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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 690648, 18 pages
http://dx.doi.org/10.1155/2012/690648
Research Article

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

1Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran
2Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Dongeui University, Pusan 614–714, Republic of Korea

Received 2 October 2011; Accepted 11 November 2011

Academic Editor: Rudong Chen

Copyright © 2012 Javad Balooee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [24] 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 [137] 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 [57, 1618, 29, 30, 3537].

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 be a real Hilbert space which is equipped with an inner product , and corresponding norm . Let 𝐾 be a nonempty convex subset of , and, 𝐶𝐵() denote the family of all closed and bounded subsets of . We denote by 𝑑𝐾() or 𝑑(,𝐾) the usual distance function to the subset 𝐾; that is,𝑑𝐾(𝑢)=inf𝑣𝐾𝑢𝑣. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis [1921, 28].

Definition 2.1. Let 𝑢 be a point not lying in 𝐾. A point 𝑣𝐾 is called a closest point or a projection of 𝑢 onto 𝐾 if 𝑑𝐾(𝑢)=𝑢𝑣. The set of all such closest points is denoted by 𝑃𝐾(𝑢); that is, 𝑃𝐾(𝑢)=𝑣𝐾𝑑𝐾(𝑢)=𝑢𝑣.(2.1)

Definition 2.2. The proximal normal cone of 𝐾 at a point 𝑢 with 𝑢𝐾 is given by 𝑁𝑃𝐾(𝑢)=𝜉𝑢𝑃𝐾(𝑢+𝛼𝜉),forsome𝛼>0.(2.2) Clarke et al. [20], in Proposition 1.1.5, give a characterization of 𝑁𝑃𝐾(𝑢) as the following.

Lemma 2.3. Let 𝐾 be a nonempty closed subset in . Then 𝜉𝑁𝑃𝐾(𝑢) if and only if there exists a constant 𝛼=𝛼(𝜉,𝑢)>0 such that 𝜉,𝑣𝑢𝛼𝑣𝑢2 for all 𝑣𝐾.

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.

Lemma 2.4. Let 𝐾 be a nonempty, closed, and convex subset in . Then 𝜉𝑁𝑃𝐾(𝑢) if and only if 𝜉,𝑣𝑢0, for all 𝑣𝐾.

Definition 2.5. Let 𝑋 be a real Banach space, and let 𝑓𝑋 be the Lipschitz with constant 𝜏 near a given point 𝑥𝑋; that is, for some 𝜀>0, one has |𝑓(𝑦)𝑓(𝑧)|𝜏𝑦𝑧 for all 𝑦,𝑧𝐵(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)=limsup𝑦𝑥,𝑡0𝑓(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 𝑇𝐾(𝑥) to 𝐾 at a point 𝑥 in 𝐾 is defined as follows: 𝑇𝐾(𝑥)=𝑣𝑑𝐾(𝑥;𝑣)=0.(2.4)

Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from 𝑇𝐾(𝑥) by polarity. Accordingly, we define the normal cone of 𝐾 at 𝑥 by polarity with 𝑇𝐾(𝑥) as follows:𝑁𝐾(𝑥)=𝜉𝜉,𝑣0,𝑣𝑇𝐾(𝑥).(2.5)

Definition 2.7. The Clarke normal cone, denoted by 𝑁𝐶𝐾(𝑥), is given by 𝑁𝐶𝐾(𝑥)=co[𝑁𝑃𝐾(𝑥)], where co[𝑆] means the closure of the convex hull of 𝑆. It is clear that one always has 𝑁𝑃𝐾(𝑥)𝑁𝐶𝐾(𝑥). The converse is not true in general. Note that 𝑁𝐶𝐾(𝑥) is always closed and convex cone, whereas 𝑁𝑃𝐾(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 [911, 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 𝐾𝑟of is called normalized uniformly prox-regular (or uniformly 𝑟-prox-regular [21]) if every nonzero proximal normal to𝐾𝑟 can be realized by an 𝑟-ball.
This means that, for all 𝑥𝐾𝑟 and all 0𝜉𝑁𝑃𝐾𝑟(𝑥) with 𝜉=1, 𝜉,𝑥𝑥12𝑟𝑥𝑥2,𝑥𝐾𝑟.(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 𝐶1,1 submanifolds (possibly with boundary) of , the images under a 𝐶1,1 diffeomorphism of convex sets and many other nonconvex sets, see [14, 21].

