Hybrid Steepest Descent Viscosity Method for Triple Hierarchical Variational Inequalities

Abstract

We consider a triple hierarchical variational inequality problem (in short, THVIP). By combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP.

1. Introduction and Formulations

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻 and let Γ∶𝐶→𝐻 be a nonlinear mapping. The variational inequality problem (for short, VIP) is to find 𝑥^∗∈𝐶 such that Γ𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶.(1.1) The inequality (1.1) is called a variational inequality (in short, VI). If the mapping Γ is a monotone operator, then the inequality (1.1) is called a monotone variational inequality. The theory of variational inequalities is well established in the literature because of its applications in science, engineering, social sciences, and so forth. For further detail on variational inequalities and their applications, we refer to [1–10] and the references therein.

It is well known that the VI (1.1) is equivalent to the fixed point equation 𝑥^∗=𝑃_𝐶(𝐼−𝜆Γ)𝑥^∗,(1.2) where 𝜆>0 and 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶 which assigns to each 𝑥∈𝐻 the only point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦.(1.3) It is known that the fixed point methods can be implemented to find a solution of the VI (1.1) provided Γ satisfies some conditions and 𝜆>0 is chosen appropriately. For instance, if Γ is Lipschitzian and strongly monotone (i.e., ⟨Γ𝑥−Γ𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,forall𝑥,𝑦∈𝐶 for some 𝜂>0) and 𝜆>0 is small enough, then the mapping determined by the right-hand side of (1.2) is a contraction. Hence, the Banach contraction principle guarantees that the sequence {𝑥_𝑛} of Picard iterates, given by 𝑥_𝑛=𝑃_𝐶(𝐼−𝜆Γ)𝑥_𝑛−1(𝑛≥1), converges strongly to a unique solution of the VI (1.1).

Furthermore, it is also known that if Γ is inverse strongly monotone (i.e., there is a constant 𝛼>0 such that ⟨Γ𝑥−Γ𝑦,𝑥−𝑦⟩≥𝛼‖Γ𝑥−Γ𝑦‖²,forall𝑥,𝑦∈𝐶), then the mapping 𝑃_𝐶(𝐼−𝜆Γ) is an averaged mapping (namely, there are 𝛽∈(0,1) and a nonexpansive mapping 𝑇 such that 𝑃_𝐶(𝐼−𝜆Γ)=(1−𝛽)𝐼+𝛽𝑇), then the sequence of Picard iterates, {(𝑃_𝐶(𝐼−𝜆Γ))^𝑛𝑥₀}, converges weakly to a solution of the VI (1.1) (if such a solution exists).

In the last decade, the variational inequality problem is considered over the set of fixed points of a nonexpansive mapping; see, for example, [11–15] and the reference therein. In particular, Moudafi and Maingé [12] and Xu [14] considered the following VIP over the set Fix(𝑇) of fixed points of a nonexpansive mapping 𝑇∶𝐶→𝐶 (i.e., 𝐶=Fix(𝑇)) with Γ=𝐼−𝑉, where 𝑉 is another nonexpansive self-mapping on 𝐶: find 𝑥^∗∈Fix(𝑇) such that (𝐼−𝑉)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix(𝑇),(1.4) where we assume that Fix(𝑇)≠∅. It is called hierarchical variational inequality problem (in short, HVIP). The HVIP (1.4) is equivalent to the following fixed point problem: find𝑥^∗∈𝐶suchthat𝑥^∗=𝑃_Fix(𝑇)∘𝑉𝑥^∗.(1.5) Let 𝑆 denote the solution set of the HVIP (1.4). It has been shown in [12] that the HVIP (1.4) contains the HVIP considered in [15], monotone inclusion problem, convex programming problem, minimization problem over a set of fixed points, and so forth, as special cases; see, for example, [12, 14] and the references therein. In the recent past, several kinds of approximation methods for computing the approximate solutions of HVIP are proposed; see, for example, [11–15] and the reference therein. Yamada [15] considered the so-called hybrid steepest descent method for solving the VIP over the set of fixed points of a nonexpansive mapping. Moudafi [11] proposed the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping which is also a solution of a variational inequality problem. Subsequently, this method was developed by Xu [13]. Moudafi and Maingé [12] and Xu [14] further studied the viscosity method for HVIP.

Very recently, Iiduka [16, 17] considered a variational inequality problem with variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with hierarchical constrained optimization problems or hierarchical fixed point problem, it is referred as triple hierarchical-constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms is also studied in [16, 17]. Since the original problem is a variational inequality problem, in this paper, we call it the triple hierarchical variational inequality problem (THVIP).

Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitzian with constant 𝐿≥0 and let 𝑇,𝑉∶𝐶→𝐶 be nonexpansive mappings with Fix(𝑇)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). We consider the following triple hierarchical variational inequality problem (for short, THVIP): find 𝑥^∗∈𝑆 such that 𝜇𝐹−𝛾𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆,(1.6) where 𝑆 denotes the solution set of the hierarchical variational inequality problem (1.4) which is assumed to be nonempty.

Recall the function 𝑔∶𝐶→ℝ is said to be convex if for all 𝑥,𝑦∈𝐶 and for all 𝜆∈[0,1],𝑔(𝜆𝑥+(1−𝜆)𝑦)≤𝜆𝑔(𝑥)+(1−𝜆)𝑔(𝑦). It is said to be 𝛼-strongly convex if there exists 𝛼>0 such that for all 𝑥,𝑦∈𝐾 and for all 𝜆∈[0,1], 𝑔(𝜆𝑥+(1−𝜆)𝑦)≤𝜆𝑔(𝑥)+(1−𝜆)𝑔(𝑦)−(1/2)𝛼𝜆(1−𝜆)‖𝑥−𝑦‖². It is easy to see that if 𝑔 is Fréchet differential and 𝛼-strongly convex, then the gradient ∇𝑔 is 𝛼-strongly monotone.

Now, we illustrate the triple hierarchical variational inequality problem (for short, THVIP) by an example which is closely related to [17, Example 3.1].

