem 13. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻,  ℎ∶𝐶→𝐻 a contraction with 𝜌∈(0,1), 𝑈∩𝐸𝑃(𝜙)≠∅, and 𝜙 a bifunction from 𝐶×𝐶 into ℝ satisfying (𝐴1), (𝐴2), (𝐴3), and (𝐴4). Let {𝑥_𝑛} be sequence generated by 𝑥₁∈𝐻 and 𝜙𝑢_𝑛+1,𝑦𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛, (57) where 𝑃_𝑐[𝐼−𝛾∇𝑓_𝛼𝑛]=𝜆_𝑛𝐼+(1−𝜆_𝑛)𝑇_𝑛, 0<𝛾<2/𝐿 and 𝜆_𝑛=(2−𝛾(𝐿+𝛼_𝑛))/4, 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛. Let {𝛼_𝑛} and {𝜃_𝑛} satisfy the following conditions:(i){𝜃_𝑛}⊂(0,1), lim_𝑛→∞𝜃_𝑛=0, ∑^∞_𝑛=1𝜃_𝑛=∞, ∑^∞_𝑛=1|𝜃_𝑛+1−𝜃_𝑛|<∞; (ii)𝛼_𝑛=𝑜(𝜃_𝑛)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞; (iii){𝛽_𝑛}∈(0,+∞), lim_𝑛→∞𝛽_𝑛>0 and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞.
Then, {𝑥_𝑛} and {𝑢_𝑛} converge strongly to a point 𝑞∈𝑈∩𝐸𝑃(𝜙) which solves the variational inequality (32).