Lemma 2.9 (see [21]). A closed set 𝐾 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 𝐾𝑟 is equivalent to the convexity of 𝐾𝑟, which makes this class of great importance. For the case of that 𝑟=+, we set 𝐾𝑟=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 𝐾𝑟 be a nonempty closed and uniformly 𝑟-prox-regular subset of . Set 𝑈(𝑟)={𝑢0<𝑑𝐾𝑟(𝑢)<𝑟}. Then the following statements hold.(a)For all 𝑥𝑈(𝑟), one has 𝑃𝐾𝑟(𝑥).(b)For all 𝑟(0,𝑟), 𝑃𝐾𝑟 is Lipschitz continuous with constant 𝑟/(𝑟𝑟) on 𝑈(𝑟)={𝑢0<𝑑𝐾𝑟(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 𝑁𝐶𝐾𝑟(𝑥)=𝑁𝑃𝐾𝑟(𝑥). Therefore, we will define 𝑁𝐾𝑟(𝑥)=𝑁𝐶𝐾𝑟(𝑥)=𝑁𝑃𝐾𝑟(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=(𝑐𝑏)/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 is called(a)monotone if (𝑥)(𝑦),𝑥𝑦0,𝑥,𝑦,(2.7)(b)𝑟-strongly monotone if, there exists a constant 𝑟>0 such that (𝑥)(𝑦),𝑥𝑦𝑟𝑥𝑦2,𝑥,𝑦,(2.8)(c)𝛾-Lipschitz continuous if there exists a constant 𝛾>0 such that (𝑥)(𝑦)𝛾𝑥𝑦,𝑥,𝑦.(2.9)

Definition 2.12. Let 𝑇 be a set-valued operator, and let 𝑔 be a single-valued operator. Then 𝑇 is said to be(a)monotone if 𝑢𝑣,𝑥𝑦0,𝑥,𝑦,𝑢𝑇(𝑥),𝑣𝑇(𝑦),(2.10)(b)𝜅-strongly monotone with respect to 𝑔 if there exists a constant 𝜅>0 such that 𝑢𝑣,𝑔(𝑥)𝑔(𝑦)𝜅𝑥𝑦2,𝑥,𝑦,𝑢𝑇(𝑥),𝑣𝑇(𝑦).(2.11)

Definition 2.13. A two-variable set-valued operator 𝑇× is called 𝜉-𝐻-Lipschitz continuous in the first variable, if there exists a constant 𝜉>0 such that, for all 𝑥,𝑥, 𝐻𝑥𝑇(𝑥,𝑦),𝑇,𝑦𝜉𝑥𝑥,𝑦,𝑦,(2.12) where 𝐻 is the Hausdorff pseudo-metric, that is, for any two nonempty subsets 𝐴 and 𝐵 of , 𝐻(𝐴,𝐵)=maxsup𝑥𝐴𝑑(𝑥,𝐵),sup𝑦𝐵𝑑(𝑦,𝐴).(2.13)

It should be pointed that if the domain of 𝐻 is restricted to the family closed bounded subsets of (denoted by 𝐶𝐵()), 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 𝑇𝑖×𝐶𝐵()(𝑖=1,2) be two nonlinear set-valued operators, and let 𝑔𝑖,𝑖(𝑖=1,2) be four nonlinear single-valued operators such that 𝐾𝑟𝑔𝑖(), for each 𝑖=1,2. For any constants 𝜌>0 and 𝜂>0, we consider the problem of finding 𝑥,𝑦 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) such that 1(𝑥),2(𝑦)𝐾𝑟 and𝜌𝑢+1𝑥𝑔1𝑦,𝑔1(𝑥)1𝑥+1𝑔2𝑟1(𝑥)1𝑥20,𝑥𝑔1(𝑥)𝐾𝑟,𝜂𝑤+2𝑦𝑔2𝑥,𝑔2(𝑥)2𝑦+1𝑔2𝑟2(𝑥)2𝑦20,𝑥𝑔2(𝑥)𝐾𝑟.(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 𝐾𝑟 is a uniformly prox-regular set, then the problem (3.1) is equivalent to that of finding 𝑥,𝑦 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) such that 1(𝑥),2(𝑦)𝐾𝑟 and 0𝜌𝑢+1𝑥𝑔1𝑦+𝑁𝑃𝐾𝑟1𝑥,0𝜂𝑤+2𝑦𝑔2𝑥+𝑁𝑃𝐾𝑟2𝑦,(3.2) where 𝑁𝑃𝐾𝑟(𝑠) denotes the 𝑃-normal cone of 𝐾𝑟 at 𝑠 in the sense of nonconvex analysis.