Example 1.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitz continuous with constant 𝐿>0. Suppose that 𝑔₀∶𝐻→ℝ is a convex function with a 1/𝛼₀-Lipschitz continuous gradient, 𝑔₁∶𝐻→ℝ is a convex function with a 1/𝛼₁-Lipschitz continuous gradient, and 𝑔₂∶𝐻→ℝ is an 𝛼-strongly convex function with an 𝛼₂-Lipschitz continuous gradient. Define 𝑇∶=𝑃_𝐶(𝐼−𝜆∇𝑔₀)(𝜆∈(0,2𝛼₀]), 𝑉∶=𝑃_𝐶(𝐼−𝜆∇𝑔₁)(𝜆∈(0,2𝛼₁]) and 𝐹∶=∇𝑔₂. Then 𝑇,𝑉∶𝐶→𝐶 are nonexpansive mappings with Fix(𝑇)=Argmin_𝑧∈𝐶𝑔₀(𝑧) and Fix(𝑉)=Argmin_𝑧∈𝐶𝑔₁(𝑧), and 𝐹 is 𝜅-Lipschitzian and 𝜂-strongly monotone with 𝜅=1/𝛼₂ and 𝜂=𝛼. Assume that Argmin_𝑧∈𝐶𝑔₀(𝑧)∩Argmin_𝑧∈𝐶𝑔₁(𝑧)≠∅. Then for the solution set 𝑆 of the hierarchical variational inequality problem (for short, HVIP), we have ∅≠Argmin_𝑧∈𝐶𝑔₀(𝑧)∩Argmin_𝑧∈𝐶𝑔₁(𝑧)(𝑇)(𝑉)⊂𝑧=Fix∩Fix_∗∈Fix(𝑇)∶(𝐼−𝑉)𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇)=𝑆.(1.7) When 0<𝜇<2𝛼𝛼²₂ and 0≤𝛾𝐿<𝜏, where 𝜏=1−1−𝜇(2𝛼−𝜇/𝛼²₂), we have 0<𝜇<2𝜂𝜅²,0≤𝛾𝐿<𝜏,(1.8) where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). In particular, when 𝜇=𝜂/𝜅²=𝛼𝛼²₂, we have 0≤𝛾𝐿<𝜏=1−𝜇1−𝜇2𝛼−𝛼²₂=1−1−𝛼²𝛼²₂.(1.9) In this case, when 𝛾=(1/2𝐿)𝛼²𝛼²₂ (obviously,1−𝛼²𝛼²₂<1−(1/2)𝛼²𝛼²₂), the following triple hierarchical variational inequality problem (for short, THVIP): find 𝑥^∗∈𝑆 such that 𝜇𝐹−𝛾𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆(1.10) reduces to the following THVIP: find 𝑥^∗∈𝑆 such that ∇𝑔₂−𝛼𝑓𝑥2𝐿^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆.(1.11)

In this paper, by combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP. The results of this paper extend and generalize the results given in [12, 14] and several others given in the literature.

2. Preliminaries

Throughout the paper, unless other specified, we assume that 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻. We use 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥 to denote strong and weak convergence to 𝑥 of the sequence {𝑥_𝑛}, respectively.

Recall that a mapping 𝑓∶𝐶→𝐻 is called 𝐿-Lipschitzian on 𝐶 if there exists 𝐿∈[0,∞) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝐿‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶. In particular, if 𝐿∈[0,1) then 𝑓 is called a contraction on 𝐶; if 𝐿=1 then 𝑓 is called a nonexpansive mapping on 𝐶.

We present some basic facts and results which will be used in the sequel.

Lemma 2.1 (see [18]). Let 𝐻 be a real Hilbert space. Then, ‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²(2.1) for all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1].

The following lemma can be easily proved, and therefore, we omit the proof.

Lemma 2.2. Let 𝑓∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping with constant 𝐿∈[0,∞), and let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Then, for 0≤𝛾𝐿<𝜇𝜂, 𝑥−𝑦,𝜇𝐹−𝛾𝑓𝑥−𝜇𝐹−𝛾𝑓𝑦≥𝜇𝜂−𝛾𝐿‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(2.2) That is, 𝜇𝐹−𝛾𝑓 is strongly monotone with constant 𝜇𝜂−𝛾𝐿>0.

Lemma 2.3 (see [18, Demiclosedness Principle]). Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} weakly converges to 𝑥 in 𝐶 and if {(𝐼−𝑇)𝑥_𝑛} strongly converges to 𝑦, then (𝐼−𝑇)𝑥=𝑦; in particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

In the following lemma, we present some properties of the projection.

Lemma 2.4. Given 𝑥∈𝐻 and 𝑧∈𝐶. Then (a)𝑧=𝑃_𝐶𝑥 if and only if there holds the relation: 𝑥−𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶.(2.3)(b)𝑧=𝑃_𝐶𝑥 if and only if there holds the relation: ‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖²,∀𝑦∈𝐶.(2.4)(c)𝑃_𝐶 is nonexpansive and monotone, that is, 𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖²,∀𝑥,𝑦∈𝐻.(2.5)

Lemma 2.5. Let 𝐻 be a real Hilbert space. Then, for all 𝑥,𝑦∈𝐻, ‖𝑥+𝑦‖²≤‖𝑥‖²+2𝑦,𝑥+𝑦.(2.6)

The following lemma plays a key role in proving the main results of this paper.

Lemma 2.6 (see [19, Lemma 3.1]). Let 𝜆∈(0,1] and 𝜇>0. Let 𝐹∶𝐶→𝐻 be an operator on 𝐶 such that, for some constants 𝜅,𝜂>0, 𝐹 is 𝜅-Lipschitzian and 𝜂-strongly monotone. Associating with a nonexpansive mapping 𝑇∶𝐶→𝐶, define the mapping 𝑇^𝜆∶𝐶→𝐻 by 𝑇^𝜆𝑥∶=𝑇𝑥−𝜆𝜇𝐹(𝑇𝑥),∀𝑥∈𝐶.(2.7) Then 𝑇^𝜆 is a contraction provided 𝜇<2𝜂/𝜅², that is, 𝑇^𝜆𝑥−𝑇^𝜆𝑦≤(1−𝜆𝜏)𝑥−𝑦,∀𝑥,𝑦∈𝐶,(2.8) where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²)∈(0,1].

Remark 2.7. If 𝐹=𝐼, where 𝐼 is the identity operator of 𝐻. Then 𝜅=𝜂=1 and hence 𝜇<2𝜂/𝜅²=2. Also, if 𝜇=1, then it is easy to see that 𝜏=1−1−𝜇2𝜂−𝜇𝜅²=1.(2.9) In particular, whenever 𝜆>0, we have 𝑇^𝜆𝑥∶=𝑇𝑥−𝜆𝜇𝐹(𝑇𝑥)=(1−𝜆)𝑇𝑥.

3. Approximation Methods and Convergence Results

Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be a 𝐿-Lipschitzian mapping with constant 𝐿≥0 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). We consider the hierarchical variational inequality problem (in short, HVIP) of finding 𝑧^∗∈Fix(𝑇) such that 𝜇𝐹−𝛾𝑓𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇).(3.1) We denote by Ω the solution set of the HVIP (3.1).

When 𝜇=1, 𝐹=𝐼, 𝛾=𝜏=1 and 𝑓=𝑉 are a nonexpansive self-mapping on 𝐶, the HVIP (3.1) reduces to the following hierarchical variational inequality problem of finding 𝑧^∗∈Fix(𝑇) such that (𝐼−𝑉)𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇).(3.2) It is considered and studied in [12, 14].

