Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

Abstract

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

1. Introduction and Formulations

Throughout the paper, we will adopt the following terminology and notations. ℋ is a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. The strong (resp., weak) convergence of the sequence {𝑥_𝑛} to 𝑥 will be denoted by 𝑥_𝑛→𝑥 (resp., 𝑥_𝑛⇀𝑥). We shall use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}; namely, 𝜔_𝑤𝑥_𝑛∶=𝑥∶𝑥_𝑛𝑘⇀𝑥forsomesubsequence𝑥_𝑛𝑘of𝑥_𝑛. (1)

Throughout the paper, unless otherwise specified, we assume that 𝐶 is a nonempty, closed, and convex subset of a Hilbert space ℋ and 𝐴∶𝐶→ℋ is a nonlinear mapping. The variational inequality problem (VIP) on 𝐶 is defined as follows: find𝑥^∗∈𝐶suchthat𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (2)

We denote by Γ the set of solutions of VIP. In particular, if 𝐶 is the set of fixed points of a nonexpansive mapping 𝑇, denoted by Fix(𝑇), then (VIP) is called a hierarchical variational inequality problem (HVIP), also known as a hierarchical fixed point problem (HFPP). If we replace the mapping 𝐴 by 𝐼−𝑆, where 𝐼 is the identity mapping and 𝑆 is a nonexpansive mapping (not necessarily with fixed points), then the VIP becomes as follows: find𝑥^∗∈Fix(𝑇)suchthat(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗(𝑇).≥0,∀𝑥∈Fix (3)

This problem, first introduced and studied in [1, 2], is called a hierarchical variational inequality problem, also known as a hierarchical fixed point problem. Observe that if 𝑆 has fixed points, then they are solutions of VIP (3). It is worth mentioning that many practical problems can be written in the form of a hierarchical variational inequality problem; see, for example, [1–18] and the references therein.

If 𝑆 is a 𝜌-contraction with coefficient 𝜌∈[0,1) (i.e., ‖𝑆𝑥−𝑆𝑦‖≤𝜌‖𝑥−𝑦‖ for some 𝜌∈[0,1)), then the set of solutions of VIP (3) is a singleton, and it is well known as a viscosity problem, which was first introduced by Moudafi [19] and then developed by Xu [20]. It is not hard to verify that solving VIP (3) is equivalent to finding a fixed point of the nonexpansive mapping 𝑃_Fix(𝑇)𝑆, where 𝑃_Fix(𝑇) is the metric projection on the closed and convex set Fix(𝑇).

Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone, where 𝜅>0, 𝜂>0 are constants, that is, for all 𝑥,𝑦∈𝐶𝐹𝑥−𝐹𝑦≤𝜅𝑥−𝑦,𝐹𝑥−𝐹𝑦,𝑥−𝑦≥𝜂𝑥−𝑦². (4)

A mapping 𝑇∶𝐶→𝐶 is called 𝜁-strictly pseudocontractive if there exists a constant 𝜁∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜁(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶; (5)

see [21] for more details. We denote by Fix(𝑇) the fixed point set of 𝑇; that is, Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

We introduce and consider the following triple hierarchical variational inequality problem (THVIP).

Problem I. Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone on the nonempty, closed, and convex subset 𝐶 of ℋ, where 𝜅 and 𝜂 are positive constants. Let 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and, for 𝑖=1,2, 𝑇_𝑖∶𝐶→𝐶 be 𝜁_𝑖-strictly pseudocontractive mapping with Fix(𝑇₁)∩Fix(𝑇₂)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Then the objective is to find 𝑥^∗∈Ξ such that 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ, (6) where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding 𝑧^∗∈Fix(𝑇) such that 𝜇𝐹−𝛾𝑆𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix𝑇₁∩Fix𝑇₂. (7)
In particular, whenever 𝑇₁=𝑇 a nonexpansive mapping, and 𝑇₂=𝐼 an identity mapping, Problem I reduces to the THVIP considered by Ceng et al. [22]. By combining the regularization method, the hybrid steepest-descent method, and the projection method, they proposed an iterative algorithm that generates a sequence via the explicit scheme and studied the convergence analysis of the sequences generated by the proposed method.
We consider and study the following triple hierarchical variational inequality problem.

Problem II. Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone on the nonempty, closed, and convex subset 𝐶 of ℋ, where 𝜅 and 𝜂 are positive constants. Let 𝐴∶𝐶→ℋ be a monotone and 𝐿-Lipschitzian mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping with Fix(𝑇)∩Γ≠∅. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Then the objective is to find 𝑥^∗∈Ξ such that 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ, (8) where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding 𝑧^∗∈Fix(𝑇)∩Γ such that 𝜇𝐹−𝛾𝑆𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇)∩Γ. (9)

We remark that Problem II is a generalization of Problem I. Indeed, in Problem II we put 𝑇=𝑇₁ and 𝐴=𝐼−𝑇₂, where 𝑇_𝑖∶𝐶→𝐶 is a 𝜁_𝑖-strictly pseudocontractive mapping for 𝑖=1,2. Then from the definition of strictly pseudocontractive mapping, it follows that 𝑇₂𝑥−𝑇₂≤𝑦,𝑥−𝑦𝑥−𝑦²−1−𝜁₂2𝐼−𝑇₂𝑥−𝐼−𝑇₂𝑦²,∀𝑥,𝑦∈𝐶. (10)

It is clear that the mapping 𝐴=𝐼−𝑇₂ is (1−𝜁₂)/2-inverse strongly monotone. Taking 𝐿=2/(1−𝜁₂), we know that 𝐴∶𝐶→ℋ is a monotone and 𝐿-Lipschitzian mapping. In this case, Γ=Fix(𝑇₂). Therefore, Problem II reduces to Problem I.

Motivated and inspired by Korpelevich’s extragradient method [23] and the iterative method proposed in [22], we propose the following multistep hybrid extragradient method for solving Problem II.

Algorithm I. Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone on the nonempty, closed, and convex subset 𝐶 of ℋ, 𝐴∶𝐶→ℋ be a monotone and 𝐿-Lipschitzian mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping. Let {𝛼_𝑛}⊂[0,∞), {𝜈_𝑛}⊂(0,1/𝐿), {𝛾_𝑛}⊂[0,1) and {𝛽_𝑛}, {𝛿_𝑛}, {𝜎_𝑛}, {𝜆_𝑛}⊂(0,1), 0<𝜇<2𝜂/𝜅², and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). The sequence {𝑥_𝑛} is generated by the following iterative scheme: 𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0, (11) where 𝐴_𝑛=𝛼_𝑛𝐼+𝐴 for all 𝑛≥0. In particular, if 𝑉≡0, then (11) reduces to the following iterative scheme: 𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛1−𝛿_𝑛𝛾𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0. (12)
Further, if 𝑆=𝑉, then (11) reduces to the following iterative scheme: 𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0; (13) moreover, if 𝑆=𝑉≡0, then (12) reduces to the following iterative scheme: 𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0. (14)
We prove that under appropriate conditions the sequence {𝑥_𝑛} generated by Algorithm I converges strongly to a unique solution of Problem II. Our result improves and extends Theorem  4.1 in [22] in the following aspects.(a)Problem II generalizes Problem I from the fixed point set Fix(𝑇) of a nonexpansive mapping 𝑇 to the intersection Fix(𝑇)∩Γ of the fixed point set of a strictly pseudocontractive mapping 𝑇 and the solution set Γ of VIP (2).(b)The Korpelevich extragradient algorithm is extended to develop the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving Problem II by virtue of the iterative schemes in Theorem  4.1 in [22].(c)The strong convergence of the sequence {𝑥_𝑛} generated by Algorithm I holds under the lack of the same restrictions as those in Theorem  4.1 in [22].(d)The boundedness requirement of the sequence {𝑥_𝑛} in Theorem  4.1 in [22] is replaced by the boundedness requirement of the sequence {𝑆𝑥_𝑛}.

