Projection Methods and a New System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities

Abstract

A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.

1. Introduction

Variational inequality theory, introduced by Stampacchia [1], has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, finance, transportation, network and structural analysis, elasticity, and optimization. Many research papers have been written lately, both on the theory and applications of this field. Important connections with main areas of pure and applied sciences have been made. (See, for example, [2–4] and the references cited therein.) The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solution to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods to solve, for example, obstacle, unilateral, free, moving and the complex equilibrium problems. One of the most interesting and important problems in variational inequality theory is the development of an efficient numerical method. There is a substantial number of numerical methods including projection method and its variant forms, Wiener-Holf (normal) equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems. For applications on physical formulations, numerical methods and other aspects of variational inequalities, see [1–37] and the references therein.

Projection method and its variant forms represent important tool for finding the approximate solution of various types of variational and quasi-variational inequalities, the origin of which can be traced back to Lions and Stampacchia [23]. The projection type methods were developed in 1970s and 1980s. The main idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problem using the concept of projection. This alternate formulation enables us to suggest some iterative methods for computing the approximate solution; for example, see [5–7, 16–18, 29, 30, 35–37].

It should be pointed that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting. Consequently, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. It is known that the uniformly prox-regular sets are nonconvex and include the convex sets as special cases. For more details, see, for example, [12, 20, 21, 28]. In recent years, Bounkhel et al. [12], Moudafi [24], Noor [25, 26], and Pang et al. [27] have considered variational inequalities in the context of uniformly prox-regular sets.

In this paper, we introduce and consider a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities. We establish the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems, and then, by this equivalent formulation, we suggest and analyze a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system. We also prove the convergence of the suggested iterative algorithm under some suitable conditions.

2. Preliminaries

Throughout this paper, we will let ℋ be a real Hilbert space which is equipped with an inner product ⟨⋅,⋅⟩ and corresponding norm ‖⋅‖. Let 𝐾 be a nonempty convex subset of ℋ, and, 𝐶𝐵(ℋ) denote the family of all closed and bounded subsets of ℋ. We denote by 𝑑_𝐾(⋅) or 𝑑(⋅,𝐾) the usual distance function to the subset 𝐾; that is,𝑑_𝐾(𝑢)=inf_𝑣∈𝐾‖𝑢−𝑣‖. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis [19–21, 28].

Definition 2.1. Let 𝑢∈ℋ be a point not lying in 𝐾. A point 𝑣∈𝐾 is called a closest point or a projection of 𝑢 onto 𝐾 if 𝑑_𝐾(𝑢)=‖𝑢−𝑣‖. The set of all such closest points is denoted by 𝑃_𝐾(𝑢); that is, 𝑃_𝐾(𝑢)∶=𝑣∈𝐾∶𝑑_𝐾(𝑢)=‖𝑢−𝑣‖.(2.1)

Definition 2.2. The proximal normal cone of 𝐾 at a point 𝑢∈ℋ with 𝑢∉𝐾 is given by 𝑁^𝑃𝐾(𝑢)∶=𝜉∈ℋ∶𝑢∈𝑃_𝐾(𝑢+𝛼𝜉),forsome𝛼>0.(2.2) Clarke et al. [20], in Proposition 1.1.5, give a characterization of 𝑁^𝑃𝐾(𝑢) as the following.

Lemma 2.3. Let 𝐾 be a nonempty closed subset in ℋ. Then 𝜉∈𝑁^𝑃𝐾(𝑢) if and only if there exists a constant 𝛼=𝛼(𝜉,𝑢)>0 such that ⟨𝜉,𝑣−𝑢⟩≤𝛼‖𝑣−𝑢‖² for all 𝑣∈𝐾.

The above inequality is called the proximal normal inequality. The special case in which 𝐾 is closed and convex is an important one. In Proposition 1.1.10 of [20], the authors give the following characterization of the proximal normal cone, the closed and convex subset 𝐾⊂ℋ.

Lemma 2.4. Let 𝐾 be a nonempty, closed, and convex subset in ℋ. Then 𝜉∈𝑁^𝑃𝐾(𝑢) if and only if ⟨𝜉,𝑣−𝑢⟩≤0, for all 𝑣∈𝐾.

Definition 2.5. Let 𝑋 be a real Banach space, and let 𝑓∶𝑋→ℝ be the Lipschitz with constant 𝜏 near a given point 𝑥∈𝑋; that is, for some 𝜀>0, one has |𝑓(𝑦)−𝑓(𝑧)|≤𝜏‖𝑦−𝑧‖ for all 𝑦,𝑧∈𝐵(𝑥;𝜀) where 𝐵(𝑥;𝜀) denotes the open ball of radius 𝑟>0 and centered at 𝑥. The generalized directional derivative of 𝑓 at 𝑥 in the direction 𝑣, denoted as 𝑓^∘(𝑥;𝑣), is defined as follows: 𝑓^∘(𝑥;𝑣)=limsup_{𝑦→𝑥,𝑡↓0}𝑓(𝑦+𝑡𝑣)−𝑓(𝑦)𝑡,(2.3) where 𝑦 is a vector in 𝑋 and 𝑡 is a positive scalar.

The generalized directional derivative defined earlier can be used to develop a notion of tangency that does not require 𝐾 to be smooth or convex.

Definition 2.6. The tangent cone 𝑇_𝐾(𝑥) to 𝐾 at a point 𝑥 in 𝐾 is defined as follows: 𝑇_𝐾(𝑥)∶=𝑣∈ℋ∶𝑑^∘𝐾(𝑥;𝑣)=0.(2.4)

Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from 𝑇_𝐾(𝑥) by polarity. Accordingly, we define the normal cone of 𝐾 at 𝑥 by polarity with 𝑇_𝐾(𝑥) as follows:𝑁_𝐾(𝑥)∶=𝜉∶⟨𝜉,𝑣⟩≤0,∀𝑣∈𝑇_𝐾(𝑥).(2.5)

Definition 2.7. The Clarke normal cone, denoted by 𝑁^𝐶𝐾(𝑥), is given by 𝑁^𝐶𝐾(𝑥)=co[𝑁^𝑃𝐾(𝑥)], where co[𝑆] means the closure of the convex hull of 𝑆. It is clear that one always has 𝑁^𝑃𝐾(𝑥)⊆𝑁^𝐶𝐾(𝑥). The converse is not true in general. Note that 𝑁^𝐶𝐾(𝑥) is always closed and convex cone, whereas 𝑁^𝑃𝐾(𝑥) is always convex but may not be closed (see [19, 20, 28]).