We consider a mapping Θ_𝑡 on 𝐶 defined by Θ_𝑡𝑥=𝑃_𝐶𝑡𝛾𝑓(𝑥)+𝐼−𝑡𝜇𝐹𝑇𝑥,∀𝑥∈𝐶.(3.3) It is easy to see that Θ_𝑡 is a nonexpansive mapping. Indeed, we have Θ_𝑡𝑥−Θ_𝑡𝑦=𝑃_𝐶𝑡𝛾𝑓(𝑥)+𝐼−𝑡𝜇𝐹𝑇𝑥−𝑃_𝐶𝑦+𝑓(𝑥)𝑦++(=𝑡.𝑡𝛾𝑓𝐼−𝑡𝜇𝐹𝑇𝑦≤𝑡𝛾−𝑓𝐼−𝑡𝜇𝐹𝑇𝑥−𝐼−𝑡𝜇𝐹𝑇𝑦≤𝑡𝛾𝐿𝑥−𝑦1−𝑡𝜏)𝑥−𝑦1−𝜏−𝛾𝐿𝑥−𝑦(3.4) Since 0≤𝛾𝐿≤𝜏, it is known that Θ_𝑡 is nonexpansive on 𝐶.

Proposition 3.1. Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitzian with constant 𝐿≥0 and let 𝑇∶𝐶→𝐶 be nonexpansive with Fix(𝑇)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Let 𝑡∈(0,1) and 𝑧_𝑡 be a fixed point of the mapping Θ_𝑡=𝑃_𝐶[𝑡𝛾𝑓+(𝐼−𝑡𝜇𝐹)𝑇], that is, 𝑧_𝑡=𝑃_𝐶[𝑡𝛾𝑓(𝑧_𝑡)+(𝐼−𝑡𝜇𝐹)𝑇𝑧_𝑡]. Assume {𝑧_𝑡} remains bounded as 𝑡→0, then the following conclusions hold. (a)The solution set Ω of the HVIP (3.1) is nonempty and each weak limit point (as 𝑡→0) of {𝑧_𝑡} solves the HVIP (3.1).(b)If 𝜇𝐹−𝛾𝑓 is strictly monotone, then the net {𝑧_𝑡} converges weakly to the (unique) solution of the HVIP (3.1).(c)If 𝜇𝐹−𝛾𝑓 is strongly monotone (e.g., 𝜇𝜂>𝛾𝐿), then the net {𝑧_𝑡} converges strongly to a solution of the HVIP (3.1).

Proof. Let 𝑊 be the set of all weak accumulation points of {𝑧_𝑡} as 𝑡→0; that is, 𝑊=𝑧∶𝑧_𝑡𝑛⇀𝑧forsomesequence𝑡_𝑛in(0,1)suchthat𝑡_𝑛→0.(3.5) Then, 𝑊≠∅ because {𝑧_𝑡} is bounded.
To prove (a), we notice that the boundedness of {𝑧_𝑡} implies that 𝑊≠∅ and 𝑧_𝑡−𝑇𝑧_𝑡=𝑃_𝐶𝑡𝛾𝑓𝑧_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝑃_𝐶𝑇𝑧_𝑡≤𝑡𝛾𝑓𝑧_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝑇𝑧_𝑡=𝑡𝛾𝑓𝑧_𝑡−𝜇𝐹𝑇𝑧_𝑡→0as𝑡→0.(3.6) It thus follows from Lemma 2.3 that 𝑊⊂Fix(𝑇). Take a fixed 𝑥∈Fix(𝑇) arbitrarily and set 𝑤_𝑡=𝑡𝛾𝑓𝑧_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡,∀𝑡∈(0,1).(3.7) Then 𝑧_𝑡=𝑃_𝐶𝑤_𝑡 and 𝑧_𝑡−𝑥=𝑃_𝐶𝑤_𝑡−𝑤_𝑡+𝑡𝛾𝑓𝑧_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝑥=𝑃_𝐶𝑤_𝑡−𝑤_𝑡+𝑡𝛾𝑓𝑧_𝑡−𝜇𝐹𝑥+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝐼−𝑡𝜇𝐹𝑥=𝑃_𝐶𝑤_𝑡−𝑤_𝑡+𝑡𝛾𝑓𝑧_𝑡(𝑥)(𝑥)𝑥+−𝑓+𝑡𝛾𝑓−𝜇𝐹𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝐼−𝑡𝜇𝐹𝑇𝑥.(3.8) Since 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶, utilizing Lemma 2.4, we have 𝑃_𝐶𝑤_𝑡−𝑤_𝑡,𝑃_𝐶𝑤_𝑡−𝑥≤0.(3.9) Hence, utilizing 𝜅-Lipschitzian property of 𝐹, we get 