We also consider and study the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving the following system of hierarchical variational inequalities (SHVI).

Problem III. Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone on the nonempty, closed, and convex subset 𝐶 of ℋ, where 𝜅 and 𝜂 are positive constants. Let 𝐴∶𝐶→ℋ be a monotone and 𝐿-Lipschitzian mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping with Fix(𝑇)∩Γ≠∅. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Then the objective is to find 𝑥^∗∈Fix(𝑇)∩Γ such that 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗(𝑇)𝑥≥0,∀𝑥∈Fix∩Γ,𝜇𝐹−𝛾𝑆^∗,𝑦−𝑥^∗≥0,∀𝑦∈Fix(𝑇)∩Γ. (15)

In particular, if 𝑇=𝑇₁ and 𝐴=𝐼−𝑇₂ where 𝑇_𝑖∶𝐶→𝐶 is 𝜁_𝑖-strictly pseudocontractive for 𝑖=1,2, Problem III reduces to the following Problem IV.

Problem IV. Let 𝐹∶𝐶→ℋ be 𝜅-Lipschitzian and 𝜂-strongly monotone on the nonempty, closed, and convex subset 𝐶 of ℋ, where 𝜅 and 𝜂 are positive constants. Let 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and, for 𝑖=1,2, 𝑇_𝑖∶𝐶→𝐶 be 𝜁_𝑖-strictly pseudocontractive mapping with Fix(𝑇₁)∩Fix(𝑇₂)≠∅. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Then the objective is to find 𝑥^∗∈Fix(𝑇₁)∩Fix(𝑇₂) such that 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix𝑇₁∩Fix𝑇₂,𝜇𝐹−𝛾𝑆𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈Fix𝑇₁∩Fix𝑇₂. (16)
We prove that under very mild conditions the sequence {𝑥_𝑛} generated by Algorithm I converges strongly to a unique solution of Problem III.

2. Preliminaries

Let 𝐶 be a nonempty, closed, and convex subset of ℋ and 𝑉∶𝐶→ℋ be a (possibly nonself) 𝜌-contraction mapping with coefficient 𝜌∈[0,1); that is, there exists a constant 𝜌∈[0,1) such that ‖𝑉𝑥−𝑉𝑦‖≤𝜌‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶. Now we present some known results and definitions which will be used in the sequel.

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). (17)

The following properties of projections are useful and pertinent to our purpose.

Proposition 1 (see [21]). Given any 𝑥∈ℋ and 𝑧∈𝐶. One has(i) 𝑧=𝑃_𝐶𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0,forall𝑦∈𝐶,(ii) 𝑧=𝑃_𝐶𝑥⇔‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖², for all 𝑦∈𝐶;(iii)⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖², for all 𝑥,𝑦∈ℋ, which hence implies that 𝑃_𝐶 is nonexpansive and monotone.

Definition 2. A mapping 𝑇∶ℋ→ℋ is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈ℋ; (18)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or, equivalently, 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈ℋ; (19)alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (20) where 𝑆∶ℋ→ℋ is nonexpansive; projections are firmly nonexpansive.

Definition 3. Let 𝑇 be a nonlinear operator whose domain is 𝐷(𝑇)⊆ℋ and whose range is 𝑅(𝑇)⊆ℋ.(a)𝑇 is said to be monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥0,∀𝑥,𝑦∈𝐷(𝑇). (21)(b)Given a number 𝛽>0, 𝑇 is said to be 𝛽-strongly monotone if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (22)(c)Given a number 𝜈>0, 𝑇 is said to be 𝜈-inverse strongly monotone (𝜈-ism) if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (23)
It can be easily seen that if 𝑇 is nonexpansive, then 𝐼−𝑇 is monotone. It is also easy to see that a projection 𝑃_𝐶 is 1-ism.
Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see [24, 25].

Definition 4. 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, (24) where 𝛼∈(0,1) and 𝑆∶ℋ→ℋ are nonexpansive. More precisely, when the last equality holds, we say that 𝑇 is 𝛼-averaged. Thus firmly nonexpansive mappings (in particular, projections) are 1/2-averaged maps.

Proposition 5 (see [26]). Let 𝑇∶ℋ→ℋ be a given mapping.(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.

Proposition 6 (see [26, 27]). Let 𝑆,𝑇,𝑉∶ℋ→ℋ be given operators.(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.(iv)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {𝑇_𝑖}^𝑁_𝑖=1 is averaged, then so is the composite 𝑇₁∘⋯∘𝑇_𝑁. In particular, if 𝑇₁ is 𝛼₁-averaged and 𝑇₂ is 𝛼₂-averaged, where 𝛼₁,𝛼₂∈(0,1), then the composite 𝑇₁∘𝑇₂ is 𝛼-averaged, where 𝛼=𝛼₁+𝛼₂−𝛼₁𝛼₂.

On the other hand, it is clear that, in a real Hilbert space ℋ, 𝑇∶𝐶→𝐶 is 𝜁-strictly pseudocontractive if and only if there holds the following inequality: 𝑇𝑥−𝑇𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝜁2(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (25)

This immediately implies that if 𝑇 is a 𝜁-strictly pseudocontractive mapping, then 𝐼−𝑇 is (1−𝜁)/2-inverse strongly monotone; for further detail, we refer to [21] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. The so-called demiclosedness principle for strict pseudocontractive mappings in the following lemma will often be used.

Lemma 7 (see [21, Proposition 2.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ and 𝑇∶𝐶→𝐶 be a mapping.(i)If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then 𝑇 satisfies the Lipschitz condition: 𝑇𝑥−𝑇𝑦≤1+𝜁1−𝜁𝑥−𝑦,∀𝑥,𝑦∈𝐶. (26)(ii)If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then the mapping 𝐼−𝑇 is semiclosed at 0; that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑥^∼ and (𝐼−𝑇)𝑥_𝑛→0, then (𝐼−𝑇)𝑥^∼=0.(iii)If 𝑇 is a 𝜁-(quasi-)strict pseudocontraction, then the fixed point set Fix(𝑇) of 𝑇 is closed and convex so that the projection 𝑃_Fix(𝑇) is well defined.

The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.

Lemma 8 (see [28]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property: 𝑎_𝑛+1≤1−𝑠_𝑛𝑎_𝑛+𝑠_𝑛𝑏_𝑛+𝑟_𝑛,𝑛≥0, (27) where {𝑠_𝑛}⊂(0,1] and {𝑏_𝑛} are such that(i)∑^∞_𝑛=0𝑠_𝑛=∞;(ii)either limsup_𝑛→∞𝑏_𝑛≤0 or ∑^∞_𝑛=0|𝑠_𝑛𝑏_𝑛|<∞;(iii)∑^∞_𝑛=0𝑟_𝑛<∞ where 𝑟_𝑛≥0, for all 𝑛≥0.
Then, lim_𝑛→∞𝑎_𝑛=0.

The following lemma is not hard to prove.

Lemma 9 (see [20]). Let 𝑉∶𝐶→ℋ be a 𝜌-contraction with 𝜌∈[0,1) and 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then(i)𝐼−𝑉 is (1−𝜌)-strongly monotone: (𝐼−𝑉)𝑥−(𝐼−𝑉)𝑦,𝑥−𝑦≥1−𝜌𝑥−𝑦²,∀𝑥,𝑦∈𝐶; (28)(ii)𝐼−𝑇 is monotone: (𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (29)

The following lemma plays an important role in proving strong convergence of the sequences generated by our algorithm.