In 1995, Clarke et al. [21] introduced and studied a new class of nonconvex sets, called proximally smooth sets; subsequently, Poliquin et al. in [28] investigated the aforementioned sets, under the name of uniformly prox-regular sets. These have been successfully used in many nonconvex applications in areas such as optimizations, economic models, dynamical systems, differential inclusions, and so forth. For such applications see [9–11, 13]. This class seems particularly well suited to overcome the difficulties which arise due to the nonconvexity assumptions on 𝐾. We take the following characterization proved in [21] as a definition of this class. We point out that the original definition was given in terms of the differentiability of the distance function (see [21]).

Definition 2.8. For any 𝑟∈(0,+∞], a subset 𝐾_𝑟ofℋ is called normalized uniformly prox-regular (or uniformly 𝑟-prox-regular [21]) if every nonzero proximal normal to𝐾_𝑟 can be realized by an 𝑟-ball.
This means that, for all 𝑥∈𝐾_𝑟 and all 0≠𝜉∈𝑁^𝑃𝐾_𝑟(𝑥) with ‖𝜉‖=1, 𝜉,𝑥−𝑥≤21𝑟‖‖𝑥−𝑥‖‖²,∀𝑥∈𝐾_𝑟.(2.6)

Obviously, the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, 𝑝-convex sets, and 𝐶^1,1 submanifolds (possibly with boundary) of ℋ, the images under a 𝐶^1,1 diffeomorphism of convex sets and many other nonconvex sets, see [14, 21].

Lemma 2.9 (see [21]). A closed set 𝐾⊆ℋ is convex if and only if it is proximally smooth of radius 𝑟 for every 𝑟>0.

If 𝑟=+∞, then, in view of Definition 2.8 and Lemma 2.9, the uniform 𝑟-prox-regularity of 𝐾_𝑟 is equivalent to the convexity of 𝐾_𝑟, which makes this class of great importance. For the case of that 𝑟=+∞, we set 𝐾_𝑟=𝐾.

The following proposition summarizes some important consequences of the uniform prox-regularity needed in the sequel. The proof of this results can be found in [21, 28].

Proposition 2.10. Let 𝑟>0, and let 𝐾_𝑟 be a nonempty closed and uniformly 𝑟-prox-regular subset of ℋ. Set 𝑈(𝑟)={𝑢∈ℋ∶0<𝑑_𝐾𝑟(𝑢)<𝑟}. Then the following statements hold.(a)For all 𝑥∈𝑈(𝑟), one has 𝑃_𝐾𝑟(𝑥)≠∅.(b)For all 𝑟^∈(0,𝑟), 𝑃_𝐾𝑟 is Lipschitz continuous with constant 𝑟/(𝑟−𝑟^) on 𝑈(𝑟^)={𝑢∈ℋ∶0<𝑑_𝐾𝑟(𝑢)<𝑟^}.(c)The proximal normal cone is closed as a set-valued mapping.

As a direct consequent of part (c) of Proposition 2.10, we have 𝑁^𝐶𝐾_𝑟(𝑥)=𝑁^𝑃𝐾_𝑟(𝑥). Therefore, we will define 𝑁_𝐾𝑟(𝑥)∶=𝑁^𝐶𝐾_𝑟(𝑥)=𝑁^𝑃𝐾_𝑟(𝑥) for such a class of sets.

In order to make clear the concept of 𝑟-prox-regular sets, we state the following concrete example. The union of two disjoint intervals [𝑎,𝑏] and [𝑐,𝑑] is 𝑟-prox-regular with 𝑟=(𝑐−𝑏)/2. The finite union of disjoint intervals is also 𝑟-prox-regular, and 𝑟 depends on the distances between the intervals.

Definition 2.11. The single-valued operator ℎ∶ℋ→ℋ is called(a)monotone if ⟨ℎ(𝑥)−ℎ(𝑦),𝑥−𝑦⟩≥0,∀𝑥,𝑦∈ℋ,(2.7)(b)𝑟-strongly monotone if, there exists a constant 𝑟>0 such that ⟨ℎ(𝑥)−ℎ(𝑦),𝑥−𝑦⟩≥𝑟‖𝑥−𝑦‖²,∀𝑥,𝑦∈ℋ,(2.8)(c)𝛾-Lipschitz continuous if there exists a constant 𝛾>0 such that ‖ℎ(𝑥)−ℎ(𝑦)‖≤𝛾‖𝑥−𝑦‖,∀𝑥,𝑦∈ℋ.(2.9)

Definition 2.12. Let 𝑇∶ℋ⊸ℋ be a set-valued operator, and let 𝑔∶ℋ→ℋ be a single-valued operator. Then 𝑇 is said to be(a)monotone if ⟨𝑢−𝑣,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈ℋ,𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦),(2.10)(b)𝜅-strongly monotone with respect to 𝑔 if there exists a constant 𝜅>0 such that ⟨𝑢−𝑣,𝑔(𝑥)−𝑔(𝑦)⟩≥𝜅‖𝑥−𝑦‖²,∀𝑥,𝑦∈ℋ,𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦).(2.11)

Definition 2.13. A two-variable set-valued operator 𝑇∶ℋ×ℋ⊸ℋ is called 𝜉-𝐻-Lipschitz continuous in the first variable, if there exists a constant 𝜉>0 such that, for all 𝑥,𝑥^∈ℋ, 𝐻𝑇(𝑥,𝑦),𝑇𝑥^,𝑦^≤𝜉‖‖𝑥−𝑥^‖‖,∀𝑦,𝑦^∈ℋ,(2.12) where 𝐻 is the Hausdorff pseudo-metric, that is, for any two nonempty subsets 𝐴 and 𝐵 of ℋ, 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝑦,𝐴).(2.13)

It should be pointed that if the domain of 𝐻 is restricted to the family closed bounded subsets of ℋ (denoted by 𝐶𝐵(ℋ)), then 𝐻 is the Hausdorff metric.

3. System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities

In this section, we introduce a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities and a new system of extended general nonlinear set-valued variational inequalities in Hilbert spaces and investigate their relations.