𝑧_𝑡−𝑥²=𝑃_𝐶𝑤_𝑡−𝑤_𝑡,𝑃_𝐶𝑤_𝑡−𝑥+𝑤_𝑡−𝑥,𝑧_𝑡−𝑥≤𝑤_𝑡−𝑥,𝑧_𝑡−𝑥=𝑡𝛾𝑓𝑧_𝑡−𝜇𝐹𝑥,𝑧_𝑡−𝑥+𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝐼−𝑡𝜇𝐹𝑥,𝑧_𝑡−𝑥≤𝑡𝛾𝑓−𝜇𝐹𝑧_𝑡,𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑥,𝑧_𝑡−𝑥+12𝐼−𝑡𝜇𝐹𝑇𝑧_𝑡−𝐼−𝑡𝜇𝐹𝑥²+𝑧_𝑡−𝑥²=𝑡𝛾𝑓−𝜇𝐹𝑧_𝑡,𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑥,𝑇𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑥,𝑧_𝑡−𝑇𝑧_𝑡+12𝑇𝑧_𝑡−𝑥²−2𝑡𝜇𝐹𝑇𝑧_𝑡−𝐹𝑥,𝑇𝑧_𝑡−𝑥+𝑡²𝜇²𝐹𝑇𝑧_𝑡−𝐹𝑥²+𝑧_𝑡−𝑥²=𝑡𝛾𝑓−𝜇𝐹𝑧_𝑡,𝑧_𝑡−𝑥+𝜇𝐹𝑇𝑧_𝑡−𝐹𝑥,𝑇𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑇𝑧_𝑡,𝑇𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑥,𝑧_𝑡−𝑇𝑧_𝑡+12𝑇𝑧_𝑡−𝑥²−2𝑡𝜇𝐹𝑇𝑧_𝑡−𝐹𝑥,𝑇𝑧_𝑡−𝑥+𝑡²𝜇²𝐹𝑇𝑧_𝑡−𝐹𝑥²+𝑧_𝑡−𝑥²=𝑡𝛾𝑓−𝜇𝐹𝑧_𝑡,𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑇𝑧_𝑡,𝑇𝑧_𝑡−𝑥+𝜇𝐹𝑧_𝑡−𝐹𝑥,𝑧_𝑡−𝑇𝑧_𝑡+12𝑇𝑧_𝑡−𝑥²+𝑡²𝜇²𝐹𝑇𝑧_𝑡−𝐹𝑥²+𝑧_𝑡−𝑥²≤𝑡𝛾𝑓−𝜇𝐹𝑧_𝑡,𝑧_𝑡−𝑥+2𝑡𝜇𝜅𝑧_𝑡−𝑇𝑧_𝑡𝑧_𝑡−𝑥+𝑡1+²𝜇²𝜅²2𝑧_𝑡−𝑥².(3.10) It follows that 𝜇𝐹−𝛾𝑓𝑧_𝑡,𝑧_𝑡−𝑥≤𝑡𝜇²𝜅²2𝑧_𝑡−𝑥²+2𝜇𝜅𝑧_𝑡−𝑇𝑧_𝑡𝑧_𝑡−𝑥.(3.11) Note that 0≤𝛾𝐿≤𝜏 and 𝜇𝜂≥𝜏⟺𝜇𝜂≥1−1−𝜇2𝜂−𝜇𝜅²⟺1−𝜇2𝜂−𝜇𝜅²≥1−𝜇𝜂⟺1−2𝜇𝜂+𝜇²𝜅²≥1−2𝜇𝜂+𝜇²𝜂²⟺𝜅²≥𝜂²⟺𝜅≥𝜂.(3.12) Since 0≤𝛾𝐿≤𝜏≤𝜇𝜂, we have 𝜇𝜂−𝛾𝐿≥0. Thus, utilizing the 𝜂-strong monotonicity of 𝐹 and 𝐿-Lipschitzian property of 𝑓, we know that 𝜇𝐹−𝛾𝑓 is monotone because the following inequality holds: 𝜇𝐹−𝛾𝑓𝑥−𝜇𝐹−𝛾𝑓𝑦,𝑥−𝑦≥𝜇𝜂−𝛾𝐿𝑥−𝑦²,∀𝑥,𝑦∈𝐶.(3.13) Consequently, we have 𝜇𝐹−𝛾𝑓𝑧_𝑡,𝑧_𝑡−𝑥≥𝜇𝐹−𝛾𝑓𝑥,𝑧_𝑡−𝑥.(3.14) This together with (3.11) implies that 𝜇𝐹−𝛾𝑓𝑥,𝑧_𝑡−𝑥≤𝑡𝜇²𝜅²2𝑧_𝑡−𝑥²+2𝜇𝜅𝑧_𝑡−𝑇𝑧_𝑡𝑧_𝑡−𝑥.(3.15) Now, if 𝑥∈𝑊⊂Fix(𝑇) and if 𝑡_𝑛→0 is such that 𝑧_𝑡𝑛⇀𝑥, then we obtain from (3.15) and ‖𝑧_𝑡−𝑇𝑧_𝑡‖→0 that 𝜇𝐹−𝛾𝑓𝑥,𝑥−𝑥≤0,∀𝑥∈Fix(𝑇).(3.16) Replacing 𝑥 by 𝑥+𝜆(𝑥−𝑥)∈Fix(𝑇) in (3.16), where 𝜆∈(0,1) and 𝑥∈Fix(𝑇), we get 𝜇𝐹−𝛾𝑓(𝑥+𝜆(𝑥−𝑥)),𝑥−𝑥≤0.(3.17) Letting 𝜆→0 yields 𝜇𝐹−𝛾𝑓𝑥,𝑥−𝑥≤0,∀𝑥∈Fix(𝑇).(3.18) Consequently, 𝑥∈Ω.
To see (b), we assume that {𝑡^′_𝑛} is another null sequence in (0,1) such that 𝑥_𝑡′𝑛𝑥. Then 𝑥∈Fix(𝑇) and by replacing 𝑥 by 𝑥 in (3.18), we get 𝜇𝐹−𝛾𝑓𝑥,𝑥−𝑥≤0.(3.19) By interchanging 𝑥 and 𝑥, we get 𝜇𝐹−𝛾𝑓𝑥,𝑥−𝑥≤0.(3.20) Adding up (3.19) and (3.20) yields 𝜇𝐹−𝛾𝑓𝑥−𝜇𝐹−𝛾𝑓𝑥,𝑥−𝑥≤0.(3.21) So the strict monotonicity of 𝜇𝐹−𝛾𝑓 implies that 𝑥=𝑥 and {𝑧_𝑡} converges weakly.
Finally, to prove (c), we observe that the strong monotonicity of 𝜇𝐹−𝛾𝑓 and (3.11) implies that 𝛼𝑧_𝑡−𝑥²+𝜇𝐹−𝛾𝑓𝑥,𝑧_𝑡−𝑥≤𝑡𝜇²𝜅²2𝑧_𝑡−𝑥²+2𝜇𝜅𝑧_𝑡−𝑇𝑧_𝑡𝑧_𝑡−𝑥,(3.22) where 𝛼>0 is the strong monotonicity constant of 𝜇𝐹−𝛾𝑓; that is, 𝜇𝐹−𝛾𝑓𝑥−𝜇𝐹−𝛾𝑓𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶.(3.23) A straightforward consequence of (3.22) is that if 𝑥∈𝑊 and if 𝑧_𝑡𝑛⇀𝑥 for some null sequence {𝑡_𝑛} in (0,1), then we must have 𝑧_𝑡𝑛→𝑥. This shows that {𝑧_𝑡} is relatively compact in the norm topology, and each of its limit points solves the HVIP (3.1). Finally repeating the argument in the weak convergence case of (b), we see that {𝑧_𝑡} can have exactly one limit point; hence, {𝑧_𝑡} converges in norm.