Lemma 10 (see [29]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space ℋ. Let 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudo-contractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers such that (𝛾+𝛿)𝜁≤𝛾. Then 𝛾𝑥−𝑦+𝛿𝑇𝑥−𝑇𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (30)

Lemma 11 (see [30, Lemma 3.1]). Let 𝜆 be a number in (0,1] and let 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. (31)
Then 𝑇^𝜆 is a contraction provided that 𝜇<2𝜂/𝜅², that is, 𝑇^𝜆𝑥−𝑇^𝜆𝑦≤(1−𝜆𝜏)𝑥−𝑦,∀𝑥,𝑦∈𝐶, (32) where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²)∈(0,1]. In particular, if  𝑇 is the identity mapping 𝐼, then 𝐼−𝜆𝜇𝐹𝑥−𝐼−𝜆𝜇𝐹𝑦≤(1−𝜆𝜏)𝑥−𝑦,∀𝑥,𝑦∈𝐶. (33)

The following lemma appears implicitly in Reineermann [31].

Lemma 12. Let ℋ be a Hilbert space. Then 𝜆𝑥+(1−𝜆)𝑦−𝑧²=𝜆‖𝑥−𝑧‖²+(1−𝜆)𝑦−𝑧²−𝜆(1−𝜆)𝑥−𝑦²[].,∀𝑥,𝑦,𝑧∈ℋ,∀𝜆∈0,1 (34)

The following lemma is not difficult to prove.

Lemma 13 (see [32]). Let {𝛼_𝑛} and {𝛽_𝑛} be a sequence of nonnegative real numbers and a sequence of real numbers, respectively, such that limsup_𝑛→∞𝛼_𝑛<∞ and limsup_𝑛→∞𝛽_𝑛≤0. Then limsup_𝑛→∞𝛼_𝑛𝛽_𝑛≤0.

A set-valued mapping 𝑇^∼∶𝐻→2^𝐻 is called monotone if, for all 𝑥,𝑦∈𝐻,𝑓∈𝑇^∼𝑥, and 𝑔∈𝑇^∼𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑇^∼∶𝐻→2^𝐻 is maximal if its graph 𝐺(𝑇^∼) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇^∼ is maximal if and only if, for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for all (𝑦,𝑔)∈𝐺(𝑇^∼) implies 𝑓∈𝑇^∼𝑥. Let 𝐴∶𝐶→𝐻 be a monotone and 𝐿-Lipschitzian mapping, and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0, for all𝑢∈𝐶}. Define 𝑇^∼𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (35)

It is known that in this case 𝑇^∼ is maximal monotone, and 0∈𝑇^∼𝑣 if and only if 𝑣∈Γ; see [33].

3. Main Results

We are now in a position to present the convergence analysis of Algorithm I for solving Problem II.

Theorem 14. Let 𝐹∶𝐶→ℋ be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅,𝜂>0, respectively, 𝐴∶𝐶→ℋ be a 1/𝐿-inverse strongly monotone mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that the solution set Ξ of the HVIP (9) is nonempty and that the following conditions hold for the sequences {𝛼_𝑛}⊂[0,∞),{𝜈_𝑛}⊂(0,1/𝐿),{𝛾_𝑛}⊂[0,1) and {𝛽_𝑛},{𝛿_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)∑^∞_𝑛=0𝛼_𝑛<∞ and lim_𝑛→∞(𝛼_𝑛/𝜆²_𝑛)=0;(C2)0<liminf_𝑛→∞𝜈_𝑛≤limsup_𝑛→∞𝜈_𝑛<1/L;(C3)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛  for all 𝑛≥0;(C4)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝜎_𝑛>0;(C5)lim_𝑛→∞𝜆_𝑛=0,lim_𝑛→∞𝛿_𝑛=0 and ∑^∞_𝑛=0𝛿_𝑛𝜆_𝑛=∞;(C6)there are constants k,𝜃>0 satisfying ‖𝑥−𝑇𝑥‖≥𝑘[𝑑(𝑥,Fix(𝑇)∩Γ)]^𝜃 for each x∈C;(C7)lim_𝑛→∞(𝜆_𝑛^1/𝜃/𝛿_𝑛)=0.
One has the following.(a)If {𝑥_𝑛} is the sequence generated by scheme (11) and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to the point 𝑥^∗∈Fix(𝑇)∩Γ which is a unique solution of Problem II provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛).(b)If {𝑥_𝑛} is the sequence generated by the scheme (12) and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution 𝑥^∗ of the following VIP provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛): 𝑓𝑖𝑛𝑑𝑥^∗∈Ξ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ. (14)