Let 𝑇_𝑖∶ℋ×ℋ→𝐶𝐵(ℋ)(𝑖=1,2) be two nonlinear set-valued operators, and let 𝑔_𝑖,ℎ_𝑖∶ℋ→ℋ(𝑖=1,2) be four nonlinear single-valued operators such that 𝐾_𝑟⊆𝑔_𝑖(ℋ), for each 𝑖=1,2. For any constants 𝜌>0 and 𝜂>0, we consider the problem of finding 𝑥^∗,𝑦^∗∈ℋ and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) such that ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟 and𝜂𝜌𝑤𝑢^∗_∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗,𝑔₁(𝑥)−ℎ₁𝑥^∗+21𝑟‖‖𝑔¹(𝑥)−ℎ₁𝑥^∗‖‖²≥0,∀𝑥∈ℋ∶𝑔₁(𝑥)∈𝐾_𝑟,+ℎ₂𝑦^∗−𝑔₂𝑥^∗,𝑔₂(𝑥)−ℎ₂𝑦^∗+12𝑟‖‖𝑔²(𝑥)−ℎ₂𝑦^∗‖‖²≥0,∀𝑥∈ℋ∶𝑔₂(𝑥)∈𝐾_𝑟.(3.1)

The problem (3.1) is called the system of extended general nonlinear regularized nonconvex set-valued variational inequalities involving six different nonlinear operators.

Lemma 3.1. If 𝐾_𝑟 is a uniformly prox-regular set, then the problem (3.1) is equivalent to that of finding 𝑥^∗,𝑦^∗∈ℋ and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) such that ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟 and 00∈∈𝜂𝜌𝑤𝑢^∗_∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗+𝑁^𝑃𝐾_𝑟ℎ₁𝑥^∗,+ℎ₂𝑦^∗−𝑔₂𝑥^∗+𝑁^𝑃𝐾_𝑟ℎ₂𝑦^∗,(3.2) where 𝑁^𝑃𝐾_𝑟(𝑠) denotes the 𝑃-normal cone of 𝐾_𝑟 at 𝑠 in the sense of nonconvex analysis.

Proof. Let (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) with 𝑥^∗,𝑦^∗∈ℋ, ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟, and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) be a solution of the system (3.1). If 𝜌𝑢^∗+ℎ₁(𝑥^∗)−𝑔₁(𝑦^∗)=0, because the vector zero always belongs to any normal cone, then 0∈𝜌𝑢^∗+ℎ₁(𝑥^∗)−𝑔₁(𝑦^∗)+𝑁^𝑃𝐾_𝑟(ℎ₁(𝑥^∗)). If 𝜌𝑢^∗+ℎ₁(𝑥^∗)−𝑔₁(𝑦^∗)≠0, then, for all 𝑥∈ℋ with 𝑔₁(𝑥)∈𝐾_𝑟, one has −𝜌𝑢^∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗,𝑔₁(𝑥)−ℎ₁𝑥^∗≤21𝑟‖‖𝑔¹(𝑥)−ℎ₁𝑥^∗‖‖².(3.3) Now, by Lemma 2.3, one gets −(𝜌𝑢^∗+ℎ₁(𝑥^∗)−𝑔₁(𝑦^∗))∈𝑁^𝑃𝐾_𝑟(ℎ₁(𝑥^∗)), and so 0∈𝜌𝑢^∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗+𝑁^𝑃𝐾_𝑟ℎ₁𝑥^∗.(3.4) Similarly, one can establish 0∈𝜂𝑤^∗+ℎ₂𝑦^∗−𝑔₂𝑥^∗+𝑁^𝑃𝐾_𝑟ℎ₂𝑦^∗.(3.5) Conversely, if (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) with 𝑥^∗,𝑦^∗∈ℋ, ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟, and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) is a solution of the system (3.2), then, in view of Definition 2.8, 𝑥^∗,𝑦^∗∈ℋ and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) with ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟 are a solution of the system (3.1).
The problem (3.2) is called the extended general nonlinear nonconvex set-valued variational inclusion system associated with the system of extended general nonlinear regularized nonconvex set-valued variational inequalities (3.1).

Some special cases of the system (3.1) are as follows.

Case 1. If 𝑟=∞; that is, 𝐾_𝑟=𝐾, the convex set in ℋ, then the system (3.1) collapses to the following system.
Find 𝑥^∗,𝑦^∗∈ℋ and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) such that ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾 and 𝜂𝜌𝑤𝑢^∗_∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗,𝑔₁(𝑥)−ℎ₁𝑥^∗≥0,∀𝑥∈ℋ∶𝑔₁(𝑥)∈𝐾,+ℎ₂𝑦^∗−𝑔₂𝑥^∗,𝑔₂(𝑥)−ℎ₂𝑦^∗≥0,∀𝑥∈ℋ∶𝑔₂(𝑥)∈𝐾,(3.6) which is called the system of extended general nonlinear set-valued variational inequalities in the sense of convex analysis.

Case 2. If 𝑇₁,𝑇₂∶ℋ×ℋ→ℋ are two nonlinear single-valued operators, ℎ_𝑖≡𝐼, the identity operator, and 𝑔_𝑖=𝑔(𝑖=1,2), then the system (3.6) reduces to the system of finding 𝑥^∗,𝑦^∗∈𝐾 such that 𝜌𝜂𝑇𝑇₁₂𝑦^∗,𝑥^∗+𝑥^∗−𝑔𝑦^∗,𝑔(𝑥)−𝑥^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,𝑥^∗,𝑦^∗+𝑦^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑦^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,(3.7) which has been considered and studied by Noor [26].

Case 3. If 𝑟=∞; that is, 𝐾_𝑟=𝐾, 𝑇₁,𝑇₂∶ℋ→ℋ are two univariate nonlinear single-valued operators, and ℎ_𝑖=𝑔_𝑖=𝑔(𝑖=1,2), then the system (3.1) changes into that of finding 𝑥^∗,𝑦^∗∈𝐾 such that 𝑔(𝑥^∗),𝑔(𝑦^∗)∈𝐾 and 𝜌𝜂𝑇𝑇₁₂𝑦^∗+𝑔𝑥^∗−𝑔𝑦^∗,𝑔(𝑥)−𝑔𝑥^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,𝑥^∗+𝑔𝑦^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑔𝑦^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,(3.8) which has been introduced and studied by Yang et al. [34].

Case 4. If 𝑇₁=𝑇₂=𝑇, then the problem (3.7) is equivalent to finding 𝑥^∗,𝑦^∗∈𝐾 such that 𝜌𝜂𝑇𝑇𝑦𝑥^∗_∗,𝑥^∗+𝑥^∗−𝑔𝑦^∗,𝑔(𝑥)−𝑥^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,,𝑦^∗+𝑦^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑦^∗≥0,∀𝑥∈ℋ∶𝑔(𝑥)∈𝐾,(3.9) which was considered and investigated by Noor [26].