Proof. Let (𝑥,𝑦,𝑢,𝑤) with 𝑥,𝑦, 1(𝑥),2(𝑦)𝐾𝑟, and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) be a solution of the system (3.1). If 𝜌𝑢+1(𝑥)𝑔1(𝑦)=0, because the vector zero always belongs to any normal cone, then 0𝜌𝑢+1(𝑥)𝑔1(𝑦)+𝑁𝑃𝐾𝑟(1(𝑥)). If 𝜌𝑢+1(𝑥)𝑔1(𝑦)0, then, for all 𝑥 with 𝑔1(𝑥)𝐾𝑟, one has 𝜌𝑢+1𝑥𝑔1𝑦,𝑔1(𝑥)1𝑥1𝑔2𝑟1(𝑥)1𝑥2.(3.3) Now, by Lemma 2.3, one gets (𝜌𝑢+1(𝑥)𝑔1(𝑦))𝑁𝑃𝐾𝑟(1(𝑥)), and so 0𝜌𝑢+1𝑥𝑔1𝑦+𝑁𝑃𝐾𝑟1𝑥.(3.4) Similarly, one can establish 0𝜂𝑤+2𝑦𝑔2𝑥+𝑁𝑃𝐾𝑟2𝑦.(3.5) Conversely, if (𝑥,𝑦,𝑢,𝑤) with 𝑥,𝑦, 1(𝑥),2(𝑦)𝐾𝑟, and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) is a solution of the system (3.2), then, in view of Definition 2.8, 𝑥,𝑦 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) with 1(𝑥),2(𝑦)𝐾𝑟 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, 𝐾𝑟=𝐾, the convex set in , then the system (3.1) collapses to the following system.
Find 𝑥,𝑦 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) such that 1(𝑥),2(𝑦)𝐾 and 𝜌𝑢+1𝑥𝑔1𝑦,𝑔1(𝑥)1𝑥0,𝑥𝑔1(𝑥)𝐾,𝜂𝑤+2𝑦𝑔2𝑥,𝑔2(𝑥)2𝑦0,𝑥𝑔2(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× are two nonlinear single-valued operators, 𝑖𝐼, the identity operator, and 𝑔𝑖=𝑔(𝑖=1,2), then the system (3.6) reduces to the system of finding 𝑥,𝑦𝐾 such that 𝜌𝑇1𝑦,𝑥+𝑥𝑦𝑔,𝑔(𝑥)𝑥0,𝑥𝑔(𝑥)𝐾,𝜂𝑇2𝑥,𝑦+𝑦𝑥𝑔,𝑔(𝑥)𝑦0,𝑥𝑔(𝑥)𝐾,(3.7) which has been considered and studied by Noor [26].

Case 3. If 𝑟=; that is, 𝐾𝑟=𝐾, 𝑇1,𝑇2 are two univariate nonlinear single-valued operators, and 𝑖=𝑔𝑖=𝑔(𝑖=1,2), then the system (3.1) changes into that of finding 𝑥,𝑦𝐾 such that 𝑔(𝑥),𝑔(𝑦)𝐾 and 𝜌𝑇1𝑦𝑥+𝑔𝑦𝑔𝑥,𝑔(𝑥)𝑔0,𝑥𝑔(𝑥)𝐾,𝜂𝑇2𝑥𝑦+𝑔𝑥𝑔𝑦,𝑔(𝑥)𝑔0,𝑥𝑔(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,𝑦𝐾 such that 𝑦𝜌𝑇,𝑥+𝑥𝑦𝑔,𝑔(𝑥)𝑥𝑥0,𝑥𝑔(𝑥)𝐾,𝜂𝑇,𝑦+𝑦𝑥𝑔,𝑔(𝑥)𝑦0,𝑥𝑔(𝑥)𝐾,(3.9) which was considered and investigated by Noor [26].

Case 5. If 𝑔𝐼, then the system (3.7) reduces to the system of finding 𝑥,𝑦𝐾 such that 𝜌𝑇1𝑦,𝑥+𝑥𝑦,𝑥𝑥0,𝑥𝐾,𝜂𝑇2𝑥,𝑦+𝑦𝑥,𝑥𝑦0,𝑥𝐾,(3.10) which has been considered and studied by Huang and Noor [22].

Case 6. If 𝑔𝐼, then the system (3.9) changes into that of finding 𝑥,𝑦𝐾 such that 𝑦𝜌𝑇,𝑥+𝑥𝑦,𝑥𝑥𝑥0,𝑥𝐾,𝜂𝑇,𝑦+𝑦𝑥,𝑥𝑦0,𝑥𝐾.(3.11) The system (3.11) has been studied and investigated by Chang et al. [15] and Verma [33].