Proof. We treat only case (a); that is, the sequence {𝑥_𝑛} is generated by the scheme (11). Obviously, from the condition Ξ≠∅ it follows that Fix(𝑇)∩Γ≠∅. In addition, in terms of conditions (C2) and (C4), without loss of generality, we may assume that {𝜈_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1/𝐿), {𝛽_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1).
First of all, we observe (see, e.g., [34]) that 𝑃_𝐶(𝐼−𝜈(𝛼𝐼+𝐴)) and 𝑃_𝐶(𝐼−𝜈_𝑛𝐴_𝑛) are nonexpansive for all 𝑛≥0.
Next we divide the remainder of the proof into several steps.
Step  1 ({𝑥_𝑛} is bounded). Indeed, take a fixed 𝑝∈Fix(𝑇)∩Γ arbitrarily. Then, we get 𝑇𝑝=𝑝 and 𝑃_𝐶(𝐼−𝜈𝐴)𝑝=𝑝 for 𝜈∈(0,2/𝐿). From (11), it follows that 𝑦_𝑛−𝑝=𝑃_𝐶𝐼−𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜈_𝑛𝐴𝑝≤𝑃_𝐶𝐼−𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝑃_𝐶𝐼−𝜈_𝑛𝐴_𝑛𝑝+𝑃_𝐶𝐼−𝜈_𝑛𝐴_𝑛𝑝−𝑃_𝐶𝐼−𝜈_𝑛𝐴𝑝≤𝑥_𝑛−𝑝+𝐼−𝜈_𝑛𝐴_𝑛𝑝−𝐼−𝜈_𝑛𝐴𝑝=𝑥_𝑛−𝑝+𝜈_𝑛𝛼_𝑛𝑝. (37)
Put 𝑡_𝑛=𝑃_𝐶(𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛) for each 𝑛≥0. Then, by Proposition 1 (ii), we have 𝑡_𝑛−𝑝²≤𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑝²−𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑡_𝑛²=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑡_𝑛²+2𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑝−𝑡_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑡_𝑛²+2𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝐴_𝑛𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑡_𝑛²+2𝜈_𝑛𝐴_𝑛𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑡_𝑛²+2𝜈_𝑛𝛼_𝑛𝐼+𝐴𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑡_𝑛²+2𝜈_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−2𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑡_𝑛−𝑦_𝑛−𝑡_𝑛²+2𝜈_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛+𝐴_𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+2𝜈_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛. (38)
Further, by Proposition 1 (i), we have 𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛=𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑡_𝑛−𝑦_𝑛≤𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑡_𝑛−𝑦_𝑛≤𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑦_𝑛𝑡_𝑛−𝑦_𝑛≤𝜈_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛. (39)
So, we obtain 𝑡_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+2𝜈_𝑛𝛼_𝑛𝑝,𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝜈_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+𝜈²_𝑛𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²+𝑦_𝑛−𝑡_𝑛²+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛=𝑥_𝑛−𝑝²+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑝²+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛. (40)
Since (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛, utilizing Lemmas 10 and 12, from (37) and the last inequality, we conclude that 𝑧_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛+𝜎_𝑛1𝛾_𝑛+𝜎_𝑛×𝛾_𝑛𝑡_𝑛−𝑝+𝜎_𝑛𝑇𝑡_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛+𝜎_𝑛×1𝛾_𝑛+𝜎_𝑛𝛾_𝑛𝑡_𝑛−𝑝+𝜎_𝑛𝑇𝑡_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛+𝜎_𝑛×1𝛾_𝑛+𝜎_𝑛𝛾_𝑛𝑡_𝑛−𝑥_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑡_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×𝑥_𝑛−𝑝²+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²+2𝜈_𝑛𝛼_𝑛𝑝𝑝−𝑦_𝑛+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²+2𝜈_𝑛𝛼_𝑛𝑝𝑥_𝑛−𝑝+𝜈_𝑛𝛼_𝑛𝑝+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²+2𝑥_𝑛−𝑝√2𝜈_𝑛𝛼_𝑛𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²=𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝². (41)
Noticing the boundedness of {𝑆𝑥_𝑛}, we get sup_𝑛≥0‖𝛾𝑆𝑥_𝑛−𝜇𝐹𝑝‖≤𝑀 for some 𝑀≥0. Moreover, utilizing Lemma 11 we have from (11) 𝑥_𝑛+1=𝑃−𝑝_𝐶𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝑃_𝐶𝑝≤𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛=𝜆−𝑝_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝜆_𝑛𝜇𝐹𝑝+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝑝≤𝜆𝜇𝐹_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝜆_𝑛+𝜇𝐹𝑝𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑝=𝜆_𝑛𝛿_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+1−𝛿_𝑛𝛾𝑆𝑥_𝑛+−𝜇𝐹𝑝𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑝≤𝜆_𝑛𝛿_𝑛𝛾𝑉𝑥_𝑛+−𝜇𝐹𝑝1−𝛿_𝑛𝛾𝑆𝑥_𝑛+−𝜇𝐹𝑝1−𝜆_𝑛𝜏𝑧_𝑛−𝑝≤𝜆_𝑛𝛿_𝑛𝛾𝑉𝑥_𝑛−𝛾𝑉𝑝+𝛾𝑉𝑝−𝜇𝐹𝑝+1−𝛿_𝑛𝑀+1−𝜆_𝑛𝜏𝑧_𝑛−𝑝≤𝜆_𝑛𝛿_𝑛𝛾𝜌𝑥_𝑛−𝑝+𝛿_𝑛𝛾𝑉𝑝−𝜇𝐹𝑝+1−𝛿_𝑛𝑀+1−𝜆_𝑛𝜏𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝≤𝜆_𝑛𝛿_𝑛𝛾𝜌𝑥_𝑛+−𝑝+max𝑀,𝛾𝑉𝑝−𝜇𝐹𝑝1−𝜆_𝑛𝜏𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝≤𝜆_𝑛𝛾𝜌𝑥_𝑛−𝑝+𝜆_𝑛+max𝑀,𝛾𝑉𝑝−𝜇𝐹𝑝1−𝜆_𝑛𝜏𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝=1−𝜏−𝛾𝜌𝜆_𝑛𝑥_𝑛−𝑝+𝜆_𝑛max𝑀,𝛾𝑉𝑝−𝜇𝐹𝑝+√2𝜈_𝑛𝛼_𝑛𝑝. (42)
So, calling 𝑀^∼=max𝑥₀,𝑀−𝑝,𝜏−𝛾𝜌𝛾𝑉𝑝−𝜇𝐹𝑝,𝜏−𝛾𝜌 (43)
we claim that 𝑥_𝑛+1−𝑝≤𝑀^∼+^𝑛_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝,∀𝑛≥0. (44)
Indeed, when 𝑛=0, it is clear from (42) that (44) is valid, that is, 𝑥₁−𝑝≤𝑀^∼+⁰_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝. (45)
Assume that (44) is valid for 𝑛(≥1), that is, 𝑥_𝑛−𝑝≤𝑀^∼+^𝑛−1_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝. (46)
Then from (42) and (46) it follows that 𝑥_𝑛+1−𝑝≤1−𝜏−𝛾𝜌𝜆_𝑛𝑥_𝑛−𝑝+𝜆_𝑛max𝑀,𝛾𝑉𝑝−𝜇𝐹𝑝+√2𝜈_𝑛𝛼_𝑛𝑝≤1−𝜏−𝛾𝜌𝜆𝑀_𝑛^∼+^𝑛−1_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝+𝜆_𝑛max𝑀,𝛾𝑉𝑝−𝜇𝐹𝑝+√2𝜈_𝑛𝛼_𝑛𝑝≤1−𝜏−𝛾𝜌𝜆𝑀_𝑛^∼+^𝑛−1_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝+𝜏−𝛾𝜌𝜆_𝑛𝑀max,𝜏−𝛾𝜌𝛾𝑉𝑝−𝜇𝐹𝑝+√𝜏−𝛾𝜌2𝜈_𝑛𝛼_𝑛𝑝≤𝑀^∼+^𝑛_𝑗=0√2𝜈_𝑗𝛼_𝑗𝑝. (47)
This shows that (44) is also valid for 𝑛+1. Hence, by induction we derive the claim. Consequently, {𝑥_𝑛} is bounded (due to ∑^∞_𝑛=0𝛼_𝑛<∞) and so are {𝑦_𝑛},{𝑧_𝑛},{𝐴𝑥_𝑛}, and {𝐴𝑦_𝑛}.
Step  2 (lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑡_𝑛‖=lim_𝑛→∞‖𝑡_𝑛−𝑇𝑡_𝑛‖=0). Indeed, from (11) and (41), it follows that 𝑥_𝑛+1−𝑝²≤𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝑝²=𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝜆_𝑛𝜇𝐹𝑝+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑝²≤𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝜆_𝑛+𝜇𝐹𝑝𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑝²≤𝜆_𝑛𝛿_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+1−𝛿_𝑛𝛾𝑆𝑥_𝑛+−𝜇𝐹𝑝1−𝜆_𝑛𝜏𝑧_𝑛−𝑝²≤𝜆_𝑛1𝜏𝛿_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+1−𝛿_𝑛𝛾𝑆𝑥_𝑛−𝜇𝐹𝑝²+1−𝜆_𝑛𝜏𝑧_𝑛−𝑝²≤𝜆_𝑛1𝜏𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+𝛾𝑆𝑥_𝑛−𝜇𝐹𝑝²+𝑧_𝑛−𝑝²≤𝜆_𝑛1𝜏𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝+𝛾𝑆𝑥_𝑛−𝜇𝐹𝑝²+𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+𝜆_𝑛𝑀₁+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛², (48) where 𝑀₁=sup_𝑛≥0{(1/𝜏)[‖𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝‖+‖𝛾𝑆𝑥_𝑛−𝜇𝐹𝑝‖]²}. This together with {𝜈_𝑛}⊂[𝑎,𝑏]⊂(0,1/𝐿) and {𝛽_𝑛}⊂[𝑐,𝑑]⊂(0,1) implies that (1−𝑑)1−𝑏²𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²+𝑐𝑧1−𝑐^𝑛−𝑥_𝑛²≤1−𝛽_𝑛1−𝜈²_𝑛𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²+𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²−𝑥_𝑛+1−𝑝²+𝜆_𝑛𝑀₁=𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝−𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝+𝑥_𝑛+1−𝑝+𝜆_𝑛𝑀₁≤𝑥_𝑛+1−𝑥_𝑛+√2𝜈_𝑛𝛼_𝑛𝑝×𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝+𝜆_𝑛𝑀₁≤𝑥_𝑛+1−𝑥_𝑛+√2𝑏𝛼_𝑛𝑝×𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝+√2𝑏𝛼_𝑛𝑝+𝜆_𝑛𝑀₁. (49)
Note that lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝜆_𝑛=0. Hence, taking into account the boundedness of {𝑥_𝑛} and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, we deduce from (49) that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (50)
Furthermore, we obtain 𝑦_𝑛−𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛≤𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛=𝜈_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑦_𝑛≤𝜈_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛, (51) which together with (50) implies that lim_𝑛→∞𝑦_𝑛−𝑡_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=0. (52)
So, from (11) we get 𝜎_𝑛𝑇𝑡_𝑛−𝑥_𝑛=𝑧_𝑛−𝑥_𝑛−𝛾_𝑛𝑡_𝑛−𝑥_𝑛≤𝑧_𝑛−𝑥_𝑛+𝛾_𝑛𝑡_𝑛−𝑥_𝑛⟶0, (53) which together with liminf_𝑛→∞𝜎_𝑛>0 implies that lim_𝑛→∞𝑇𝑡_𝑛−𝑥_𝑛=0. (54)
Note that 𝑡_𝑛−𝑇𝑡_𝑛≤𝑡_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑡_𝑛. (55)
This together with (50)–(54) implies that lim_𝑛→∞𝑡_𝑛−𝑇𝑡_𝑛=0. (56)
Step  3 (𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑇)∩Γ). Indeed, since 𝐴 is 𝐿-Lipschitz continuous, we have lim_𝑛→∞𝐴𝑦_𝑛−𝐴𝑡_𝑛=0. (57)
As {𝑥_𝑛} is bounded, there is a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} that converges weakly to some 𝑥. By the same argument as that in [34], we can obtain that 𝑥∈Fix(𝑇)∩Γ from which it follows that 𝜔_𝑤𝑥_𝑛⊂Fix(𝑇)∩Γ. (58)
Step  4 (𝜔_𝑤(𝑥_𝑛)⊂Ξ). Indeed, we first note that 0<𝛾≤𝜏 and 𝜇𝜂≥𝜏⟺𝜇𝜂≥1−1−𝜇2𝜂−𝜇𝜅²⟺1−𝜇2𝜂−𝜇𝜅²≥1−𝜇𝜂⟺1−2𝜇𝜂+𝜇²𝜅²≥1−2𝜇𝜂+𝜇²𝜂²⟺𝜅²≥𝜂²⟺𝜅≥𝜂. (59)
It is clear that ≥𝜇𝐹−𝛾𝑆𝑥−𝜇𝐹−𝛾𝑆𝑦,𝑥−𝑦𝜇𝜂−𝛾𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (60)
Hence, it follows from 0<𝛾≤𝜏≤𝜇𝜂 that 𝜇𝐹−𝛾𝑆 is monotone. Putting 𝑤_𝑛=𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+I−𝜆_𝑛𝜇𝐹𝑇𝑧_𝑛,∀𝑛≥0, (61)