Case 5. If 𝑔≡𝐼, then the system (3.7) reduces to the system of finding 𝑥^∗,𝑦^∗∈𝐾 such that 𝜌𝜂𝑇𝑇₁₂𝑦^∗,𝑥^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐾,𝑥^∗,𝑦^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾,(3.10) which has been considered and studied by Huang and Noor [22].

Case 6. If 𝑔≡𝐼, then the system (3.9) changes into that of finding 𝑥^∗,𝑦^∗∈𝐾 such that 𝜌𝜂𝑇𝑇𝑦𝑥^∗_∗,𝑥^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐾,,𝑦^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾.(3.11) The system (3.11) has been studied and investigated by Chang et al. [15] and Verma [33].

Case 7. If 𝑇∶ℋ→ℋ is an univariate nonlinear operator, then the system (3.11) reduces to the following system: find 𝑥^∗,𝑦^∗∈𝐾 such that 𝜌𝜂𝑇𝑇𝑥𝑦^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐾,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾,(3.12) which has been introduced and studied by Verma [31, 32].

Case 8. If 𝑥^∗=𝑦^∗, then the system (3.12) collapses to the following problem.
Find 𝑥^∗∈𝐾 such that ⟨𝑇𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐾.(3.13) Inequality of type (3.13) is called variational inequality, which was introduced and studied by Stampacchia [1] in 1964.

4. Perturbed Projection Iterative Algorithms

In this section, by using the projection operator technique, we first verify the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities (3.1) and the fixed point problems. Then, by using the obtained fixed point formulation, we construct two new perturbed projection iterative algorithms with mixed errors for solving the systems (3.1) and (3.6).

Lemma 4.1. Let 𝑇_𝑖, 𝑔_𝑖, ℎ_𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1). Then (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) with 𝑥^∗,𝑦^∗∈ℋ, ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟, and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) is a solution of the system (3.1), if and only if ℎℎ₁₂𝑥^∗=𝑃_𝐾𝑟𝑔₁𝑦^∗−𝜌𝑢^∗,𝑦^∗=𝑃_𝐾𝑟𝑔₂𝑥^∗−𝜂𝑤^∗,(4.1) where 𝑃_𝐾𝑟 is the projection of ℋ onto the uniformly prox-regular set 𝐾_𝑟.

Proof. Let (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) with 𝑥^∗,𝑦^∗∈ℋ, ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟, and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) be a solution of the system (3.1). Then, in view of Lemma 3.1, we have 00∈∈𝜂𝜌𝑤𝑢^∗_∗+ℎ₁𝑥^∗−𝑔₁𝑦^∗+𝑁^𝑃𝐾_𝑟ℎ₁𝑥^∗,+ℎ₂𝑦^∗−𝑔₂𝑥^∗+𝑁^𝑃𝐾_𝑟ℎ₂𝑦^∗,(4.2)⟺𝑔𝑔₁𝑦^∗−𝜌𝑢^∗∈𝐼+𝑁^𝑃𝐾_𝑟ℎ¹𝑥^∗₂,𝑥^∗−𝜂𝑤^∗∈𝐼+𝑁^𝑃𝐾_𝑟ℎ²𝑦^∗,(4.3)⟺ℎℎ₁₂𝑥^∗=𝑃_𝐾𝑟𝑔₁𝑦^∗−𝜌𝑢^∗,𝑦^∗=𝑃_𝐾𝑟𝑔₂𝑥^∗−𝜂𝑤^∗,(4.4) where 𝐼 is identity operator, and we have used the well-known fact that 𝑃_𝐾𝑟=(𝐼+𝑁^𝑃𝐾_𝑟)⁻¹.

Remark 4.2. The equality (4.1) can be written as follows: 𝑧ℎ₁=𝑔₁𝑦^∗−𝜌𝑢^∗,𝑡=𝑔₂𝑥^∗−𝜂𝑤^∗,𝑥^∗=𝑃_𝐾𝑟(𝑧),ℎ₂𝑦^∗=𝑃_𝐾𝑟(𝑡),(4.5) where 𝜌,𝜂>0 are two constants.

The fixed point formulation (4.5) enables us to construct the following perturbed iterative algorithms with mixed errors.

Algorithm 4.3. Let 𝑇_𝑖, 𝑔_𝑖, ℎ_𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1) such that ℎ_𝑖∶ℋ→ℋ be an onto operator for 𝑖=1,2. For arbitrary chosen initial point (𝑧₀,𝑡₀)∈ℋ×ℋ, compute the iterative sequence {(𝑥_𝑛,𝑦_𝑛,𝑢_𝑛,𝑤_𝑛)}^∞𝑛=0 by using 𝑧ℎ_1𝑛+1𝑥_𝑛=𝑃_𝐾𝑟𝑧_𝑛,ℎ₂𝑦_𝑛=𝑃_𝐾𝑟𝑡_𝑛,𝑡_𝑛+1=(1−𝛼)𝑧_𝑛+𝛼𝑔₁𝑦_𝑛−𝜌𝑢_𝑛+𝑒_𝑛+𝑟_𝑛,𝑢_𝑛=(1−𝛼)𝑡_𝑛+𝛼𝑔₂𝑥_𝑛−𝜂𝑤_𝑛+𝑝_𝑛+𝑘_𝑛,𝑤∈𝑇₁𝑦_𝑛,𝑥_𝑛,‖‖𝑢_𝑛−𝑢_𝑛+1‖‖≤1+(1+𝑛)⁻¹𝐻𝑇₁𝑦_𝑛,𝑥_𝑛,𝑇₁𝑦_𝑛+1,𝑥_𝑛+1_𝑛,∈𝑇₂𝑥_𝑛,𝑦_𝑛,‖‖𝑤_𝑛−𝑤_𝑛+1‖‖≤1+(1+𝑛)⁻¹𝐻𝑇₂𝑥_𝑛,𝑦_𝑛,𝑇₂𝑥_𝑛+1,𝑦_𝑛+1,(4.6) where initial points 𝑢₀∈𝑇₁(𝑦₀,𝑥₀) and 𝑤₀∈𝑇₂(𝑥₀,𝑦₀) are chosen arbitrary, 0<𝛼≤1 is a parameter and {𝑒_𝑛}^∞𝑛=0, {𝑝_𝑛}^∞𝑛=0, {𝑟_𝑛}^∞𝑛=0 and {𝑘_𝑛}^∞𝑛=0 are four sequences in ℋ to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: lim_𝑛→∞𝑒_𝑛=lim_𝑛→∞𝑝_𝑛=lim_𝑛→∞𝑟_𝑛=lim_𝑛→∞𝑘_𝑛=0,_∞_𝑛=1‖‖𝑒_𝑛−𝑒_𝑛−1‖‖<∞,^∞_𝑛=1‖‖𝑝_𝑛−𝑝_𝑛−1‖‖<∞,_∞_𝑛=1‖‖𝑟_𝑛−𝑟_𝑛−1‖‖<∞,^∞_𝑛=1‖‖𝑘_𝑛−𝑘_𝑛−1‖‖<∞.(4.7)