Corollary 3.2 (see [14, Proposition 3.1]). Let 𝑉,𝑇∶𝐶→𝐶 be nonexpansive mappings with Fix(𝑇)≠∅. Let 𝑡∈(0,1) and 𝑧_𝑡 be a fixed point of the mapping 𝑊_𝑡=𝑡𝑉+(1−𝑡)𝑇, that is, 𝑧_𝑡=𝑡𝑉𝑧_𝑡+(1−𝑡)𝑇𝑧_𝑡. Assume {𝑧_𝑡} remains bounded as 𝑡→0, then the following conclusions hold. (a)The solution set 𝑆 of the HVIP (1.4) is nonempty and each weak limit point (as 𝑡→0) of {𝑧_𝑡} solves the HVIP (1.4).(b)If 𝐼−𝑉 is strictly monotone, then the net {𝑧_𝑡} converges weakly to the solution of the HVIP (1.4).(c)If 𝐼−𝑉 is strongly monotone (e.g., 𝑉 is a contraction), then the net {𝑧_𝑡} converges strongly to a solution of the HVIP (1.4).

Now by combining hybrid steepest descent method, viscosity method, and projection method, we define, for each 𝑠,𝑡∈(0,1), two mappings 𝑊_𝑡 and 𝑓_𝑠,𝑡 by 𝑊_𝑡=𝑡𝑉+(1−𝑡)𝑇,𝑓_𝑠,𝑡=𝑃_𝐶𝑠𝛾𝑓+𝐼−𝑠𝜇𝐹𝑊_𝑡.(3.24) It is easy to see that 𝑊_𝑡 is a nonexpansive self-mapping on 𝐶. Moreover, utilizing Lemma 2.6, we can see that 𝑓_𝑠,𝑡 is a (1−(𝜏−𝛾𝐿)𝑠)-contraction. Indeed, observe that 𝑓_𝑠,𝑡(𝑥)−𝑓_𝑠,𝑡𝑦=𝑃_𝐶𝑠𝛾𝑓(𝑥)+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑥−𝑃_𝐶𝑠𝛾𝑓𝑦+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑦≤𝑠𝛾𝑓(𝑥)+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑥−𝑠𝛾𝑓𝑦+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑦≤𝑠𝛾𝑓(𝑥)−𝑓𝑦+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑥−𝐼−𝑠𝜇𝐹𝑊_𝑡𝑦+(=.≤𝑠𝛾𝐿𝑥−𝑦1−𝑠𝜏)𝑥−𝑦1−𝑠𝜏−𝛾𝐿𝑥−𝑦(3.25) Let 𝑥_𝑠,𝑡 be the unique fixed point of 𝑓_𝑠,𝑡. Namely, 𝑥_𝑠,𝑡 is the unique solution in 𝐶 to the following: 𝑥_𝑠,𝑡=𝑃_𝐶𝑠𝛾𝑓𝑥_𝑠,𝑡+𝐼−𝑠𝜇𝐹𝑊_𝑡𝑥_𝑠,𝑡=𝑃_𝐶𝑠𝛾𝑓𝑥_𝑠,𝑡+𝐼−𝑠𝜇𝐹𝑡𝑉𝑥_𝑠,𝑡+(1−𝑡)𝑇𝑥_𝑠,𝑡.(3.26)

Theorem 3.3. Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitzian with constant 𝐿≥0 and let 𝑉,𝑇∶𝐶→𝐶 be nonexpansive with Fix(𝑇)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). For each 𝑠,𝑡∈(0,1), let 𝑥_𝑠,𝑡 be the unique solution to (3.26). Assume also that, for each 𝑡∈(0,1), Fix(𝑊_𝑡) is nonempty (but not necessarily bounded), and the following assumption holds: ∅≠𝑆⊂‖⋅‖−liminf_𝑡→0Fix𝑊_𝑡∶=𝑧∶∃𝑧_𝑡∈Fix𝑊_𝑡𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑧_𝑡→𝑧.(A) Then the strong lim_𝑠→0𝑥_𝑠,𝑡=∶𝑥_𝑡 exists for each 𝑡∈(0,1). Moreover, the strong lim_𝑡→0𝑥_𝑡=∶𝑥_∞ exists and solves the THVIP (1.6). Hence, for any null sequence {𝑠_𝑛} in (0,1), there is another null sequence {𝑡_𝑛} in (0,1) such that 𝑥_{𝑠𝑛,𝑡𝑛}→𝑥_∞ in norm, as 𝑛→∞.

Proof. Observe that the condition 0≤𝛾𝐿<𝜏 and the fact 𝜏≤𝜇𝜂 imply that 0≤𝛾𝐿<𝜏≤𝜇𝜂.(3.27) Therefore, 𝜇𝐹−𝛾𝑓 is a strongly monotone operator with constant 𝜇𝜂−𝛾𝐿>0. Since, for each fixed 𝑡∈(0,1), the fixed point set Fix(𝑊_𝑡) of 𝑊_𝑡 is nonempty, we can apply Proposition 3.1 (c) to get that 𝑥_𝑡∶=‖⋅‖−lim_𝑠→0𝑥_𝑠,𝑡(3.28) exists in Fix(𝑊_𝑡) and solves the following hierarchical variational inequality problem of finding 𝑥_𝑡∈Fix(𝑊_𝑡) such that 𝜇𝐹−𝛾𝑓𝑥_𝑡,𝑥−𝑥_𝑡≥0,∀𝑥∈Fix𝑊_𝑡.(3.29) Equivalently, 𝑥_𝑡=𝑃_{Fix(𝑊𝑡)}(𝐼−𝜇𝐹+𝛾𝑓)𝑥_𝑡, where 𝑃_{Fix(𝑊𝑡)} is the metric projection from 𝐻 onto Fix(𝑊_𝑡).
Utilizing the strong monotonicity of 𝜇𝐹−𝛾𝑓, we conclude from (3.29) that for each 𝑧∈Fix(𝑊_𝑡), 𝜇𝜂−𝛾𝐿𝑥_𝑡−𝑧²≤𝜇𝐹−𝛾𝑓𝑥_𝑡−𝜇𝐹−𝛾𝑓𝑧,𝑥_𝑡𝑥−𝑧=−𝜇𝐹−𝛾𝑓_𝑡,𝑧−𝑥_𝑡+𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑡≤−𝑧𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑡−𝑧.(3.30) Hence, 𝑥_𝑡−𝑧²≤1𝜇𝜂−𝛾𝐿𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑡−𝑧,∀𝑧∈Fix𝑊_𝑡.(3.31) This implies that 𝑥_𝑡≤1−𝑧𝑧𝑊𝜇𝜂−𝛾𝐿𝛾𝑓−𝜇𝐹,∀𝑧∈Fix_𝑡.(3.32) The inequality (3.32) is yet to imply the boundedness of {𝑥_𝑡} since 𝑧 may depend on 𝑡. However, since the solution set 𝑆 of the HVIP (1.4) is nonempty, we can take (an arbitrary) 𝑣∈𝑆 and use assumption (A) to find 𝑧_𝑡∈Fix(𝑊_𝑡) such that 𝑧_𝑡→𝑣 in norm as 𝑡→0. Hence, {𝑧_𝑡} must be bounded (as 𝑡→0). The inequality (3.32) implies 𝑥_𝑡≤𝑥_𝑡−𝑧_𝑡+𝑧_𝑡+‖𝑣‖≤1−𝑣𝑧𝜇𝜂−𝛾𝐿𝛾𝑓−𝜇𝐹_𝑡+𝑧_𝑡−𝑣+‖𝑣‖,(3.33) and this is sufficient to ensure that {𝑥_𝑡} is bounded (as 𝑡 closes 0).
Now, the boundedness of {𝑥_𝑡} allows us to apply Corollary 3.2 (a) to conclude that every weak limit point 𝑥 of {𝑥_𝑡} belongs to the solution set 𝑆 of the HVIP (1.4). Then (3.31) guarantees that every such weak limit point 𝑥 of {𝑥_𝑡} is also a strong limit point of {𝑥_𝑡}. Indeed, if {𝑡_𝑛} is a null sequence in (0,1) and if 𝑥_𝑡𝑛⇀𝑥, then 𝑥∈𝑆. By assumption (A), we get a sequence {𝑧_𝑛} such that 𝑧_𝑛∈Fix(𝑊_𝑡𝑛) for all 𝑛 and 𝑧_𝑛→𝑥 in norm. From (3.31) we derive 𝑥_𝑡𝑛−𝑥²=𝑥_𝑡𝑛−𝑧_𝑛+𝑧_𝑛−𝑥²≤2𝑥_𝑡𝑛−𝑧_𝑛²+𝑧_𝑛−𝑥²≤2𝑧𝜇𝜂−𝛾𝐿𝛾𝑓−𝜇𝐹_𝑛,𝑥_𝑡𝑛−𝑧_𝑛+2𝑧_𝑛−𝑥².(3.34) However, ⟨(𝛾𝑓−𝜇𝐹)𝑧_𝑛,𝑥_𝑡𝑛−𝑧_𝑛⟩→0 since (𝛾𝑓−𝜇𝐹)𝑧_𝑛→(𝛾𝑓−𝜇𝐹)𝑥 in norm and 𝑥_𝑡𝑛−𝑧_𝑛→0 weakly, and we find that the right-hand side of (3.34) tends to zero. Hence, 𝑥_𝑡𝑛→𝑥 in norm.
So to prove the strong convergence of the entire net {𝑥_𝑡}, it remains to prove that {𝑥_𝑡} can have only one strong limit point. Let 𝑥 and 𝑥^′ be two strong limit points of {𝑥_𝑡} and assume that 𝑥_𝑡𝑛→𝑥 and 𝑥_𝑡′𝑛→𝑥^′ both in norm, where {𝑡_𝑛} and {𝑡^′_𝑛} are null sequences in (0,1). It remains to verify that 𝑥=𝑥^′.
Since 𝑥^′∈𝑆, by assumption (A), we can find 𝑧_𝑡∈Fix(𝑊_𝑡) such that 𝑧_𝑡→𝑥^′ in norm as 𝑡→0. The HVIP (3.29) implies 𝜇𝐹−𝛾𝑓𝑥_𝑡𝑛,𝑧_𝑡𝑛−𝑥_𝑡𝑛≥0.(3.35) Taking the limit as 𝑛→∞ yields 𝜇𝐹−𝛾𝑓𝑥,𝑥^′−𝑥≥0.(3.36) Similarly, we have 𝜇𝐹−𝛾𝑓𝑥^′,𝑥−𝑥^′≥0.(3.37) Adding up (3.36) and (3.37) gives 𝜇𝐹−𝛾𝑓𝑥−𝜇𝐹−𝛾𝑓𝑥^′,𝑥−𝑥^′≤0.(3.38) Utilizing Lemma 2.2, we know that 𝜇𝐹−𝛾𝑓 is strongly monotone with constant 𝜇𝜂−𝛾𝐿>0. Hence, from (3.38) it follows that 𝑥=𝑥^′ and so {𝑥_𝑡} converges in norm to (say) 𝑥_∞.
Now, for any 𝑣∈𝑆, since by assumption (A), we can find 𝑧_𝑡∈Fix(𝑊_𝑡) such that 𝑧_𝑡→𝑣 in norm, (3.29) then implies 𝜇𝐹−𝛾𝑓𝑥_𝑡,𝑣−𝑥_𝑡≥𝜇𝐹−𝛾𝑓𝑥_𝑡,𝑣−𝑧_𝑡→0(3.39) which in turns implies 𝜇𝐹−𝛾𝑓𝑥_∞,𝑣−𝑥_∞≥0,∀𝑣∈𝑆,(3.40) that is, 𝑥_∞=𝑃_𝑆(𝐼−𝜇𝐹+𝛾𝑓)𝑥_∞, the unique fixed point of the contraction 𝑃_𝑆(𝐼−𝜇𝐹+𝛾𝑓). Finally, for any null sequence {𝑠_𝑛} in (0,1), using a diagonalization argument (cf. [1]), we can find another null sequence {𝑡_𝑛} in (0,1) such that 𝑥_{𝑠𝑛,𝑡𝑛}→𝑥_∞ in norm, as 𝑛→∞.