and noticing from (11) 𝑥_𝑛+1=𝑃_𝐶𝑤_𝑛−𝑤_𝑛+𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛, (62)
we obtain 𝑥_𝑛−𝑥_𝑛+1=𝑤_𝑛−𝑃_𝐶𝑤_𝑛+𝛿_𝑛𝜆_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛+𝜆_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛. (63)
Set 𝑒_𝑛=𝑥_𝑛−𝑥_𝑛+1𝜆_𝑛1−𝛿_𝑛,∀𝑛≥0. (64)
It can be easily seen from (63) that 𝑒_𝑛=𝑤_𝑛−𝑃_𝐶𝑤_𝑛𝜆_𝑛1−𝛿_𝑛+𝜇𝐹−𝛾𝑆𝑥_𝑛+𝛿_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛𝜆_𝑛1−𝛿_𝑛. (65)
This yields that, for all 𝑤∈Fix(𝑇)∩Γ (noticing 𝑥_𝑛=𝑃_𝐶𝑤_𝑛−1), 𝑒_𝑛,𝑥_𝑛=1−𝑤𝜆_𝑛1−𝛿_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1+𝑥−𝑤𝜇𝐹−𝛾𝑆_𝑛,𝑥_𝑛+𝛿−𝑤_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛,𝑥_𝑛+1−𝑤𝜆_𝑛1−𝛿_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,𝑥_𝑛=1−𝑤𝜆_𝑛1−𝛿_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛+1−𝑤𝜆_𝑛1−𝛿_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛+𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛+𝑥−𝑤𝜇𝐹−𝛾𝑆_𝑛−𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛+𝛿−𝑤_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛,𝑥_𝑛+1−𝑤𝜆_𝑛1−𝛿_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,𝑥_𝑛−𝑤. (66)
In (66), the first term is nonnegative due to Proposition 1, and the fourth term is also nonnegative due to the monotonicity of 𝜇𝐹−𝛾𝑆. We, therefore, deduce from (66) that (noticing again 𝑥_𝑛+1=𝑃_𝐶𝑤_𝑛) 𝑒_𝑛,𝑥_𝑛≥1−𝑤𝜆_𝑛1−𝛿_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛+𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛+𝛿−𝑤_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛,𝑥_𝑛+1−𝑤𝜆_𝑛1−𝛿_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,𝑥_𝑛=𝑤−𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑒_𝑛+𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛+𝛿−𝑤_𝑛1−𝛿_𝑛𝜇𝐹−𝛾𝑉𝑥_𝑛,𝑥_𝑛+1−𝑤𝜆_𝑛1−𝛿_𝑛𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,𝑥_𝑛−𝑤. (67)
Note that 𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛≤1−𝜆_𝑛𝜏𝑥_𝑛−𝑧_𝑛. (68)
Hence it follows from ‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆_𝑛) that 𝐼−𝜆_𝑛𝜇𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛𝜆_𝑛⟶0. (69)
Also, since 𝑒_𝑛→0 (due to ‖𝑥_𝑛+1−𝑥_𝑛‖=𝑜(𝜆_𝑛)), 𝛿_𝑛→0 and {𝑥_𝑛} is bounded by Step 1 which implies that {𝑤_𝑛} is bounded, we obtain from (67) that limsup_𝑛→∞𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛−𝑤≤0,∀𝑤∈Fix(𝑇)∩Γ. (70)
This suffices to guarantee that 𝜔_𝑤(𝑥_𝑛)⊂Ξ; namely, every weak limit point of {𝑥_𝑛} solves the HVIP (9). As a matter of fact, if 𝑥_𝑛𝑖⇀𝑥^∼∈𝜔_𝑤(𝑥_𝑛) for some subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛}, then we deduce from (70) that 𝜇𝐹−𝛾𝑆𝑤,𝑥^∼−𝑤≤limsup_𝑛→∞𝜇𝐹−𝛾𝑆𝑤,𝑥_𝑛(𝑇)−𝑤≤0,∀𝑤∈Fix∩Γ, (71)
that is, 𝜇𝐹−𝛾𝑆𝑤,𝑤−𝑥^∼≥0,∀𝑤∈Fix(𝑇)∩Γ. (72)
In addition, note that 𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑇)∩Γ by Step 3. Since 𝜇𝐹−𝛾𝑆 is monotone and Lipschitz continuous and Fix(𝑇)∩Γ is nonempty, closed, and convex, by the Minty lemma [1] the last inequality is equivalent to (9). Thus, we get 𝑥^∼∈Ξ.
Step  5 ({𝑥_𝑛} converges strongly to a unique solution 𝑥^∗ of Problem II.) Indeed, we now take a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} satisfying limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝜇𝐹−𝛾𝑉𝑥^∗,𝑥_𝑛𝑖−𝑥^∗. (73)
Without loss of generality, we may further assume that 𝑥_𝑛𝑖⇀𝑥^∼; then 𝑥^∼∈Ξ as we just proved. Since 𝑥^∗ is a solution of the THVIP (8), we get limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥^∗,𝑥_𝑛−𝑥^∗=𝜇𝐹−𝛾𝑉𝑥^∗,𝑥^∼−𝑥^∗≥0. (74)
From (11) and (41), it follows that (noticing that 𝑥_𝑛+1=𝑃_𝐶𝑤_𝑛 and 0<𝛾≤𝜏) 𝑥_𝑛+1−𝑥^∗2=𝑤_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝑃_𝐶𝑤_𝑛−𝑤_𝑛,𝑃_𝐶𝑤_𝑛−𝑥^∗≤𝑤_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝐼−𝜆_𝑛𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛿_𝑛𝜆_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆𝑥_𝑛−𝑆𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝜆_𝑛𝜏𝑧_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛿_𝑛𝜆_𝑛𝛾𝜌+𝜆_𝑛1−𝛿_𝑛𝛾×𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝜆_𝑛𝜏12𝑧_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝜌+𝜆_𝑛1−𝛿_𝑛𝛾×12𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝜆_𝑛𝜏12𝑥_𝑛−𝑥^∗+√2𝜈_𝑛𝛼_𝑛𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝜌+𝜆_𝑛1−𝛿_𝑛𝛾×12𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝜆_𝑛𝜏12𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝑀₂+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝜌+𝜆_𝑛1−𝛿_𝑛𝛾×12𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛿_𝑛𝜆_𝑛𝛾×11−𝜌2𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛿_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑀₂, (75) where 𝑀₂=sup_𝑛≥0{2𝜈_𝑛‖𝑥^∗‖(√2‖𝑥_𝑛−𝑥^∗‖+𝜈_𝑛𝛼_𝑛‖𝑥^∗‖)}<∞. It turns out that 𝑥_𝑛+1−𝑥^∗2≤1−𝛿_𝑛𝜆_𝑛𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾𝑥1−𝜌^𝑛−𝑥^∗2+21+𝛿_𝑛𝜆_𝑛𝛾×𝛿1−𝜌_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛×𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑀₂≤1−𝛿_𝑛𝜆_𝑛𝛾1−𝜌𝑥_𝑛−𝑥^∗2+21+𝛿_𝑛𝜆_𝑛𝛾×𝛿1−𝜌_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝑀₂. (76)
However, from 𝑥^∗∈Ξ and condition (C6) we obtain that 𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑃_{Fix(𝑇)∩Γ}𝑥_𝑛+1+𝛾𝑆−𝜇𝐹𝑥^∗,𝑃_{Fix(𝑇)∩Γ}𝑥_𝑛+1−𝑥^∗≤𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑃_{Fix(𝑇)∩Γ}𝑥_𝑛+1≤𝛾𝑆−𝜇𝐹𝑥^∗𝑑𝑥_𝑛+1(𝑇)≤𝑥,Fix∩Γ𝛾𝑆−𝜇𝐹^∗1𝑘𝑥_𝑛+1−𝑇𝑥_𝑛+1^1/𝜃. (77)
On the other hand, from (41) we have 𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²+1−𝛽_𝑛𝜈²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛², (78) which, together with ‖𝑥_𝑛−𝑧_𝑛‖+𝛼_𝑛=o(𝜆²_𝑛), implies that (1−𝑑)1−𝑏²𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²𝜆²_𝑛≤1−𝛽_𝑛1−𝜈²_𝑛𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²𝜆²_𝑛≤𝑥_𝑛−𝑝+√2𝜈_𝑛𝛼_𝑛𝑝²−𝑧_𝑛−𝑝²𝜆²_𝑛≤𝑥_𝑛−𝑧_𝑛+√2𝑏𝛼_𝑛𝑝𝜆²_𝑛×𝑥_𝑛−𝑝+𝑧_𝑛−𝑝+√2𝑏𝛼_𝑛𝑝⟶0as𝑛⟶∞. (79)
That is, ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝜆_𝑛). Observe that 𝑧_𝑛−𝑥_𝑛=𝛾_𝑛𝑡_𝑛−𝑥_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑥_𝑛=𝛾_𝑛𝑡_𝑛−𝑥_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑇𝑥_𝑛+𝜎_𝑛𝑇𝑥_𝑛−𝑥_𝑛. (80)
Since (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛, utilizing Lemma 10 we get 𝜎_𝑛𝑇𝑥_𝑛−𝑥_𝑛𝜆_𝑛=𝑧_𝑛−𝑥_𝑛−𝛾_𝑛𝑡_𝑛−𝑥_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑇𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+𝛾_𝑛𝑡_𝑛−𝑥_𝑛+𝜎_𝑛𝑇𝑡_𝑛−𝑇𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+𝑡_𝑛−𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+𝑡_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛𝜆_𝑛=𝑧_𝑛−𝑥_𝑛+𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛𝜆_𝑛+𝑦_𝑛−𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+𝜈_𝑛𝐴_𝑛𝑦_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛+𝑦_𝑛−𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+𝜈_𝑛𝛼_𝑛+𝐿𝑦_𝑛−𝑥_𝑛+𝑦_𝑛−𝑥_𝑛𝜆_𝑛≤𝑧_𝑛−𝑥_𝑛+2𝑦_𝑛−𝑥_𝑛𝜆_𝑛⟶0as𝑛⟶∞. (81)
That is, ‖𝜎_𝑛(𝑇𝑥_𝑛−𝑥_𝑛)‖=𝑜(𝜆_𝑛). Taking into account liminf_𝑛→∞𝜎_𝑛>0, we have ‖𝑥_𝑛−𝑇𝑥_𝑛‖=𝑜(𝜆_𝑛). Furthermore, utilizing Lemma 7 (i), we have 𝑥_𝑛+1−𝑇𝑥_𝑛+1≤𝑥_𝑛+1−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑇𝑥_𝑛+1≤1+𝜁𝑥1−𝜁^𝑛−𝑥_𝑛+1+𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛−𝑇𝑥_𝑛≤1+𝜁𝑥1−𝜁^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑇𝑥_𝑛+𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝜇𝐹𝑧_𝑛≤1+𝜁𝑥1−𝜁^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛+𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛−𝑆𝑥_𝑛+𝛾𝑆𝑥_𝑛−𝜇𝐹𝑧_𝑛≤1+𝜁𝑥1−𝜁^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛+𝜆_𝑛𝑀₀, (82) where 𝑀₀=sup_𝑛≥0‖𝛾𝛿_𝑛(𝑉𝑥_𝑛−𝑆𝑥_𝑛)+𝛾𝑆𝑥_𝑛−𝜇𝐹𝑧_𝑛‖<∞. Hence, for a big enough constant 𝑘₁>0, from (77), we have 𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑘₁𝜆_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛^1/𝜃≤𝑘₁𝜆_𝑛^1/𝜃𝑥1+^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛𝜆_𝑛^1/𝜃. (83)
Combining (76)–(83), we get 𝑥_𝑛+1−𝑥^∗2≤1−𝛿_𝑛𝜆_𝑛𝛾1−𝜌𝑥_𝑛−𝑥^∗2+21+𝛿_𝑛𝜆_𝑛𝛾×𝛿1−𝜌_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝑀₂≤1−𝛿_𝑛𝜆_𝑛𝛾1−𝜌𝑥_𝑛−𝑥^∗2+2𝛿_𝑛𝜆_𝑛1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑉−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗+𝑘₁𝜆_𝑛^1/𝜃𝛿_𝑛×𝑥1+^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛𝜆_𝑛^1/𝜃+2𝛼_𝑛𝑀₂=1−𝑠_𝑛𝑥_𝑛−𝑥^∗2+𝜇_𝑛+2𝛼_𝑛𝑀₂, (84) where 𝑠_𝑛=𝛿_𝑛𝜆_𝑛𝛾(1−𝜌) and 𝜇_𝑛=2𝛿_𝑛𝜆_𝑛1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑉−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗+𝑘₁𝜆_𝑛^1/𝜃𝛿_𝑛×𝑥1+^𝑛−𝑥_𝑛+1+𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇𝑥_𝑛𝜆_𝑛^1/𝜃. (85)
Now condition (C5) implies that ∑^∞_𝑛=0𝑠_𝑛=∞. Moreover, since ‖𝑥_𝑛+1−𝑥_𝑛‖ + ‖𝑧_𝑛−𝑥_𝑛‖ + ‖𝑥_𝑛−𝑇𝑥_𝑛‖ = 𝑜(𝜆_𝑛), conditions (C7) and (74) imply that limsup_𝑛→∞𝜇_𝑛𝑠_𝑛≤0. (86)
Therefore, we can apply Lemma 8 to (84) to conclude that 𝑥_𝑛→𝑥^∗. The proof of part (a) is complete. It is easy to see that part (b) now becomes a straightforward consequence of part (a) since, if 𝑉=0, THVIP (8) reduces to the VIP in part (b). This completes the proof.