Algorithm 4.4. Let 𝑇_𝑖, 𝑔_𝑖, ℎ_𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.6) such that ℎ_𝑖∶ℋ→ℋ is an onto operator for 𝑖=1,2. For arbitrary chosen initial point (𝑧₀,𝑡₀)∈ℋ×ℋ, compute the iterative sequence {(𝑥_𝑛,𝑦_𝑛,𝑢_𝑛,𝑤_𝑛)}^∞𝑛=0 by using 𝑧ℎ_1𝑛+1𝑥_𝑛=𝑃_𝐾𝑧_𝑛,ℎ₂𝑦_𝑛=𝑃_𝐾𝑡_𝑛,𝑡_𝑛+1=(1−𝛼)𝑧_𝑛+𝛼𝑔₁𝑦_𝑛−𝜌𝑢_𝑛+𝑒_𝑛+𝑟_𝑛,𝑢_𝑛=(1−𝛼)𝑡_𝑛+𝛼𝑔₂𝑥_𝑛−𝜂𝑤_𝑛+𝑝_𝑛+𝑘_𝑛,𝑤∈𝑇₁𝑦_𝑛,𝑥_𝑛,‖‖𝑢_𝑛−𝑢_𝑛+1‖‖≤1+(1+𝑛)⁻¹𝐻𝑇₁𝑦_𝑛,𝑥_𝑛,𝑇₁𝑦_𝑛+1,𝑥_𝑛+1_𝑛,∈𝑇₂𝑥_𝑛,𝑦_𝑛,‖‖𝑤_𝑛−𝑤_𝑛+1‖‖≤1+(1+𝑛)⁻¹𝐻𝑇₂𝑥_𝑛,𝑦_𝑛,𝑇₂𝑥_𝑛+1,𝑦_𝑛+1,(4.8) where initial points 𝑢₀∈𝑇₁(𝑦₀,𝑥₀) and 𝑤₀∈𝑇₂(𝑥₀,𝑦₀) are chosen arbitrary, the parameter 𝛼 and the sequences {𝑒_𝑛}^∞𝑛=0, {𝑝_𝑛}^∞𝑛=0, {𝑟_𝑛}^∞𝑛=0, and {𝑘_𝑛}^∞𝑛=0 are the same as in Algorithm 4.3.

Remark 4.5. It should be pointed that(a)when 𝑟_𝑛=𝑘_𝑛=0, for all 𝑛≥0, Algorithms 4.3 and 4.4 reduce to the perturbed iterative process with mean errors;(b)if 𝑒_𝑛=𝑝_𝑛=𝑟_𝑛=𝑘_𝑛=0, for all 𝑛≥0, then Algorithms 4.3 and 4.4 change into the perturbed iterative process without error.

5. Main Results

In this section, we establish the strongly convergence of the sequence generated by the perturbed projection iterative Algorithms 4.3 and 4.4.

Theorem 5.1. Let 𝑇_𝑖, 𝑔_𝑖, ℎ_𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1) such that, for each 𝑖=1,2,(a)𝑇_𝑖 is 𝜃_𝑖-strongly monotone with respect to 𝑔_𝑖 and 𝛾_𝑖-𝐻-Lipschitz continuous in the first variable;(b)ℎ_𝑖 is 𝜋_𝑖-strongly monotone and 𝛿_𝑖-Lipschitz continuous;(c)𝑔_𝑖 is 𝜎_𝑖-Lipschitz continuous.
If the constants 𝜌 and 𝜂 satisfy the following conditions: ||||𝜌−𝜃₁𝛾²¹||||<𝑟²𝜃²¹−𝛾²¹𝑟²𝜎²¹−(𝑟−𝑟^)²1−𝜇₂²𝑟𝛾²¹||||𝜂−𝜃₂,𝛾²²||||<𝑟²𝜃²²−𝛾²²𝑟²𝜎²²−(𝑟−𝑟^)²1−𝜇₁²𝑟𝛾²²𝑟𝜃₁,𝑟𝜃₂>𝛾₁𝑟²𝜎²¹−(𝑟−𝑟^)²1−𝜇₂²,𝑟𝜎₁>𝛾₂𝑟²𝜎²²−(𝑟−𝑟^)²1−𝜇₁²,𝜇_𝑖>𝑟−𝑟^1−𝜇₂,𝑟𝜎₂>𝑟−𝑟^1−𝜇₁,=1−2𝜋_𝑖−𝛿^2𝑖<1,2𝜋_𝑖<1+𝛿^2𝑖,(𝑖=1,2),(5.1) where 𝑟^∈(0,𝑟), then there exist 𝑥^∗,𝑦^∗∈ℋ with ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾_𝑟 and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) such that (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) is a solution of the system (3.1), and the sequence {(𝑥_𝑛,𝑦_𝑛,𝑢_𝑛,𝑤_𝑛)}^∞𝑛=0 generated by Algorithm 4.3 converges strongly to (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗).