Remark 3.4. Theorem 3.3 shows that for any null sequence {𝑠_𝑛} in (0,1), there is another null sequence {𝑡_𝑛} in (0,1) such that 𝑥_{𝑠𝑛,𝑡𝑛}→𝑥_∞ in norm, as 𝑛→∞, and 𝑥_∞ is a solution to the HVIP (3.40). Theorem 3.3 is the main result of the present paper in which we improve the result of Moudafi and Maingé [12] by proving that {𝑥_𝑡} actually converges strongly and also by removing the boundedness of the set {Fix(𝑊_𝑡)∶0<𝑡<1}. Our proof is different from that of [12]. In the meantime, Theorem 3.3 covers [14, Theorem 3.2] as a special case. For instance, whenever we put 𝜇=1, 𝐹=𝐼, 𝛾=𝜏=1, and let the 𝐿-Lipschitzian mapping 𝑓∶𝐶→𝐻 be a (self-) contraction with coefficient 𝜌∈[0,1), our Theorem 3.3 reduces to [14, Theorem 3.2].
Now, we present a general result. We show that as long as 𝑡_𝑠 is taken so that 𝑡_𝑠=𝑜(𝑠) (i.e., lim_𝑠→0𝑡_𝑠/𝑠=0), then 𝑥_{𝑠,𝑡𝑠}→𝑧_∞ in norm, and moreover, 𝑧_∞ solves the HVIP (3.40) on the larger set Fix(𝑇) (i.e., 𝑧_∞ is the unique fixed point in Fix(𝑇) of the contraction 𝑃_Fix(𝑇)(𝐼−𝜇𝐹+𝛾𝑓)), without the assumption (A). However, for such a general choice of {𝑡_𝑠}, this solution 𝑧_∞ may differ from the solution 𝑥_∞ of the HVIP (3.40) on the smaller set 𝑆 (i.e., 𝑥_∞ is the unique fixed point in 𝑆 of the contraction 𝑃_𝑆(𝐼−𝜇𝐹+𝛾𝑓)). We will verify this by taking 𝑡_𝑠=𝑠² for simplicity (the argument, however, works for any net {𝑡_𝑠} in (0,1) such that lim_𝑠→0𝑡_𝑠/𝑠=0).

Theorem 3.5. Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitzian with constant 𝐿≥0 and let 𝑉,𝑇∶𝐶→𝐶 be nonexpansive with Fix(𝑇)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). For each 𝑠∈(0,1), let 𝑥_𝑠 be the unique solution in 𝐶 to the following: 𝑥_𝑠=𝑃_𝐶𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑠²𝑉𝑥_𝑠+1−𝑠²𝑇𝑥_𝑠.(3.41) Then, as 𝑠→0, 𝑥_𝑠 converges in norm to the solution of the HVIP of finding 𝑧_∞∈Fix(𝑇) such that 𝜇𝐹−𝛾𝑓𝑧_∞,𝑧−𝑧_∞≥0,∀𝑧∈Fix(𝑇);(3.42) equivalently, 𝑧_∞=𝑃_Fix(𝑇)(𝐼−𝜇𝐹+𝛾𝑓)𝑧_∞.