Next we consider a special case of Problem II. In Problem II, put 𝜇=2,𝐹=(1/2)𝐼 and 𝛾=𝜏=1. In this case, the objective is to find 𝑥^∗∈Ξ such that (𝐼−𝑉)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ, (87) where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding 𝑧^∗∈Fix(𝑇)∩Γ such that (𝐼−𝑆)𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇)∩Γ. (88)

Corollary 15. Let 𝐴 be a 1/L-inverse strongly monotone mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping. Assume that the solution set Ξ of the HVIP (88) is nonempty and that the following conditions hold for the sequences {𝛼_𝑛}⊂[0,∞),{𝜈_𝑛}⊂(0,1/L),{𝛾_𝑛}⊂[0,1), and {𝛽_𝑛},{𝛿_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)∑^∞_𝑛=0𝛼_𝑛<∞ and lim_𝑛→∞(𝛼_𝑛/𝜆²_𝑛)=0;(C2)0<liminf_𝑛→∞𝜈_𝑛≤limsup_𝑛→∞𝜈_𝑛<1/𝐿;(C3)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0;(C4)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝜎_𝑛>0;(C5)lim_𝑛→∞𝜆_𝑛=0,lim_𝑛→∞𝛿_𝑛=0 and ∑^∞_𝑛=0𝛿_𝑛𝜆_𝑛=∞;(C6)there are constants 𝑘,𝜃>0 satisfying ‖𝑥−𝑇𝑥‖≥𝑘[𝑑(𝑥,Fix(𝑇)∩Γ)]^𝜃 for each 𝑥∈C;(C7)lim_𝑛→∞(𝜆_𝑛^1/𝜃/𝛿_𝑛)=0.
One has(a)If {𝑥_𝑛} is the sequence generated by the iterative scheme 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+1−𝜆_𝑛𝑧_𝑛,∀𝑛≥0 (89) and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to the point 𝑥^∗∈Fix(𝑇)∩Γ which is a unique solution of THVIP (87) provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛).(b)If {𝑥_𝑛} is the sequence generated by the iterative scheme 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛1−𝛿_𝑛𝑆𝑥_𝑛+1−𝜆_𝑛𝑧_𝑛,∀𝑛≥0 (90)
and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution 𝑥^∗ of the following VIP provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛): 𝑓𝑖𝑛𝑑𝑥^∗∈Ξ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ; (91)
that is, 𝑥^∗ is the minimum-norm solution of HVIP (88).