Proof. It follows from (4.6) that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤+(1𝛼−‖‖𝑒𝛼_𝑛)‖‖𝑧_𝑛−𝑧_𝑛−1‖‖+𝛼‖‖𝑔₁𝑦_𝑛−𝑔₁𝑦_𝑛−1−𝜌𝑢_𝑛−𝑢_𝑛−1‖‖−𝑒_𝑛−1‖‖+‖‖𝑟_𝑛−𝑟_𝑛−1‖‖.(5.2) Since 𝑇₁ is 𝜃₁-strongly monotone with respect to 𝑔₁ and 𝛾₁-𝐻-Lipschitz continuous in the first variable and 𝑔₁ is 𝜎₁-Lipschitz continuous, we conclude that ‖‖𝑔₁𝑦_𝑛−𝑔₁𝑦_𝑛−1−𝜌𝑢_𝑛−𝑢_𝑛−1‖‖²=‖‖𝑔₁𝑦_𝑛−𝑔₁𝑦_𝑛−1‖‖²−2𝜌𝑢_𝑛−𝑢_𝑛−1,𝑔₁𝑦_𝑛−𝑔₁𝑦_𝑛−1+𝜌²‖‖𝑢_𝑛−𝑢_𝑛−1‖‖²≤𝜎²¹−2𝜌𝜃₁‖‖𝑦_𝑛−𝑦_𝑛−1‖‖²+𝜌²1+𝑛⁻¹²𝐻𝑇¹𝑦_𝑛,𝑥_𝑛,𝑇₁𝑦_𝑛−1,𝑥_𝑛−1²≤𝜎²¹−2𝜌𝜃₁+𝜌²1+𝑛⁻¹²𝛾²¹‖‖𝑦^𝑛−𝑦_𝑛−1‖‖².(5.3) Substituting (5.3) in (5.2), we get ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤+(1𝛼−‖‖𝑒𝛼_𝑛)‖‖𝑧_𝑛−𝑧_𝑛−1‖‖+𝛼𝜎²¹−2𝜌𝜃₁+𝜌²1+𝑛⁻¹²𝛾²¹‖‖𝑦_𝑛−𝑦_𝑛−1‖‖−𝑒_𝑛−1‖‖+‖‖𝑟_𝑛−𝑟_𝑛−1‖‖.(5.4) Like the proof (5.4), by using (4.6), we can prove that ‖‖𝑡_𝑛+1−𝑡_𝑛‖‖≤+(1𝛼−‖‖𝑝𝛼_𝑛)‖‖𝑡_𝑛−𝑡_𝑛−1‖‖+𝛼𝜎²²−2𝜂𝜃₂+𝜂²1+𝑛⁻¹²𝛾²²‖‖𝑥_𝑛−𝑥_𝑛−1‖‖−𝑝_𝑛−1‖‖+‖‖𝑘_𝑛−𝑘_𝑛−1‖‖.(5.5) On the other hand, by using (4.6) and Proposition 2.10, we find that ‖‖𝑥_𝑛−𝑥_𝑛−1‖‖≤=‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛−1−ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖+‖‖ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖≤‖‖𝑥_𝑛−𝑥_𝑛−1−ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖+‖‖𝑃_𝐾𝑟𝑧_𝑛−𝑃_𝐾𝑟𝑧_𝑛−1‖‖−𝑥_𝑛−1−ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖+𝑟𝑟−𝑟^‖‖𝑧_𝑛−𝑧_𝑛−1‖‖.(5.6) From 𝜋₁-strongly monotonicity and 𝛿₁-Lipschitz continuity of ℎ₁, we have ‖‖𝑥_𝑛−𝑥_𝑛−1−ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖²=‖‖𝑥_𝑛−𝑥_𝑛−1‖‖²−2ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1,𝑥_𝑛−𝑥_𝑛−1+‖‖ℎ₁𝑥_𝑛−ℎ₁𝑥_𝑛−1‖‖²≤1−2𝜋₁+𝛿²¹‖‖𝑥_𝑛−𝑥_𝑛−1‖‖².(5.7) Substituting (5.7) in (5.6), we obtain ‖‖𝑥_𝑛−𝑥_𝑛−1‖‖≤1−2𝜋₁+𝛿²¹‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑟𝑟−𝑟^‖‖𝑧_𝑛−𝑧_𝑛−1‖‖,(5.8) which leads to ‖‖𝑥_𝑛−𝑥_𝑛−1‖‖≤𝑟(𝑟−𝑟^)1−1−2𝜋₁+𝛿²¹‖‖𝑧^𝑛−𝑧_𝑛−1‖‖.(5.9) In similar way to the proofs (5.6)–(5.9), we can prove that ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖≤𝑟(𝑟−𝑟^)1−1−2𝜋₂+𝛿²²‖‖𝑡^𝑛−𝑡_𝑛−1‖‖.(5.10) It follows from (5.4) and (5.10) that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤(1−𝛼)‖‖𝑧_𝑛−𝑧_𝑛−1‖‖+𝛼𝑟𝜎²¹−2𝜌𝜃₁+𝜌²1+𝑛⁻¹²𝛾²¹(𝑟−𝑟^)1−1−2𝜋₂+𝛿²²‖‖𝑡^𝑛+𝛼‖‖𝑒_𝑛−𝑡_𝑛−1‖‖−𝑒_𝑛−1‖‖+‖‖𝑟_𝑛−𝑟_𝑛−1‖‖.(5.11) From (5.5) and (5.9), it follows that ‖‖𝑡_𝑛+1−𝑡_𝑛‖‖≤(1−𝛼)‖‖𝑡_𝑛−𝑡_𝑛−1‖‖+𝛼𝑟𝜎²²−2𝜂𝜃₂+𝜂²1+𝑛⁻¹²𝛾²²(𝑟−𝑟^)1−1−2𝜋₁+𝛿²¹‖‖𝑧^𝑛+𝛼‖‖𝑝_𝑛−𝑧_𝑛−1‖‖−𝑝_𝑛−1‖‖+‖‖𝑘_𝑛−𝑘_𝑛−1‖‖.(5.12) Now define ‖⋅‖_∗ on ℋ×ℋ by ‖(𝑥,𝑦)‖_∗=‖𝑥‖+‖𝑦‖, for all (𝑥,𝑦)∈ℋ×ℋ. It is obvious that (ℋ×ℋ,‖⋅‖_∗) is a Hilbert space. Applying (5.11) and (5.12), one has ‖‖𝑧_𝑛+1,𝑡_𝑛+1−𝑧_𝑛,𝑡_𝑛‖‖_∗≤(1−𝛼)‖‖𝑧_𝑛,𝑡_𝑛−𝑧_𝑛−1,𝑡_𝑛−1‖‖_∗+𝛼𝜗(𝑛)‖‖𝑧_𝑛,𝑡_𝑛−𝑧_𝑛−1,𝑡_𝑛−1‖‖_∗+𝛼‖‖𝑒_𝑛,𝑝_𝑛−𝑒_𝑛−1,𝑝_𝑛−1‖‖_∗+‖‖𝑟_𝑛,𝑘_𝑛−𝑟_𝑛−1,𝑘_𝑛−1‖‖_∗,(5.13) where 𝜗(𝑛)=max⎧⎪⎪⎨⎪⎪⎩𝑟𝜎²¹−2𝜌𝜃₁+𝜌²1+𝑛⁻¹²𝛾²¹(𝑟−𝑟^)1−1−2𝜋₂+𝛿²²,𝑟𝜎²²−2𝜂𝜃₂+𝜂²1+𝑛⁻¹²𝛾²²(𝑟−𝑟^)1−1−2𝜋₁+𝛿²¹⎫⎪⎪⎬⎪⎪⎭.(5.14) Obviously, 𝜗(𝑛)→𝜗, as 𝑛→∞, where 𝜗=max⎧⎪⎪⎨⎪⎪⎩𝑟𝜎²¹−2𝜌𝜃₁+𝜌²𝛾²¹(𝑟−𝑟^)1−1−2𝜋₂+𝛿²²,𝑟𝜎²²−2𝜂𝜃₂+𝜂²𝛾²²(𝑟−𝑟^)1−1−2𝜋₁+𝛿²¹⎫⎪⎪⎬⎪⎪⎭.