Case 7. If 𝑇 is an univariate nonlinear operator, then the system (3.11) reduces to the following system: find 𝑥,𝑦𝐾 such that 𝑦𝜌𝑇+𝑥𝑦,𝑥𝑥𝑥0,𝑥𝐾,𝜂𝑇+𝑦𝑥,𝑥𝑦0,𝑥𝐾,(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 𝑥𝐾 such that 𝑇𝑥,𝑥𝑥0,𝑥𝐾.(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 𝑇𝑖, 𝑔𝑖, 𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1). Then (𝑥,𝑦,𝑢,𝑤) with 𝑥,𝑦, 1(𝑥),2(𝑦)𝐾𝑟, and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) is a solution of the system (3.1), if and only if 1𝑥=𝑃𝐾𝑟𝑔1𝑦𝜌𝑢,2𝑦=𝑃𝐾𝑟𝑔2𝑥𝜂𝑤,(4.1) where 𝑃𝐾𝑟 is the projection of onto the uniformly prox-regular set 𝐾𝑟.

Proof. Let (𝑥,𝑦,𝑢,𝑤) with 𝑥,𝑦, 1(𝑥),2(𝑦)𝐾𝑟, and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) be a solution of the system (3.1). Then, in view of Lemma 3.1, we have 0𝜌𝑢+1𝑥𝑔1𝑦+𝑁𝑃𝐾𝑟1𝑥,0𝜂𝑤+2𝑦𝑔2𝑥+𝑁𝑃𝐾𝑟2𝑦,(4.2)𝑔1𝑦𝜌𝑢𝐼+𝑁𝑃𝐾𝑟1𝑥,𝑔2𝑥𝜂𝑤𝐼+𝑁𝑃𝐾𝑟2𝑦,(4.3)1𝑥=𝑃𝐾𝑟𝑔1𝑦𝜌𝑢,2𝑦=𝑃𝐾𝑟𝑔2𝑥𝜂𝑤,(4.4) where 𝐼 is identity operator, and we have used the well-known fact that 𝑃𝐾𝑟=(𝐼+𝑁𝑃𝐾𝑟)1.

Remark 4.2. The equality (4.1) can be written as follows: 𝑧=𝑔1𝑦𝜌𝑢,𝑡=𝑔2𝑥𝜂𝑤,1𝑥=𝑃𝐾𝑟(𝑧),2𝑦=𝑃𝐾𝑟(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 𝑇𝑖, 𝑔𝑖, 𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1) such that 𝑖 be an onto operator for 𝑖=1,2. For arbitrary chosen initial point (𝑧0,𝑡0)×, compute the iterative sequence {(𝑥𝑛,𝑦𝑛,𝑢𝑛,𝑤𝑛)}𝑛=0 by using 1𝑥𝑛=𝑃𝐾𝑟𝑧𝑛,2𝑦𝑛=𝑃𝐾𝑟𝑡𝑛,𝑧𝑛+1=(1𝛼)𝑧𝑛𝑔+𝛼1𝑦𝑛𝜌𝑢𝑛+𝑒𝑛+𝑟𝑛,𝑡𝑛+1=(1𝛼)𝑡𝑛𝑔+𝛼2𝑥𝑛𝜂𝑤𝑛+𝑝𝑛+𝑘𝑛,𝑢𝑛𝑇1𝑦𝑛,𝑥𝑛,𝑢𝑛𝑢𝑛+11+(1+𝑛)1𝐻𝑇1𝑦𝑛,𝑥𝑛,𝑇1𝑦𝑛+1,𝑥𝑛+1,𝑤𝑛𝑇2𝑥𝑛,𝑦𝑛,𝑤𝑛𝑤𝑛+11+(1+𝑛)1𝐻𝑇2𝑥𝑛,𝑦𝑛,𝑇2𝑥𝑛+1,𝑦𝑛+1,(4.6) where initial points 𝑢0𝑇1(𝑦0,𝑥0) and 𝑤0𝑇2(𝑥0,𝑦0) are chosen arbitrary, 0<𝛼1 is a parameter and {𝑒𝑛}𝑛=0, {𝑝𝑛}𝑛=0, {𝑟𝑛}𝑛=0 and {𝑘𝑛}𝑛=0 are four sequences in to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: lim𝑛𝑒𝑛=lim𝑛𝑝𝑛=lim𝑛𝑟𝑛=lim𝑛𝑘𝑛=0,𝑛=1𝑒𝑛𝑒𝑛1<,𝑛=1𝑝𝑛𝑝𝑛1<,𝑛=1𝑟𝑛𝑟𝑛1<,𝑛=1𝑘𝑛𝑘𝑛1<.(4.7)