Proof. First we prove that {𝑥_𝑛} is bounded.
Taking any 𝑝∈𝑈∩EP(𝜙), we have 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛,𝑝=𝐺_𝛽𝑛𝑝. (58) So, ‖𝑢_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, 𝑥_𝑛+1=𝜃−𝑝_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛=𝜃−𝑝_𝑛ℎ𝑥_𝑛−ℎ𝑝+𝜃_𝑛ℎ𝑝+−𝑝1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑝≤𝜃_𝑛𝜌𝑥_𝑛−𝑝+𝜃_𝑛ℎ𝑝+−𝑝1−𝜃_𝑛𝑥_𝑛−𝑝+𝑇_𝑛𝑝𝑝≤𝜃−𝑇1−1−𝜌_𝑛𝑥_𝑛−𝑝+𝜃_𝑛ℎ𝑝−𝑝+𝛼_𝑛𝑀₃≤max𝑥_𝑛;1−𝑝ℎ𝑝1−𝜌−𝑝+𝑀₃. (59) So, {𝑥_𝑛} is bounded.
Next we prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0, 𝑥_𝑛+1−𝑥_𝑛=𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1+1−𝜃_𝑛−1𝑇_𝑛−1𝑢_𝑛−1=𝜃_𝑛ℎ𝑥_𝑛−ℎ𝑥_𝑛−1+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑢_𝑛−1+𝜃_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1−𝑇_𝑛𝑢_𝑛−1+1−𝜃_𝑛−1𝑇_𝑛𝑢_𝑛−1−𝑇_𝑛−1𝑢_𝑛−1≤𝜃_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝜃_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1+𝑇_𝑛𝑢_𝑛−1+1−𝜃_𝑛𝑢_𝑛−𝑢_𝑛−1+1−𝜃_𝑛−1𝑇_𝑛𝑢_𝑛−1−𝑇_𝑛−1𝑢_𝑛−1. (60) From 𝑢_𝑛+1=𝐺_𝛽𝑛+1𝑥_𝑛+1 and 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛, we note that 𝜙𝑢_𝑛+1+1,𝑦𝛽_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1𝜙𝑢≥0,∀𝑦∈𝐶,_𝑛+1,𝑦𝛽_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (61)(62) Putting 𝑦=𝑢_𝑛 in (61) and 𝑦=𝑢_𝑛+1 in (62), we have 𝜙𝑢_𝑛+1,𝑢_𝑛+1𝛽_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1𝜙𝑢≥0,∀𝑦∈𝐶,_𝑛,𝑢_𝑛+1+1𝛽_𝑛⟨𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶. (63)
So, from (A2), we have 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝛽_𝑛−𝑢_𝑛+1−𝑥_𝑛+1𝛽_𝑛+1≥0, (64) and hence 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝛽_𝑛𝛽_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≥0. (65)
Since lim_𝑛→∞𝛽_𝑛>0, without loss of generality, let us assume that there exists a real number 𝑎 such that 𝛽_𝑛>𝑎>0 for all 𝑛∈ℕ. Thus, we have 𝑢_𝑛+1−𝑢_𝑛²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+𝛽1−^𝑛𝛽_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≤𝑢_𝑛+1−𝑢_𝑛×𝑥_𝑛+1−𝑥_𝑛+𝛽1−^𝑛𝛽_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1; (66) thus, 𝑢_𝑛+1−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+1𝑎𝛽_𝑛+1−𝛽_𝑛𝑀₄, (67) where 𝑀₄=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈ℕ}, 𝑇_𝑛𝑢_𝑛−1−𝑇_𝑛−1𝑢_𝑛−1=4𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛−2−𝛾𝐿+𝛼_𝑛𝐼2+𝛾𝐿+𝛼_𝑛𝑢_𝑛−1−4𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛−1−2−𝛾𝐿+𝛼_𝑛−1𝐼2+𝛾𝐿+𝛼_𝑛−1𝑢_𝑛−1≤4𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛2+𝛾𝐿+𝛼_𝑛𝑢_𝑛−1−4𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛−12+𝛾𝐿+𝛼_𝑛−1𝑢_𝑛−1+−2−𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛𝑢_𝑛−1+2−𝛾𝐿+𝛼_𝑛−12+𝛾𝐿+𝛼_𝑛−1𝑢_𝑛−1=42+𝛾𝐿+𝛼_𝑛−1𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−1−42+𝛾𝐿+𝛼_𝑛𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛−1𝑢_𝑛−1×[2+𝛾(𝐿+𝛼_𝑛)][2+𝛾(𝐿+𝛼_𝑛−1)]⁻¹+4𝛾𝛼_𝑛−𝛼_𝑛−1𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1=4𝛾𝛼_𝑛−1−𝛼_𝑛𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1+42+𝛾𝐿+𝛼_𝑛×𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−1−𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛−1𝑢_𝑛−1×[2+𝛾(𝐿+𝛼_𝑛)][2+𝛾(𝐿+𝛼_𝑛−1)]⁻¹+4𝛾𝛼_𝑛−𝛼_𝑛−1𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1≤4𝛾𝛼_𝑛−1−𝛼_𝑛𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1+4𝛾𝛼_𝑛−1−𝛼_𝑛2+𝛾𝐿+𝛼_𝑛𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1+4𝛾𝛼_𝑛−𝛼_𝑛−1𝑢_𝑛−12+𝛾𝐿+𝛼_𝑛2+𝛾𝐿+𝛼_𝑛−1≤𝛼_𝑛−1−𝛼_𝑛𝛾𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−1+5𝛾𝑢_𝑛−1≤𝑀₅𝛼_𝑛−1−𝛼_𝑛. (68)