(5.15) In view of the condition (5.1), we know that 0≤𝜗<1. Then, for 𝜗̂=(1/2)(𝜗+1)∈(𝜗,1), there exists 𝑛₀≥1 such that 𝜗(𝑛)<𝜗̂ for each 𝑛≥𝑛₀. Thus, it follows from (5.13) that, for each 𝑛≥𝑛₀, ‖‖𝑧_𝑛+1,𝑡_𝑛+1−𝑧_𝑛,𝑡_𝑛‖‖_∗≤(1−𝛼)‖‖𝑧_𝑛,𝑡_𝑛−𝑧_𝑛−1,𝑡_𝑛−1‖‖_∗+𝛼𝜗̂‖‖𝑧_𝑛,𝑡_𝑛−𝑧_𝑛−1,𝑡_𝑛−1‖‖_∗+𝛼‖‖𝑒_𝑛,𝑝_𝑛−𝑒_𝑛−1,𝑝_𝑛−1‖‖_∗+‖‖𝑟_𝑛,𝑘_𝑛−𝑟_𝑛−1,𝑘_𝑛−1‖‖_∗=1−𝛼1−𝜗̂‖‖𝑧_𝑛,𝑡_𝑛−𝑧_𝑛−1,𝑡_𝑛−1‖‖_∗+𝛼‖‖𝑒_𝑛,𝑝_𝑛−𝑒_𝑛−1,𝑝_𝑛−1‖‖_∗+‖‖𝑟_𝑛,𝑘_𝑛−𝑟_𝑛−1,𝑘_𝑛−1‖‖_∗≤1−𝛼1−𝜗̂1−𝛼1−𝜗̂‖‖𝑧_𝑛−1,𝑡_𝑛−1−𝑧_𝑛−2,𝑡_𝑛−2‖‖_∗+𝛼‖‖𝑒_𝑛+𝛼‖‖𝑒_𝑛−1,𝑝_𝑛−1−𝑒_𝑛−2,𝑝_𝑛−2‖‖_∗+‖‖𝑟_𝑛−1,𝑘_𝑛−1−𝑟_𝑛−2,𝑘_𝑛−2‖‖_∗,𝑝_𝑛−𝑒_𝑛−1,𝑝_𝑛−1‖‖_∗+‖‖𝑟_𝑛,𝑘_𝑛−𝑟_𝑛−1,𝑘_𝑛−1‖‖_∗=1−𝛼1−𝜗̂²‖‖𝑧_𝑛−1,𝑡_𝑛−1−𝑧_𝑛−2,𝑡_𝑛−2‖‖_∗++𝛼1−1𝛼−𝛼1−1𝜗−̂𝜗̂‖‖‖‖𝑟𝑒_{𝑛−1𝑛−1},𝑝_𝑛−1−𝑒_𝑛−2,𝑝_𝑛−2‖‖_∗+‖‖𝑒_𝑛,𝑝_𝑛−𝑒_𝑛−1,𝑝_𝑛−1‖‖_∗,𝑘_𝑛−1−𝑟_𝑛−2,𝑘_𝑛−2‖‖_∗+‖‖𝑟_𝑛,𝑘_𝑛−𝑟_𝑛−1,𝑘_𝑛−1‖‖_∗≤≤⋮1−𝛼1−𝜗̂_𝑛−𝑛0‖‖𝑧_𝑛0+1,𝑡_𝑛0+1−𝑧_𝑛0,𝑡_𝑛0‖‖_∗+𝛼^𝑛−𝑛0_𝑖=11−𝛼1−𝜗̂^𝑖−1‖‖𝑒_{𝑛−(𝑖−1)},𝑝_{𝑛−(𝑖−1)}−𝑒_𝑛−𝑖,𝑝_𝑛−𝑖‖‖_∗+^𝑛−𝑛0_𝑖=11−𝛼1−𝜗̂^𝑖−1‖‖𝑟_{𝑛−(𝑖−1)},𝑘_{𝑛−(𝑖−1)}−𝑟_𝑛−𝑖,𝑘_𝑛−𝑖‖‖_∗.(5.16) Hence, for any 𝑚≥𝑛>𝑛₀, we have ‖‖𝑧_𝑚,𝑡_𝑚−𝑧_𝑛,𝑡_𝑛‖‖_∗≤^𝑚−1_𝑗=𝑛‖‖𝑧_𝑗+1,𝑡_𝑗+1−𝑧_𝑗,𝑡_𝑗‖‖_∗≤^𝑚−1_𝑗=𝑛1−𝛼1−𝜗̂^𝑗−𝑛0‖‖𝑧_𝑛0+1,𝑡_𝑛0+1−𝑧_𝑛0,𝑡_𝑛0‖‖_∗+𝛼^𝑚−1_{𝑗=𝑛𝑗−𝑛}⁰_𝑖=11−𝛼1−𝜗̂^𝑖−1‖‖𝑒_{𝑛−(𝑖−1)},𝑝_{𝑛−(𝑖−1)}−𝑒_𝑛−𝑖,𝑝_𝑛−𝑖‖‖_∗+^𝑚−1_{𝑗=𝑛𝑗−𝑛}⁰_𝑖=11−𝛼1−𝜗̂^𝑖−1‖‖𝑟_{𝑛−(𝑖−1)},𝑘_{𝑛−(𝑖−1)}−𝑟_𝑛−𝑖,𝑘_𝑛−𝑖‖‖_∗.(5.17) Since 1−𝛼(1−𝜗̂)∈(0,1), it follows from (4.7) and (5.17) that ‖(𝑧_𝑚,𝑡_𝑚)−(𝑧_𝑛,𝑡_𝑛)‖_∗=‖𝑧_𝑚−𝑧_𝑛‖+‖𝑡_𝑚−𝑡_𝑛‖→0, as 𝑛→∞. Hence, {𝑧_𝑛} and {𝑡_𝑛} are both Cauchy sequences in ℋ, and so there exist 𝑧^∗ and 𝑡^∗∈ℋ such that 𝑧_𝑛→𝑧^∗ and 𝑡_𝑛→𝑡^∗, as 𝑛→∞. By the inequalities (5.9) and (5.10), it follows that the sequences {𝑥_𝑛} and {𝑦_𝑛} are both also Cauchy in ℋ. Thus, there exist 𝑥^∗,𝑦^∗∈ℋ such that 𝑥_𝑛→𝑥^∗ and 𝑦_𝑛→𝑦^∗, as 𝑛→∞. Since for 𝑖=1,2, 𝑇_𝑖 is 𝛾_𝑖-𝐻-Lipschitz continuous in the first variable, it follows from (4.6) that ‖‖𝑢_𝑛−𝑢_𝑛+1‖‖≤≤11++((11++𝑛𝑛))⁻¹₋₁𝐻𝑇₁𝑦_𝑛,𝑥_𝑛,𝑇₁𝑦_𝑛+1,𝑥_𝑛+1‖‖𝑤𝛾₁‖‖𝑦_𝑛−𝑦_𝑛+1‖‖⟶0,_𝑛−𝑤_𝑛+1‖‖≤≤11++((11++𝑛𝑛))⁻¹₋₁𝐻𝑇₂𝑥_𝑛,𝑦_𝑛,𝑇₂𝑥_𝑛+1,𝑦_𝑛+1𝛾₂‖‖𝑥_𝑛−𝑥_𝑛+1‖‖⟶0,(5.18) as 𝑛→∞. Hence, {𝑢_𝑛} and {𝑤_𝑛} are also both Cauchy sequences in ℋ and so there exist 𝑢^∗,𝑤^∗∈ℋ such that 𝑢_𝑛→𝑢^∗ and 𝑤_𝑛→𝑤^∗, as 𝑛→∞. Further, noting 𝑢_𝑛∈𝑇₁(𝑦_𝑛,𝑥_𝑛), we have 𝑑𝑢^∗,𝑇₁𝑦^∗,𝑥^∗≤=i‖‖n𝑢f‖𝑢^∗−𝑞‖∶𝑞∈𝑇₁𝑦^∗,𝑥^∗_∗≤‖‖𝑢−𝑢_𝑛‖‖+𝑑𝑢_𝑛,𝑇₁𝑦^∗,𝑥^∗_∗≤‖‖𝑢−𝑢_𝑛‖‖+𝐻𝑇₁𝑦_𝑛,𝑥_𝑛,𝑇₁𝑦^∗,𝑥^∗_∗−𝑢_𝑛‖‖+𝛾₁‖‖𝑦_𝑛−𝑦^∗‖‖.(5.19) Since 𝑤_𝑛∈𝑇₂(𝑥_𝑛,𝑦_𝑛), like the proof (5.19), we obtain 𝑑𝑤^∗,𝑇₂𝑥^∗,𝑦^∗≤‖‖𝑤^∗−𝑤_𝑛‖‖+𝛾₂‖‖𝑥_𝑛−𝑥^∗‖‖.(5.20) The right sides of the inequalities (5.19) and (5.20) tend to zero as 𝑛→∞. Hence, 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗) and 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗). Since the operators 𝑔₁ and 𝑔₂ are continuous, it follows from (4.6) and (4.7) that 𝑧^∗=𝑔₁𝑦^∗−𝜌𝑢^∗,𝑡^∗=𝑔₂𝑥^∗−𝜂𝑤^∗.(5.21) Since the operators ℎ₁, ℎ₂, and 𝑃_𝐾𝑟 are continuous, it follows from (4.6) and (5.21) that ℎ₁𝑥^∗=𝑃_𝐾𝑟𝑧^∗=𝑃_𝐾𝑟𝑔₁𝑦^∗−𝜌𝑢^∗,ℎ₂𝑦^∗=𝑃_𝐾𝑟𝑡^∗=𝑃_𝐾𝑟𝑔₂𝑥^∗−𝜂𝑤^∗.(5.22) Now, Lemma 4.1 guarantees that (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) is a solution of the system (3.1). This completes the proof.