Proof. Write 𝑊_𝑠 (instead of 𝑊_𝑠2) for 𝑠²𝑉+(1−𝑠²)𝑇; then 𝑥_𝑠=𝑃_𝐶𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑊_𝑠𝑥_𝑠.(3.43) Take a fixed 𝑧∈Fix(𝑇) arbitrarily and put 𝑦_𝑠=𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑊_𝑠𝑥_𝑠,∀𝑠∈(0,1).(3.44) Then from (3.41) we get 𝑥_𝑠=𝑃_𝐶𝑦_𝑠. Since 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶, we have 𝑃_𝐶𝑦_𝑠−𝑦_𝑠,𝑃_𝐶𝑦_𝑠−𝑧≤0.(3.45) Also, observe that 𝑦_𝑠−𝑧=𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑊_𝑠𝑥_𝑠𝑥−𝑧=𝑠𝛾𝑓_𝑠−𝜇𝐹𝑊_𝑠𝑧+𝐼−𝑠𝜇𝐹𝑊_𝑠𝑥_𝑠−𝐼−𝑠𝜇𝐹𝑊_𝑠𝑧+𝑊_𝑠𝑓𝑥𝑧−𝑧=𝑠𝛾_𝑠−𝑓(𝑧)+𝑠𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧+𝐼−𝑠𝜇𝐹𝑊_𝑠𝑥_𝑠−𝐼−𝑠𝜇𝐹𝑊_𝑠𝑧+𝑠²(𝑉−𝐼)𝑧.(3.46) Utilizing Lemma 2.6, we deduce from (3.41) that ‖𝑥_𝑠−𝑧‖²=𝑃_𝐶𝑦_𝑠−𝑦_𝑠,𝑃_𝐶𝑦_𝑠−𝑧+𝑦_𝑠−𝑧,𝑥_𝑠≤𝑦−𝑧_𝑠−𝑧,𝑥_𝑠𝑓𝑥−𝑧=𝑠𝛾_𝑠−𝑓(𝑧),𝑥_𝑠−𝑧+𝑠𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠+𝑊−𝑧𝐼−𝑠𝜇𝐹_𝑠𝑥_𝑠−𝐼−𝑠𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠−𝑧+𝑠²(𝑉−𝐼)𝑧,𝑥_𝑠𝑓𝑥−𝑧≤𝑠𝛾_𝑠−𝑓(𝑧)𝑥_𝑠−𝑧+𝑠𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠+𝑊−𝑧𝐼−𝑠𝜇𝐹_𝑠𝑥_𝑠−𝐼−𝑠𝜇𝐹𝑊_𝑠𝑧𝑥_𝑠−𝑧+𝑠²(𝑉−𝐼)𝑧,𝑥_𝑠𝑥−𝑧≤𝑠𝛾𝐿_𝑠−𝑧²+𝑠𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠−𝑧+(1−𝑠𝜏)𝑥_𝑠−𝑧²+𝑠²(𝑉−𝐼)𝑧,𝑥_𝑠=𝑥−𝑧1−𝑠𝜏−𝛾𝐿_𝑠−𝑧²+𝑠𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠−𝑧+𝑠²(𝑉−𝐼)𝑧,𝑥_𝑠−𝑧.(3.47) It follows that, for any fixed 𝑧∈Fix(𝑇), ‖𝑥_𝑠−𝑧‖²≤1(𝑧)𝜏−𝛾𝐿𝛾𝑓−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠−𝑧+𝑠(𝑉−𝐼)𝑧,𝑥_𝑠−𝑧.(3.48) This implies that 𝑥_𝑠≤1−𝑧(𝑧)𝜏−𝛾𝐿𝛾𝑓−𝜇𝐹𝑊_𝑠𝑧+‖(𝑉−𝐼)𝑧‖.(3.49) In particular, {𝑥_𝑠} is bounded, and from (3.41), we further get 𝑥_𝑠−𝑇𝑥_𝑠=𝑃_𝐶𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑠²𝑉𝑥_𝑠+1−𝑠²𝑇𝑥_𝑠−𝑃_𝐶𝑇𝑥_𝑠≤𝑠𝛾𝑓𝑥_𝑠+𝐼−𝑠𝜇𝐹𝑠²𝑉𝑥_𝑠+1−𝑠²𝑇𝑥_𝑠−𝑇𝑥_𝑠≤𝑠𝛾𝑓𝑥_𝑠−𝜇𝐹𝑠²𝑉𝑥_𝑠+1−𝑠²𝑇𝑥_𝑠+𝑠²𝑉𝑥_𝑠−𝑇𝑥_𝑠→0,(3.50) as 𝑠→0. Lemma 2.3 ensures that every weak limit point, as 𝑠→0, of {𝑥_𝑠} is a fixed point of 𝑇. Going back to (3.48), we find that each weak limit point of {𝑥_𝑠} is actually a strong limit point of {𝑥_𝑠} because 𝛾𝑓(𝑧)−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠=(𝑧)−𝑧𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑠−𝑧+𝜇𝐹𝑧−𝜇𝐹𝑊_𝑠𝑧,𝑥_𝑠≤(𝑧)−𝑧𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑠−𝑧+𝜇𝐹𝑧−𝐹𝑊_𝑠𝑧𝑥_𝑠≤(𝑧)−𝑧𝛾𝑓−𝜇𝐹𝑧,𝑥_𝑠−𝑧+𝜇𝜅𝑠²‖(𝑉−𝐼)𝑧‖𝑥_𝑠−𝑧.(3.51) So to prove the strong convergence of {𝑥_𝑠}, we need only to show the uniqueness of strong limit points of {𝑥_𝑠}. Assuming {𝑠_𝑛} and {𝑠^′_𝑛} are null sequences in (0,1) such that 𝑥_𝑠𝑛→𝑣 and 𝑥_𝑠′𝑛→𝑣′, both in norm. Observing that (3.41) implies 𝜇𝐹−𝛾𝑓𝑥_𝑠=1𝑠𝑃_𝐶𝑦_𝑠−𝑦_𝑠−1𝑠𝐼−𝑊_𝑠𝑥_𝑠+𝜇𝐹𝑥_𝑠−𝐹𝑊_𝑠𝑥_𝑠,(3.52) where 𝑦_𝑠=𝑠𝛾𝑓(𝑥_𝑠)+(𝐼−𝑠𝜇𝐹)𝑊_𝑠𝑥_𝑠 and 𝑥_𝑠=𝑃_𝐶𝑦_𝑠. Utilizing Lemmas 2.4 and 2.6, we deduce from the monotonicity of 𝐼−𝑊_𝑠 that for any fixed 𝑧∈Fix(𝑇), 𝜇𝐹−𝛾𝑓𝑥_𝑠,𝑥_𝑠=1−𝑧𝑠𝑃_𝐶𝑦_𝑠−𝑦_𝑠,𝑃_𝐶𝑦_𝑠−1−𝑧𝑠𝐼−𝑊_𝑠𝑥_𝑠,𝑥_𝑠−𝑧+𝜇𝐹𝑥_𝑠−𝐹𝑊_𝑠𝑥_𝑠,𝑥_𝑠≤1−𝑧𝑠𝑃_𝐶𝑦_𝑠−𝑦_𝑠,𝑃_𝐶𝑦_𝑠−1−𝑧𝑠𝐼−𝑊_𝑠𝑥_𝑠−𝐼−𝑊_𝑠𝑧,𝑥_𝑠+−𝑧𝐼−𝑊_𝑠𝑧,𝑥_𝑠−𝑧+𝜇𝐹𝑥_𝑠−𝐹𝑊_𝑠𝑥_𝑠,𝑥_𝑠)−𝑧≤−𝑠(𝐼−𝑉𝑧,𝑥_𝑠−𝑧+𝜇𝜅𝑥_𝑠−𝑊_𝑠𝑥_𝑠𝑥_𝑠)−𝑧=−𝑠(𝐼−𝑉𝑧,𝑥_𝑠−𝑧+𝜇𝜅𝑥_𝑠−𝑇𝑥_𝑠+𝑠²(𝑇−𝑉)𝑥_𝑠𝑥_𝑠−𝑧.(3.53) In particular, we have from 𝐼−𝑊_𝑠 that for any fixed 𝑧∈Fix(𝑇), 𝜇𝐹−𝛾𝑓𝑥_𝑠𝑛,𝑥_𝑠𝑛−𝑣^′≤−𝑠_𝑛(𝐼−𝑉)𝑣^′,𝑥_𝑠𝑛−𝑣^′+𝜇𝜅𝑥_𝑠𝑛−𝑇𝑥_𝑠𝑛+𝑠²_𝑛(𝑇−𝑉)𝑥_𝑠𝑛𝑥_𝑠𝑛−𝑣′.(3.54) So letting 𝑛→∞ yields 𝜇𝐹−𝛾𝑓𝑣,𝑣−𝑣^′≤0.(3.55) Repeating the above argument obtains 𝜇𝐹−𝛾𝑓𝑣^′,𝑣^′−𝑣≤0.(3.56) Adding up (3.55) and (3.56) gives us that 𝜇𝐹−𝛾𝑓𝑣−𝜇𝐹−𝛾𝑓𝑣^′,𝑣−𝑣^′≤0.(3.57) The strong monotonicity of 𝜇𝐹−𝛾𝑓 (Lemma 2.2) then implies 𝑣=𝑣′. Finally, taking the limit as 𝑠→0 in (3.53) and letting 𝑧_∞=‖⋅‖−lim_𝑠→0𝑥_𝑠, we conclude immediately that 𝑧_∞ solves the variational inequality of finding 𝑧_∞∈Fix(𝑇) such that 𝜇𝐹−𝛾𝑓𝑧_∞,𝑧_∞−𝑧≤0,∀𝑧∈Fix(𝑇).(3.58) Equivalently, 𝑧_∞=𝑃_Fix(𝑇)(𝐼−𝜇𝐹+𝛾𝑓)𝑧_∞. The proof is therefore complete.