Furthermore, applying Theorem 14 to Problem I, we get the result as below.

Corollary 16. Let 𝐹∶𝐶→ℋ be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅,𝜂>0, respectively, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇_𝑖∶𝐶→𝐶 be a 𝜁_𝑖-strictly pseudocontractive mapping for 𝑖=1,2. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that the solution set Ξ of the HVIP in Problem I is nonempty and that the following conditions hold for the sequences {𝜈_𝑛}⊂(0,(1−𝜁₂)/2),{𝛾_𝑛}⊂[0,1) and {𝛽_𝑛},{𝛿_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)0<liminf_𝑛→∞𝜈_𝑛≤limsup_𝑛→∞𝜈_n<(1−𝜁₂)/2;(C2)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0;(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝜎_𝑛>0;(C4)lim_𝑛→∞𝜆_𝑛=0,lim_𝑛→∞𝛿_𝑛=0 and ∑^∞_𝑛=0𝛿_𝑛𝜆_𝑛=∞;(C5)there are constants 𝑘,𝜃>0 satisfying ‖𝑥−𝑇₁𝑥‖≥𝑘[𝑑(𝑥,Fix(𝑇₁)∩Fix(𝑇₂))]^𝜃 for each 𝑥∈C;(C6)lim_𝑛→∞(𝜆_𝑛^1/𝜃/𝛿_𝑛)=0.
One has the following.(a)If {𝑥_𝑛} is the sequence generated by 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0, (92) such that {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to the point 𝑥^∗∈Fix(𝑇₁)∩Fix(𝑇₂) which is a unique solution of Problem I provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛).(b)If {𝑥_𝑛} is the sequence generated by 𝑥₀𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑥_𝑛,𝑧_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑦_𝑛+𝜎_𝑛𝑇𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛1−𝛿_𝑛𝛾𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛,∀𝑛≥0, (93)
such that {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution x^∗ of the following VIP provided that ‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑧_𝑛‖=𝑜(𝜆²_𝑛): 𝑓𝑖𝑛𝑑𝑥^∗∈Ξ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡F𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ. (94)

Proof. In Theorem 14, we put 𝑇=𝑇₁ and 𝐴=𝐼−𝑇₂ where 𝑇_𝑖∶𝐶→𝐶 is 𝜁_𝑖-strictly pseudocontractive for 𝑖=1,2. Taking 𝐿=2/(1−𝜁₂) and 𝛼_𝑛=0 for all 𝑛≥0, we know that 𝐴∶𝐶→ℋ is a 1/𝐿-inverse strongly monotone mapping such that Γ=Fix(𝑇₂). In the scheme (11), we have 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜈_𝑛𝐴_𝑛𝑥_𝑛=𝑃_𝐶1−𝜈_𝑛𝑥_𝑛+𝜈_𝑛𝑇₂𝑥_𝑛=𝑥_𝑛−𝜈_𝑛𝐼−𝑇₂𝑥_𝑛. (95)
Utilizing Theorem 14, we obtain desired result.

On the other hand, we also derive the following strong convergence result of Algorithm I for finding a unique solution of Problem III.

Theorem 17. Let 𝐹∶𝐶→ℋ be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅,𝜂>0, respectively, A be a 1/𝐿-inverse strongly monotone mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Problem III has a solution and that the following conditions hold for the sequences {𝛼_𝑛}⊂(0,∞),{𝜈_𝑛}⊂(0,1/𝐿),{𝛾_𝑛}⊂[0,1) and {𝛽_𝑛},{𝛿_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)∑^∞_𝑛=0𝛼_𝑛<∞;(C2)0<liminf_𝑛→∞𝜈_n≤limsup_𝑛→∞𝜈_𝑛<1/L;(C3)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛  for all 𝑛≥0;(C4)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_n<1 and liminf_𝑛→∞𝜎_𝑛>0;(C5)0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1;(C6)lim_𝑛→∞𝜆_𝑛=0 and ∑^∞_𝑛=0𝜆_𝑛=∞.
One has the following.(a)If {𝑥_𝑛} is the sequence generated by the scheme (11) and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution of Problem III provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.(b)If {𝑥_𝑛} is the sequence generated by the scheme (12) and {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution 𝑥^∗∈Fix(𝑇)∩Γ of the following system of variational inequalities provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0: 𝐹𝑥^∗,𝑥−𝑥^∗(𝑇)𝑥≥0,∀𝑥∈Fix∩Γ,𝜇𝐹−𝛾𝑆^∗,𝑦−𝑥^∗≥0,∀𝑦∈Fix(𝑇)∩Γ. (96)