So 𝑥_𝑛+1−𝑥_𝑛≤1−1−𝜌𝜃_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₆𝜃_𝑛−𝜃_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1. (69) So lim𝑥_𝑛+1−𝑥_𝑛𝑢=0,lim_𝑛+1−𝑢_𝑛=0. (70) Next, we prove that ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0, 𝑥_𝑛+1−𝑝²=𝜃_𝑛ℎ𝑥_𝑛−𝑝+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛−𝑝²≤1−𝜃_𝑛²𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑝+𝑇_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛ℎ𝑥_𝑛−𝑝,𝑥_𝑛≤−𝑝1−𝜃_𝑛²×𝑢_𝑛−𝑝²+2𝑢_𝑛×𝑇−𝑝_𝑛𝑝−𝑇𝑝+𝑇_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛ℎ𝑥_𝑛−𝑝,𝑥_𝑛≤−𝑝1−𝜃_𝑛²×𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑢_𝑛−𝑝𝑇_𝑛𝑝−𝑇𝑝+𝑇_𝑛𝑝−𝑇𝑝²+2𝜃_𝑛𝜌𝑥_𝑛−𝑝²+2𝜃_𝑛ℎ𝑝−𝑝,𝑥_𝑛−𝑝. (71) So, ‖𝑥_𝑛−𝑢_𝑛‖→0, 𝑥_𝑛−𝑇_𝑛𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑛𝑢_𝑛+𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑥_𝑛≤𝜃_𝑛ℎ𝑥_𝑛−𝑇_𝑛𝑢_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑢_𝑛−𝑥_𝑛𝑢⟶0,_𝑛−𝑇_𝑛𝑢_𝑛=𝑢_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑢_𝑛⟶0. (72) Observe that 𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛=𝜆_𝑛𝑢_𝑛+1−𝜆_𝑛𝑇_𝑛𝑢_𝑛−𝑢_𝑛=1−𝜆_𝑛𝑇_𝑛𝑢_𝑛−𝑢_𝑛≤𝑇_𝑛𝑢_𝑛−𝑢_𝑛. (73) Hence, 𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑢_𝑛≤𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛+𝑃_𝐶𝐼−𝛾∇𝑓_𝛼𝑛𝑢_𝑛−𝑢_𝑛≤𝛾𝛼_𝑛𝑢_𝑛+𝑇_𝑛𝑢_𝑛−𝑢_𝑛⟶0. (74) So, lim_𝑛→∞𝑃_𝐶𝐼−𝛾∇𝑓𝑢_𝑛−𝑢_𝑛=0. (75)
Since {𝑢_𝑛} is bounded, there exists {𝑢_𝑛𝑖} such that {𝑢_𝑛𝑖}⇀𝑞. Since 𝐶 is closed and convex, 𝐶 is weakly closed. So, we have 𝑞∈𝐶. Let us show that 𝑞∈𝑈. Assume that 𝑞∈𝑈. Since 𝑢_𝑛𝑖⇀𝑞 and 𝑞≠𝑇𝑞, it follows from the Opial’s condition that liminf_𝑛→∞𝑢_𝑛𝑖−𝑞<liminf_𝑛→∞𝑢_𝑛𝑖−𝑇𝑞≤liminf_𝑛→∞𝑢_𝑛𝑖−𝑇𝑢_𝑛𝑖+𝑇𝑢_𝑛𝑖−𝑇𝑞≤liminf_𝑛→∞𝑢_𝑛𝑖−𝑞. (76) This is a contradiction. So, we get 𝑞∈𝑈.
Next We show that 𝑞∈EP(𝜙). Since 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛, for any 𝑦∈𝐶, we obtain 𝜙𝑢_𝑛+1,𝑦𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0. (77) From (𝐴2), we have 1𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝜙𝑦,𝑢_𝑛. (78) Replacing 𝑛 with 𝑛_𝑖, we have 𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝛽_𝑛𝑖≥𝜙𝑦,𝑢_𝑛𝑖. (79) Since (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝛽_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑞, it follows from (𝐴₄) that 0≥𝜙(𝑦,𝑞), for all 𝑦∈𝐶. Let 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑞 for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑡∈𝐶 and hence 𝜙(𝑧_𝑡,𝑞)≤0. Thus, from (𝐴1) and (𝐴4), we have 0=𝜙𝑧_𝑡,𝑧_𝑡≤𝑡𝜙𝑧_𝑡,𝑦+(1−𝑡)𝜙𝑧_𝑡,𝑞≤𝑡𝜙𝑧_𝑡,𝑦 (80) and hence 0≤𝜙(𝑧_𝑡,𝑦). From (𝐴₃), we have 0≤𝜙(𝑞,𝑦) for all 𝑦∈𝐶 and hence 𝑞∈EP(𝜙). Therefore, 𝑞∈𝑈∩EP(𝜙).
We assume that; 𝑥_𝑛𝑖⇀𝑥, then 𝑥∈EP(𝜙)∩𝑈, limsup_𝑛→∞⟨(𝐼−ℎ)𝑞,𝑞−𝑥_𝑛⟩=⟨(𝐼−ℎ)𝑞,𝑞−𝑥⟩≤0. (81) Finally, we prove that 𝑥_𝑛→𝑞, 𝑥_𝑛+1−𝑞²=𝜃_𝑛(ℎ(𝑥_𝑛)−𝑞)+(1−𝜃_𝑛)(𝑇_𝑛𝑢_𝑛−𝑇𝑞)²=𝜃_𝑛ℎ𝑥_𝑛−ℎ𝑞+(1−𝜃_𝑛)(𝑇_𝑛𝑢_𝑛−𝑇𝑞)+𝜃_𝑛(ℎ(𝑞)−𝑞)²≤𝜃_𝑛(ℎ(𝑥_𝑛)−ℎ(𝑞))+(1−𝜃_𝑛)(𝑇_𝑛𝑢_𝑛−𝑇𝑞)²+2𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛+1−𝑞⟩≤𝜃_𝑛(ℎ(𝑥_𝑛)−ℎ(𝑞))²+1−𝜃_𝑛×(𝑇_𝑛𝑢_𝑛−𝑇𝑞)²+2𝜃_𝑛ℎ𝑞−𝑞,𝑥_𝑛+1−𝑞≤𝜃_𝑛𝜌²𝑥_𝑛−𝑞²+1−𝜃_𝑛×(𝑇_𝑛𝑢_𝑛−𝑇_𝑛𝑞+𝑇_𝑛𝑞−𝑇𝑞)²+2𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛+1−𝑞⟩≤𝜃_𝑛𝜌²𝑥_𝑛−𝑞²+1−𝜃_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑀₇²+2𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛+1≤𝜃−𝑞⟩1−1−𝜌_𝑛𝑥_𝑛−𝑞²+1−𝜃_𝑛2𝛼_𝑛𝑀₇𝑥_𝑛−𝑞+𝛼²_𝑛𝑀²₇+2𝜃_𝑛⟨ℎ𝑞−𝑞,𝑥_𝑛+1𝑥−𝑞⟩_𝑛+1−𝑞²=1−𝛽_𝑛𝑥_𝑛−𝑞²+𝛽_𝑛𝛿_𝑛.𝛽_𝑛=1−𝜌𝜃_𝑛,𝛿_𝑛=1𝛼1−𝜌𝜃^𝑛_𝑛1−𝜃_𝑛2𝑀₇𝑥_𝑛−𝑞+1−𝜃_𝑛𝑀²₇𝛼²_𝑛𝜃_𝑛+2ℎ𝑞−𝑞,𝑥_𝑛+1−𝑞, (82) by Lemma 9 and lim_𝑛→∞𝛽_𝑛=0, ∑^∞_𝑛=0𝛽_𝑛=∞; limsup_𝑛→∞𝛿_𝑛≤0, then 𝑥_𝑛→𝑞.