Remark 3.6. If 𝑇 and 𝑉 have a common fixed point, then it is not hard to see that Fix(𝑊_𝑡)=Fix(𝑇)∩Fix(𝑉) for all 𝑡∈(0,1). Indeed, it suffices to show the inclusion Fix(𝑊_𝑡)⊂Fix(𝑇)∩Fix(𝑉). Let 𝑧∈Fix(𝑊_𝑡). Then for any fixed 𝑝∈Fix(𝑇)∩Fix(𝑉) we have 𝑧−𝑝²=𝑡𝑉𝑧−𝑝²+(1−𝑡)𝑇𝑧−𝑝²−𝑡(1−𝑡)‖𝑉𝑧−𝑇𝑧‖²≤𝑧−𝑝²−𝑡(1−𝑡)‖𝑉𝑧−𝑇𝑧‖².(3.59) This implies 𝑉𝑧=𝑇𝑧=𝑧; that is 𝑧∈Fix(𝑇)∩Fix(𝑉). Furthermore, it is clear that Fix(𝑇)∩Fix(𝑉)⊂𝑆. In this case, assumption (A) is reduced to the assumption 𝑆⊂Fix(𝑇)∩Fix(𝑉). Therefore, assumption (A) is equivalent to the assumption 𝑆=Fix(𝑇)∩Fix(𝑉).

Corollary 3.7. Let 𝐹∶𝐶→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅 and 𝜂>0, respectively. Let 𝑓∶𝐶→𝐻 be 𝐿-Lipschitzian with constant 𝐿≥0 and let 𝑉,𝑇∶𝐶→𝐶 be nonexpansive with Fix(𝑇)∩Fix(𝑉)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0≤𝛾𝐿<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). For each 𝑠,𝑡∈(0,1), let 𝑥_𝑠,𝑡 be the unique solution to (3.26). Then the conclusion of Theorem 3.3 holds. Namely, the strong lim_𝑠→0𝑥_𝑠,𝑡=∶𝑥_𝑡 exists for each fixed 𝑡∈(0,1), and moreover, the strong lim_𝑡→0𝑥_𝑡=∶𝑥_∞ exists and solves the THVIP (1.6).

Proof. Since Fix(𝑊_𝑡)=Fix(𝑇)∩Fix(𝑉) is independent of 𝑡; the 𝑧 in both (3.31) and (3.32) does not depend on 𝑡. Hence, it is immediately clear that {𝑥_𝑡} is bounded, which then implies via (3.31) that every weak accumulation point of {𝑥_𝑡} is also a strong accumulation point of {𝑥_𝑡}. Eventually, {𝑥_𝑡} converges in norm as shown in the final part of the proof of Theorem 3.5.

Remark 3.8. Theorems 3.3 and 3.5 improve and extend [14, Theorems 3.2 and 3.4], respectively, in the following ways. (a)The contraction mapping 𝑓∶𝐶→𝐶 in [14, Theorems 3.2 and 3.4] is extended to the case of (possibly nonself) 𝐿-Lipschitzian mapping 𝑓∶𝐶→𝐻 from a nonempty closed convex subset 𝐶 to 𝐻.(b)The convex combination of (self) contraction mapping 𝑓 and nonexpansive mapping 𝑊_𝑡 in the implicit scheme in [14, Theorem 3.2] is extended to the linear combination of (possibly nonself) 𝐿-Lipschitzian mapping 𝑓 and hybrid steepest descent method involving 𝑊_𝑡. In particular, if 𝑡=𝑠², Theorem 3.5 is an extension of [14, Theorem 3.4].(c)In order to guarantee that the net {𝑥_𝑠,𝑡} generated by the implicit scheme still lies in 𝐶, the implicit scheme in [14, Theorem 3.2] is extended to develop our new implicit scheme (3.26) by virtue of the projection method. In particular, if 𝑡=𝑠², [14, Theorem 3.4] is extended to the corresponding case in our Theorem 3.5.(d)The new technique of argument is applied to derive our Theorems 3.3 and 3.5. For instance, the characteristic properties (Lemma 2.4) of the metric projection play a key role in proving the strong convergence of the nets {𝑥_𝑠,𝑡}_{𝑠,𝑡∈(0,1)} and {𝑥_𝑠}_𝑠∈(0,1) in our Theorems 3.3 and 3.5, respectively.(e)If we put 𝜇=1,𝐹=𝐼 and 𝛾=𝜏=1 and let 𝑓 be a contractive self-mapping on 𝐶 with coefficient 𝜌∈[0,1), then our Theorems 3.3 and 3.5 reduce to [14, Theorems 3.2 and 3.4], respectively. Thus, our Theorems 3.3 and 3.5 cover [14, Theorems 3.2 and 3.4] as special cases, respectively.

Acknowledgments

In this research, the first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The second author was supported by the D.S.T. Research Project no. SR/S4/MS:719/11. Third author was partially supported by Grant NSC 101-2115-M-037-001.