Algorithm 4.4. Let 𝑇𝑖, 𝑔𝑖, 𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.6) such that 𝑖 is an onto operator for 𝑖=1,2. For arbitrary chosen initial point (𝑧0,𝑡0)×, compute the iterative sequence {(𝑥𝑛,𝑦𝑛,𝑢𝑛,𝑤𝑛)}𝑛=0 by using 1𝑥𝑛=𝑃𝐾𝑧𝑛,2𝑦𝑛=𝑃𝐾𝑡𝑛,𝑧𝑛+1=(1𝛼)𝑧𝑛𝑔+𝛼1𝑦𝑛𝜌𝑢𝑛+𝑒𝑛+𝑟𝑛,𝑡𝑛+1=(1𝛼)𝑡𝑛𝑔+𝛼2𝑥𝑛𝜂𝑤𝑛+𝑝𝑛+𝑘𝑛,𝑢𝑛𝑇1𝑦𝑛,𝑥𝑛,𝑢𝑛𝑢𝑛+11+(1+𝑛)1𝐻𝑇1𝑦𝑛,𝑥𝑛,𝑇1𝑦𝑛+1,𝑥𝑛+1,𝑤𝑛𝑇2𝑥𝑛,𝑦𝑛,𝑤𝑛𝑤𝑛+11+(1+𝑛)1𝐻𝑇2𝑥𝑛,𝑦𝑛,𝑇2𝑥𝑛+1,𝑦𝑛+1,(4.8) where initial points 𝑢0𝑇1(𝑦0,𝑥0) and 𝑤0𝑇2(𝑥0,𝑦0) are chosen arbitrary, the parameter 𝛼 and the sequences {𝑒𝑛}𝑛=0, {𝑝𝑛}𝑛=0, {𝑟𝑛}𝑛=0, and {𝑘𝑛}𝑛=0 are the same as in Algorithm 4.3.

Remark 4.5. It should be pointed that(a)when 𝑟𝑛=𝑘𝑛=0, for all 𝑛0, Algorithms 4.3 and 4.4 reduce to the perturbed iterative process with mean errors;(b)if 𝑒𝑛=𝑝𝑛=𝑟𝑛=𝑘𝑛=0, for all 𝑛0, 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 𝑇𝑖, 𝑔𝑖, 𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.1) such that, for each 𝑖=1,2,(a)𝑇𝑖 is 𝜃𝑖-strongly monotone with respect to 𝑔𝑖 and 𝛾𝑖-𝐻-Lipschitz continuous in the first variable;(b)𝑖 is 𝜋𝑖-strongly monotone and 𝛿𝑖-Lipschitz continuous;(c)𝑔𝑖 is 𝜎𝑖-Lipschitz continuous.
If the constants 𝜌 and 𝜂 satisfy the following conditions: ||||𝜃𝜌1𝛾21||||<𝑟2𝜃21𝛾21𝑟2𝜎21(𝑟𝑟)21𝜇22𝑟𝛾21,||||𝜃𝜂2𝛾22||||<𝑟2𝜃22𝛾22𝑟2𝜎22(𝑟𝑟)21𝜇12𝑟𝛾22,𝑟𝜃1>𝛾1𝑟2𝜎21(𝑟𝑟)21𝜇22,𝑟𝜃2>𝛾2𝑟2𝜎22(𝑟𝑟)21𝜇12,𝑟𝜎1>𝑟𝑟1𝜇2,𝑟𝜎2>𝑟𝑟1𝜇1,𝜇𝑖=12𝜋𝑖𝛿2𝑖<1,2𝜋𝑖<1+𝛿2𝑖,(𝑖=1,2),(5.1) where 𝑟(0,𝑟), then there exist 𝑥,𝑦 with 1(𝑥),2(𝑦)𝐾𝑟 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) such that (𝑥,𝑦,𝑢,𝑤) is a solution of the system (3.1), and the sequence {(𝑥𝑛,𝑦𝑛,𝑢𝑛,𝑤𝑛)}𝑛=0 generated by Algorithm 4.3 converges strongly to (𝑥,𝑦,𝑢,𝑤).