Theorem 5.2. Let 𝑇_𝑖, 𝑔_𝑖, ℎ_𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.6) such that, for each 𝑖=1,2,(a)𝑇_𝑖 is 𝜃_𝑖-strongly monotone with respect to 𝑔_𝑖 and 𝛾_𝑖-𝐻-Lipschitz continuous in the first variable;(b)ℎ_𝑖 is 𝜋_𝑖-strongly monotone and 𝛿_𝑖-Lipschitz continuous;(c)𝑔_𝑖 is 𝜎_𝑖-Lipschitz continuous.
If the constants 𝜌 and 𝜂 satisfy the following conditions: ||||𝜌−𝜃₁𝛾²¹||||<𝜃²¹−𝛾²¹𝜎²¹−1−𝜇₂²𝛾²¹||||𝜂−𝜃₂,𝛾²²||||<𝜃²²−𝛾²²𝜎²²−1−𝜇₁²𝛾²²𝜃₁,𝜃₂>𝛾₁𝜎²¹−1−𝜇₂²,𝜎₁>𝛾₂𝜎²²−1−𝜇₁²,𝜇+𝜇₂>1,𝜎₂+𝜇₁>1,_𝑖=1−2𝜋_𝑖−𝛿^2𝑖<1,2𝜋_𝑖<1+𝛿^2𝑖,(𝑖=1,2),(5.23) then there exist 𝑥^∗,𝑦^∗∈ℋ with ℎ₁(𝑥^∗),ℎ₂(𝑦^∗)∈𝐾 and 𝑢^∗∈𝑇₁(𝑦^∗,𝑥^∗), 𝑤^∗∈𝑇₂(𝑥^∗,𝑦^∗) such that (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗) is a solution of the system (3.6) and the sequence {(𝑥_𝑛,𝑦_𝑛,𝑢_𝑛,𝑤_𝑛)}^∞𝑛=0 generated by Algorithm 4.4 converges strongly to (𝑥^∗,𝑦^∗,𝑢^∗,𝑤^∗).

Remark 5.3. Using the method presented in this paper, one can extend Theorems 5.1 and 5.2 to a system of 𝑛-generalized variational inequalities.

Acknowledgments

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2011-0021821).