4. Application of the Iterative Method

Next, we give an application of Theorem 13 to the split feasibility problem (say SFP, for short) which was introduced by Censor and Elfving [27], find𝑥∈𝐶,suchthat𝐴𝑥∈𝑄, (83) where 𝐶 and 𝑄 are nonempty closed convex subsets of Hilbert space 𝐻₁ and 𝐻₂, respectively. 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator.

It is clear that 𝑥^∗ is a solution to the split feasibility problem (83) if and only if 𝑥^∗∈𝐶 and 𝐴𝑥^∗−𝑃_𝑄𝐴𝑥^∗=0.

We define the proximity function 𝑓 by 𝑓(𝑥)=12𝐴𝑥−𝑃_𝑄𝐴𝑥², (84) and consider the convex optimization problem min_𝑥∈𝐶𝑓(𝑥)=min_𝑥∈𝐶12𝐴𝑥−𝑃_𝑄𝐴𝑥². (85)

Then, 𝑥^∗ solves the split feasibility problem (83) if and only if 𝑥^∗ solves the minimization (85) with the minimization equal to 0. Byrne [28] introduced the so-called 𝐶𝑄 algorithm to solve the (SFP), 𝑥_𝑛+1=𝑃_𝐶𝐼−𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (86) where 0<𝜇<2/‖𝐴^∗𝐴‖=2/‖𝐴‖².

He obtained that the sequence {𝑥_𝑛} generated by (86) converges weakly to a solution of the (SFP).

Now we consider the regularization technique; let 𝑓_𝛼(𝑥)=12𝐴𝑥−𝑃_𝑄𝐴𝑥²+𝛼2‖𝑥‖². (87) Applying Theorem 13, we obtain the following result.

Theorem 14. Assume that the split problem (83) is consistent. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻,  ℎ∶𝐶→𝐻 a contraction with 𝜌∈(0,1), 𝑈∩EP(𝜙)≠∅, and 𝜙 be a bifunction from 𝐶×𝐶 into ℝ satisfying (𝐴1), (𝐴2), (𝐴3), and (𝐴4). Let {𝑥_𝑛} be sequence generated by 𝑥₁∈𝐻 and 𝜙𝑢_𝑛+1,𝑦𝛽_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑇_𝑛𝑢_𝑛, (88) where 𝑃_𝐶𝐼−𝜇𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼=𝜆_𝑛𝐼+1−𝜆_𝑛𝑇_𝑛,𝜆_𝑛=2−𝜇‖𝐴‖²+𝛼_𝑛4, (89) where 𝑢_𝑛=𝐺_𝛽𝑛𝑥_𝑛; let {𝛼_𝑛}, {𝜃_𝑛} satisfy the following conditions:(i){𝜃_𝑛}⊂(0,1), lim_𝑛→∞𝜃_𝑛=0,∑^∞_𝑛=1𝜃_𝑛=∞,∑^∞_𝑛=1|𝜃_𝑛+1−𝜃_𝑛|<∞,0<𝜇<2/‖𝐴^∗𝐴‖=2/‖𝐴‖²;(ii)𝛼_𝑛=𝑜(𝜃_𝑛)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞; (iii){𝛽_𝑛}∈(0,+∞),lim_𝑛→∞𝛽_𝑛>0 and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞.
Then, {𝑥_𝑛} and {𝑢_𝑛} converge strongly to a point 𝑞∈𝑈∩𝐸𝑃(𝜙) which solves the variational inequality (32).

Proof. By the definition of the function 𝑓_𝛼, we have ∇𝑓_𝛼(𝑥)=𝐴^∗𝐼−𝑃_𝑄𝐴𝑥+𝛼𝑥, (90) and ∇𝑓_𝛼 is 1/(‖𝐴‖²+𝛼)-ism, ∇𝑓_𝛼(𝑥)−∇𝑓_𝛼𝑦≤‖A‖²+𝛼; (91) then, due to Theorem 13, we have the conclusion immediately.

Acknowledgments

The author wishes to thank the referees for their helpful comments, which notably improved the presentation of this paper. This work was supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China, no. 3122013 K004).