Proof. It follows from (4.6) that 𝑧𝑛+1𝑧𝑛𝑧(1𝛼)𝑛𝑧𝑛1𝑔+𝛼1𝑦𝑛𝑔1𝑦𝑛1𝑢𝜌𝑛𝑢𝑛1𝑒+𝛼𝑛𝑒𝑛1+𝑟𝑛𝑟𝑛1.(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 𝑔1𝑦𝑛𝑔1𝑦𝑛1𝑢𝜌𝑛𝑢𝑛12=𝑔1𝑦𝑛𝑔1𝑦𝑛12𝑢2𝜌𝑛𝑢𝑛1,𝑔1𝑦𝑛𝑔1𝑦𝑛1+𝜌2𝑢𝑛𝑢𝑛12𝜎212𝜌𝜃1𝑦𝑛𝑦𝑛12+𝜌21+𝑛12𝐻𝑇1𝑦𝑛,𝑥𝑛,𝑇1𝑦𝑛1,𝑥𝑛12𝜎212𝜌𝜃1+𝜌21+𝑛12𝛾21𝑦𝑛𝑦𝑛12.(5.3) Substituting (5.3) in (5.2), we get 𝑧𝑛+1𝑧𝑛𝑧(1𝛼)𝑛𝑧𝑛1+𝛼𝜎212𝜌𝜃1+𝜌21+𝑛12𝛾21𝑦𝑛𝑦𝑛1𝑒+𝛼𝑛𝑒𝑛1+𝑟𝑛𝑟𝑛1.(5.4) Like the proof (5.4), by using (4.6), we can prove that 𝑡𝑛+1𝑡𝑛𝑡(1𝛼)𝑛𝑡𝑛1+𝛼𝜎222𝜂𝜃2+𝜂21+𝑛12𝛾22𝑥𝑛𝑥𝑛1𝑝+𝛼𝑛𝑝𝑛1+𝑘𝑛𝑘𝑛1.(5.5) On the other hand, by using (4.6) and Proposition 2.10, we find that 𝑥𝑛𝑥𝑛1𝑥𝑛𝑥𝑛11𝑥𝑛1𝑥𝑛1+1𝑥𝑛1𝑥𝑛1=𝑥𝑛𝑥𝑛11𝑥𝑛1𝑥𝑛1+𝑃𝐾𝑟𝑧𝑛𝑃𝐾𝑟𝑧𝑛1𝑥𝑛𝑥𝑛11𝑥𝑛1𝑥𝑛1+𝑟𝑟𝑟𝑧𝑛𝑧𝑛1.(5.6) From 𝜋1-strongly monotonicity and 𝛿1-Lipschitz continuity of 1, we have 𝑥𝑛𝑥𝑛11𝑥𝑛1𝑥𝑛12=𝑥𝑛𝑥𝑛1221𝑥𝑛1𝑥𝑛1,𝑥𝑛𝑥𝑛1+1𝑥𝑛1𝑥𝑛1212𝜋1+𝛿21𝑥𝑛𝑥𝑛12.(5.7) Substituting (5.7) in (5.6), we obtain 𝑥𝑛𝑥𝑛112𝜋1+𝛿21𝑥𝑛𝑥𝑛1+𝑟𝑟𝑟𝑧𝑛𝑧𝑛1,(5.8) which leads to 𝑥𝑛𝑥𝑛1𝑟(𝑟𝑟)112𝜋1+𝛿21𝑧𝑛𝑧𝑛1.(5.9) In similar way to the proofs (5.6)–(5.9), we can prove that 𝑦𝑛𝑦𝑛1𝑟(𝑟𝑟)112𝜋2+𝛿22𝑡𝑛𝑡𝑛1.(5.10) It follows from (5.4) and (5.10) that 𝑧𝑛+1𝑧𝑛𝑧(1𝛼)𝑛𝑧𝑛1𝑟+𝛼𝜎212𝜌𝜃1+𝜌21+𝑛12𝛾21(𝑟𝑟)112𝜋2+𝛿22𝑡𝑛𝑡𝑛1𝑒+𝛼𝑛𝑒𝑛1+𝑟𝑛𝑟𝑛1.(5.11) From (5.5) and (5.9), it follows that 𝑡𝑛+1𝑡𝑛𝑡(1𝛼)𝑛𝑡𝑛1𝑟+𝛼𝜎222𝜂𝜃2+𝜂21+𝑛12𝛾22(𝑟𝑟)112𝜋1+𝛿21𝑧𝑛𝑧𝑛1𝑝+𝛼𝑛𝑝𝑛1+𝑘𝑛𝑘𝑛1.(5.12) Now define on × by (𝑥,𝑦)=𝑥+𝑦, for all (𝑥,𝑦)×. It is obvious that (×,) is a Hilbert space. Applying (5.11) and (5.12), one has 𝑧𝑛+1,𝑡𝑛+1𝑧𝑛,𝑡𝑛𝑧(1𝛼)𝑛,𝑡𝑛𝑧𝑛1,𝑡𝑛1𝑧+𝛼𝜗(𝑛)𝑛,𝑡𝑛𝑧𝑛1,𝑡𝑛1𝑒+𝛼𝑛,𝑝𝑛𝑒𝑛1,𝑝𝑛1+𝑟𝑛,𝑘𝑛𝑟𝑛1,𝑘𝑛1,(5.13) where 𝑟𝜗(𝑛)=max𝜎212𝜌𝜃1+𝜌21+𝑛12𝛾21(𝑟𝑟)112𝜋2+𝛿22,𝑟𝜎222𝜂𝜃2+𝜂21+𝑛12𝛾22(𝑟𝑟)112𝜋1+𝛿21.