Proof. We treat only case (a); that is, the sequence {𝑥_𝑛} is generated by scheme (11). First of all, it is seen easily that 0<𝛾≤𝜏 and 𝜅≥𝜂⇔𝜇𝜂≥𝜏. Hence it follows from the 𝜌-contractiveness of 𝑉 and 𝛾𝜌<𝛾≤𝜏≤𝜇𝜂 that 𝜇𝐹−𝛾𝑉 is (𝜇𝜂−𝛾𝜌)-strongly monotone and Lipschitz continuous. So, there exists a unique solution 𝑥^∗ of the following VIP: find𝑥^∗∈Fix(𝑇)∩Γsuchthat𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗(𝑇)≥0,∀𝑥∈Fix∩Γ. (97)
Consequently, it is easy to see that Problem III has a unique solution 𝑥^∗∈Fix(𝑇)∩Γ. In addition, taking into account condition (C5), without loss of generality we may assume that {𝛿_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1).
Next we divide the rest of the proof into several steps.
Step  1 ({𝑥_𝑛} is bounded). Indeed, repeating the same argument as in Step 1 of the proof of Theorem 14 we can derive the claim.
Step  2 (lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑡_𝑛‖=lim_𝑛→∞‖𝑡_𝑛−𝑇𝑡_𝑛‖=0). Indeed, repeating the same argument as in Step 2 of the proof of Theorem 14 we can derive the claim.
Step  3 (𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑇)∩Γ). Indeed, repeating the same argument as in Step 3 of the proof of Theorem 14 we can derive the claim.
Step  4 ({𝑥_𝑛} converges strongly to a unique solution 𝑥^∗ of Problem III). Indeed, according to ‖𝑥_𝑛+1−𝑥_𝑛‖→0, we can take a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} satisfying limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛𝑖−𝑥^∗. (98)
Without loss of generality, we may further assume that 𝑥_𝑛𝑖⇀𝑥^∼; then 𝑥^∼∈Fix(𝑇)∩Γ due to Step 3. Since 𝑥^∗ is a solution of Problem III, we get limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝛾𝑉−𝜇𝐹𝑥^∗,𝑥^∼−𝑥^∗≤0. (99)
Repeating the same argument as that of (99), we have limsup_𝑛→∞⟨𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩≤0. (100)
Repeating the same argument as that of (76) in the proof of Theorem 14, we obtain 𝑥_𝑛+1−𝑥^∗2≤1−𝛿_𝑛𝜆_𝑛𝛾1−𝜌𝑥_𝑛−𝑥^∗2+21+𝛿_𝑛𝜆_𝑛𝛾×𝛿1−𝜌_𝑛𝜆_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝜆_𝑛1−𝛿_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛼_𝑛𝑀₂. (101)
Put 𝑟_𝑛=2𝛼_𝑛𝑀₂,𝑠_𝑛=𝛿_𝑛𝜆_𝑛𝛾(1−𝜌) and 𝑏_𝑛=2𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑉−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗+21−𝛿_𝑛𝛿_𝑛𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑆−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗. (102)
Then (101) is rewritten as 𝑥_𝑛+1−𝑥^∗2≤1−𝑠_𝑛𝑥_𝑛−𝑥^∗2+𝑠_𝑛𝑏_𝑛+𝑟_𝑛. (103)
In terms of conditions (C5) and (C6), we conclude from 0<1−𝜌≤1 that 𝑠_𝑛⊂(0,1],^∞_𝑛=0𝑠_𝑛=∞. (104)
Note that 2𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾≤21−𝜌𝛾,21−𝜌1−𝛿_𝑛𝛿_𝑛𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾≤21−𝜌.𝑎𝛾1−𝜌 (105)
Consequently, utilizing Lemma 13 we obtain that limsup_𝑛→∞𝑏_𝑛≤limsup_𝑛→∞2𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑉−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗+limsup_𝑛→∞21−𝛿_𝑛𝛿_𝑛𝛾1−𝜌1+𝛿_𝑛𝜆_𝑛𝛾×𝑥1−𝜌𝛾𝑆−𝜇𝐹^∗,𝑥_𝑛+1−𝑥^∗≤0. (106)
So, this together with Lemma 8 leads to lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. The proof is complete.

Utilizing Theorem 17 we immediately derive the following result.

Corollary 18. Let 𝐹∶𝐶→ℋ be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅,𝜂>0, respectively, A be a 1/L-inverse strongly monotone mapping, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), and 𝑇∶𝐶→𝐶 be a 𝜁-strictly pseudocontractive mapping such that Fix(𝑇)∩Γ≠∅. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that the following conditions hold for the sequences {𝛼_𝑛}⊂(0,∞),{𝜈_𝑛}⊂(0,1/L),{𝛾_𝑛}⊂[0,1) and {𝛽_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)∑^∞_𝑛=0𝛼_𝑛<∞;(C2)0<liminf_𝑛→∞𝜈_𝑛≤limsup_𝑛→∞𝜈_𝑛<1/𝐿;(C3)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛  for all 𝑛≥0;(C4)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝜎_n>0;(C5)lim_𝑛→∞𝜆_𝑛=0 and ∑^∞_𝑛=0𝜆_𝑛=∞.
One has the following.(a)If {𝑥_𝑛} is the sequence generated by the scheme (13) and {𝑉𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution of the following VIP provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0: 𝑓𝑖𝑛𝑑𝑥^∗∈Fix(𝑇)∩Γ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗(𝑇)≥0,∀𝑥∈Fix∩Γ. (107)(b)If {𝑥_𝑛} is the sequence generated by the scheme (14), then {𝑥_𝑛} converges strongly to a unique solution of the following VIP provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0: 𝑓𝑖𝑛𝑑𝑥^∗∈Fix(𝑇)∩Γ𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐹𝑥^∗,𝑥−𝑥^∗(𝑇)≥0,∀𝑥∈Fix∩Γ. (108)

Proof. In Theorem 17, putting 𝑆=𝑉, we know that the iterative scheme (11) reduces to (13) since there holds for any {𝛿_𝑛}⊂(0,1)𝑥_𝑛+1=𝑃_𝐶𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛=𝑃_𝐶𝜆_𝑛𝛾𝛿_𝑛𝑉𝑥_𝑛+1−𝛿_𝑛𝑉𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛=𝑃_𝐶𝜆_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜆_𝑛𝜇𝐹𝑧_𝑛. (109)
In this case, the SVI (15) with VIP constraint is equivalent to the VIP (107). Thus, utilizing Theorem 17 (a) we obtain the desired conclusion (a). As for the conclusion (b), we immediately derive it from 𝑆=𝑉≡0 and Theorem 17 (b).

In addition, applying Theorem 17 to Problem IV, we derive the result as below.

Corollary 19. Let 𝐹∶𝐶→ℋ be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with constants 𝜅,𝜂>0, respectively, 𝑉∶𝐶→ℋ be a 𝜌-contraction with coefficient 𝜌∈[0,1), 𝑆∶𝐶→𝐶 be a nonexpansive mapping, and 𝑇_𝑖∶𝐶→𝐶 be a 𝜁_𝑖-strictly pseudocontractive mapping for 𝑖=1,2. Let 0<𝜇<2𝜂/𝜅² and 0<𝛾≤𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Problem IV has a solution and that the following conditions hold for the sequences {𝜈_𝑛}⊂(0,(1−𝜁₂)/2),{𝛾_𝑛}⊂[0,1) and {𝛽_𝑛},{𝛿_𝑛},{𝜎_𝑛},{𝜆_𝑛}⊂(0,1):(C1)0<liminf_𝑛→∞𝜈_𝑛≤limsup_𝑛→∞𝜈_𝑛<(1−𝜁₂)/2;(C2)𝛽_𝑛+𝛾_𝑛+𝜎_𝑛=1 and (𝛾_𝑛+𝜎_𝑛)𝜁≤𝛾_𝑛 for all 𝑛≥0;(C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and liminf_𝑛→∞𝜎_𝑛>0;(C4)0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1;(C5)lim_𝑛→∞𝜆_𝑛=0 and ∑^∞_𝑛=0𝜆_𝑛=∞.
One has the following.(a)If {𝑥_𝑛} is the sequence generated by the scheme in Corollary 16 (a) such that {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution of Problem IV provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.(b)If {𝑥_𝑛} is the sequence generated by the scheme in Corollary 16 (b) such that {𝑆𝑥_𝑛} is bounded, then {𝑥_𝑛} converges strongly to a unique solution 𝑥^∗∈Fix(𝑇₁)∩Fix(𝑇₂) of the following system of variational inequalities provided that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0: 𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix𝑇₁∩Fix𝑇₂,𝜇𝐹−𝛾𝑆𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈Fix𝑇₁∩Fix𝑇₂. (110)

Acknowledgments

In this research, first two authors were partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). The third author was partially supported by a research project no. SR/S4/MS: 719/11 of Department of Science and Technology, Govternment of India. The fourth author was supported in part by the National Science Council of the Republic of China.