(5.14) Obviously, 𝜗(𝑛)𝜗, as 𝑛, where 𝑟𝜗=max𝜎212𝜌𝜃1+𝜌2𝛾21(𝑟𝑟)112𝜋2+𝛿22,𝑟𝜎222𝜂𝜃2+𝜂2𝛾22(𝑟𝑟)112𝜋1+𝛿21.(5.15) In view of the condition (5.1), we know that 0𝜗<1. Then, for ̂𝜗=(1/2)(𝜗+1)(𝜗,1), there exists 𝑛01 such that ̂𝜗𝜗(𝑛)< for each 𝑛𝑛0. Thus, it follows from (5.13) that, for each 𝑛𝑛0, 𝑧𝑛+1,𝑡𝑛+1𝑧𝑛,𝑡𝑛𝑧(1𝛼)𝑛,𝑡𝑛𝑧𝑛1,𝑡𝑛1̂𝜗𝑧+𝛼𝑛,𝑡𝑛𝑧𝑛1,𝑡𝑛1𝑒+𝛼𝑛,𝑝𝑛𝑒𝑛1,𝑝𝑛1+𝑟𝑛,𝑘𝑛𝑟𝑛1,𝑘𝑛1=̂𝜗𝑧1𝛼1𝑛,𝑡𝑛𝑧𝑛1,𝑡𝑛1𝑒+𝛼𝑛,𝑝𝑛𝑒𝑛1,𝑝𝑛1+𝑟𝑛,𝑘𝑛𝑟𝑛1,𝑘𝑛1̂𝜗̂𝜗𝑧1𝛼11𝛼1𝑛1,𝑡𝑛1𝑧𝑛2,𝑡𝑛2𝑒+𝛼𝑛1,𝑝𝑛1𝑒𝑛2,𝑝𝑛2+𝑟𝑛1,𝑘𝑛1𝑟𝑛2,𝑘𝑛2𝑒+𝛼𝑛,𝑝𝑛𝑒𝑛1,𝑝𝑛1+𝑟𝑛,𝑘𝑛𝑟𝑛1,𝑘𝑛1=̂𝜗1𝛼12𝑧𝑛1,𝑡𝑛1𝑧𝑛2,𝑡𝑛2̂𝜗𝑒+𝛼1𝛼1𝑛1,𝑝𝑛1𝑒𝑛2,𝑝𝑛2+𝑒𝑛,𝑝𝑛𝑒𝑛1,𝑝𝑛1+̂𝜗𝑟1𝛼1𝑛1,𝑘𝑛1𝑟𝑛2,𝑘𝑛2+𝑟𝑛,𝑘𝑛𝑟𝑛1,𝑘𝑛1̂𝜗1𝛼1𝑛𝑛0𝑧𝑛0+1,𝑡𝑛0+1𝑧𝑛0,𝑡𝑛0+𝛼𝑛𝑛0𝑖=1̂𝜗1𝛼1𝑖1𝑒𝑛(𝑖1),𝑝𝑛(𝑖1)𝑒𝑛𝑖,𝑝𝑛𝑖+𝑛𝑛0𝑖=1̂𝜗1𝛼1𝑖1𝑟𝑛(𝑖1),𝑘𝑛(𝑖1)𝑟𝑛𝑖,𝑘𝑛𝑖.(5.16) Hence, for any 𝑚𝑛>𝑛0, we have 𝑧𝑚,𝑡𝑚𝑧𝑛,𝑡𝑛𝑚1𝑗=𝑛𝑧𝑗+1,𝑡𝑗+1𝑧𝑗,𝑡𝑗𝑚1𝑗=𝑛̂𝜗1𝛼1𝑗𝑛0𝑧𝑛0+1,𝑡𝑛0+1𝑧𝑛0,𝑡𝑛0+𝛼𝑚1𝑗=𝑛𝑗𝑛0𝑖=1̂𝜗1𝛼1𝑖1𝑒𝑛(𝑖1),𝑝𝑛(𝑖1)𝑒𝑛𝑖,𝑝𝑛𝑖+𝑚1𝑗=𝑛𝑗𝑛0𝑖=1̂𝜗1𝛼1𝑖1𝑟𝑛(𝑖1),𝑘𝑛(𝑖1)𝑟𝑛𝑖,𝑘𝑛𝑖.(5.17) Since ̂1𝛼(1𝜗)(0,1), it follows from (4.7) and (5.17) that (𝑧𝑚,𝑡𝑚)(𝑧𝑛,𝑡𝑛)=𝑧𝑚𝑧𝑛+𝑡𝑚𝑡𝑛0, as 𝑛. Hence, {𝑧𝑛} and {𝑡𝑛} are both Cauchy sequences in , and so there exist 𝑧 and 𝑡 such that 𝑧𝑛𝑧 and 𝑡𝑛𝑡, as 𝑛. By the inequalities (5.9) and (5.10), it follows that the sequences {𝑥𝑛} and {𝑦𝑛} are both also Cauchy in . Thus, there exist 𝑥,𝑦 such that 𝑥𝑛𝑥 and 𝑦𝑛𝑦, as 𝑛. Since for 𝑖=1,2, 𝑇𝑖 is 𝛾𝑖-𝐻-Lipschitz continuous in the first variable, it follows from (4.6) that 𝑢𝑛𝑢𝑛+11+(1+𝑛)1𝐻𝑇1𝑦𝑛,𝑥𝑛,𝑇1𝑦𝑛+1,𝑥𝑛+11+(1+𝑛)1𝛾1𝑦𝑛𝑦𝑛+1𝑤0,𝑛𝑤𝑛+11+(1+𝑛)1𝐻𝑇2𝑥𝑛,𝑦𝑛,𝑇2𝑥𝑛+1,𝑦𝑛+11+(1+𝑛)1𝛾2𝑥𝑛𝑥𝑛+10,(5.18) as 𝑛. Hence, {𝑢𝑛} and {𝑤𝑛} are also both Cauchy sequences in and so there exist 𝑢,𝑤 such that 𝑢𝑛𝑢 and 𝑤𝑛𝑤, as 𝑛. Further, noting 𝑢𝑛𝑇1(𝑦𝑛,𝑥𝑛), we have 𝑑𝑢,𝑇1𝑦,𝑥=inf𝑢𝑞𝑞𝑇1𝑦,𝑥𝑢𝑢𝑛𝑢+𝑑𝑛,𝑇1𝑦,𝑥𝑢𝑢𝑛+𝐻𝑇1𝑦𝑛,𝑥𝑛,𝑇1𝑦,𝑥𝑢𝑢𝑛+𝛾1𝑦𝑛𝑦.(5.19) Since 𝑤𝑛𝑇2(𝑥𝑛,𝑦𝑛), like the proof (5.19), we obtain 𝑑𝑤,𝑇2𝑥,𝑦𝑤𝑤𝑛+𝛾2𝑥𝑛𝑥.(5.20) The right sides of the inequalities (5.19) and (5.20) tend to zero as 𝑛. Hence, 𝑢𝑇1(𝑦,𝑥) and 𝑤𝑇2(𝑥,𝑦). Since the operators 𝑔1 and 𝑔2 are continuous, it follows from (4.6) and (4.7) that 𝑧=𝑔1𝑦𝜌𝑢,𝑡=𝑔2𝑥𝜂𝑤.(5.21) Since the operators 1, 2, and 𝑃𝐾𝑟 are continuous, it follows from (4.6) and (5.21) that 1𝑥=𝑃𝐾𝑟𝑧=𝑃𝐾𝑟𝑔1𝑦𝜌𝑢,2𝑦=𝑃𝐾𝑟𝑡=𝑃𝐾𝑟𝑔2𝑥𝜂𝑤.(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 𝑇𝑖, 𝑔𝑖, 𝑖(𝑖=1,2), 𝜌, and 𝜂 be the same as in the system (3.6) such that, for each 𝑖=1,2,(a)𝑇𝑖 is 𝜃𝑖-strongly monotone with respect to 𝑔𝑖 and 𝛾𝑖-𝐻-Lipschitz continuous in the first variable;(b)𝑖 is 𝜋𝑖-strongly monotone and 𝛿𝑖-Lipschitz continuous;(c)𝑔𝑖 is 𝜎𝑖-Lipschitz continuous.
If the constants 𝜌 and 𝜂 satisfy the following conditions: ||||𝜃𝜌1𝛾21||||<𝜃21𝛾21𝜎211𝜇22𝛾21,||||𝜃𝜂2𝛾22||||<𝜃22𝛾22𝜎221𝜇12𝛾22,𝜃1>𝛾1𝜎211𝜇22,𝜃2>𝛾2𝜎221𝜇12,𝜎1+𝜇2>1,𝜎2+𝜇1𝜇>1,𝑖=12𝜋𝑖𝛿2𝑖<1,2𝜋𝑖<1+𝛿2𝑖,(𝑖=1,2),(5.23) then there exist 𝑥,𝑦 with 1(𝑥),2(𝑦)𝐾 and 𝑢𝑇1(𝑦,𝑥), 𝑤𝑇2(𝑥,𝑦) such that (𝑥,𝑦,𝑢,𝑤) is a solution of the system (3.6) and the sequence {(𝑥𝑛,𝑦𝑛,𝑢𝑛,𝑤𝑛)}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).

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