On a Class of Quasi Variational...

On a Class of Quasi Variational Inequalities

By

AWAIS GUL KHAN

CIIT/SP12-PMT-003/ISB

PhD Thesis

In

Mathematics

COMSATS Institute of Information Technology

Islamabad-Pakistan

Fall, 2014

ii



A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad

In partial fulfillment

of the requirement for the degree of

PhD Mathematics

By

Awais Gul Khan


Fall, 2014

iii


A Post Graduate Thesis submitted to the Department of Mathematics as partial

fulfillment of the requirement for the award of Degree of PhD Mathematics.

Name Registration Number

Awais Gul Khan CIIT/SP12-PMT-003/ISB

Supervisor

Prof. Dr. Muhammad Aslam Noor

Professor, Department of Mathematics


Islamabad.

December, 2014

iv

Final Approval

This thesis titled


By

Awais Gul Khan


Has been approved

For the COMSATS Institute of Information Technology, Islamabad

External Examiner 1: __________________________________

Prof. Dr. Muhammad Ayub

Department of Mathematics

Quaid-i-Azam University, Islamabad.

External Examiner 2: __________________________________

Prof. Dr. Fazal-i-Haq

Department of Mathematics/Stats/CS

Khyber Pakhtunkhwa Agriculture University, Peshawar.

Supervisor: __________________________________



COMSATS Institute of Information Technology,

Islamabad.

Head of Department: __________________________________

Prof. Dr. habil. Shamsul Qamar



Islamabad.

Chairperson: __________________________________

Prof. Dr. Tahira Haroon



Islamabad.

Dean, Faculty of Sciences: __________________________________

Prof. Dr. Arshad Saleem Bhatti

v

Declaration

I, Awais Gul Khan, CIIT/SP12-PMT-003/ISB, hereby declare that I have produced the

work presented in this thesis, during the scheduled period of study. I also declare that I

have not taken any material from any source except referred to wherever due that amount

of plagiarism is within acceptable range. If a violation of HEC rules on research has

occurred in this thesis, I shall be liable to punishable action under the plagiarism rules of

the HEC.

Date: _______________ Signature of student:

Awais Gul Khan


vi

Certificate

It is certified that Awais Gul Khan, CIIT/SP12-PMT-003/ISB has carried out all the

work related to this thesis under my supervision at the Department of Mathematics,

COMSATS Institute of Information Technology, Islamabad and the work fulfills the

requirement for award of PhD degree.

Date: _______________

Supervisor:


Professor, CIIT, Islamabad.

Head of Department:

Prof. Dr. habil. Shamsul Qamar


vii

DEDICATED

To

My Parents and Family

For their untiring sacrifices, prayers and support to me.

viii

ACKNOWLEDGEMENTS

The oceans turns into inks and all trees become pens, even then, the praises of

ALLAH ALMIGHTY, cannot be expressed which is the most compassionate, the most

merciful. He who created there in it, hidden or evident which is beneficent, for man.

Who blessed me with health, thoughts and talented teachers to carry out this study. I

offer my humblest, sincerest and millions Darood-O-Salam to the greatest social

reformer “THE HOLY PROPHET HAZRAT MUHAMMAD (Sallallah-o-Allah-e-

Wassallum)”, the “city of knowledge” and torch of guidance for humanity as a whole,

forever.

I feel highly privileged to record my deep sense of gratitude, sincerity and thanks

for my respected supervisor Prof. Dr. Muhammad Aslam Noor, the motivation of whom

enabled me to select such a thought provoking and striking area of research, for his

personal interest, kind supervision, affectionate criticism, encouraging behavior, magical

thoughts and inspiring guidance provided me throughout the course of study. His

presence has always been a source of confidence for me. It was indeed an honor to work

under his supervision. I am also very thankful to my respectable madam, Prof. Dr.

Khalida Inayat Noor for her very helping and encouraging attitude towards me.

I also want to acknowledge the untiring struggles of Honorable Rector Dr. S.M.

Junaid Zaidi (S.I.), COMSATS Institute of Information Technology, the Dean, Faculty

of Sciences and the Head, Department of Mathematics, COMSATS Institute of

Information Technology, Islamabad for providing state of the art research facilities.

I also have immense obligation and special thanks to my parent Institute,

Government College University, Faisalabad for providing me an opportunity for

obtaining higher education.

Finally, I am pleased to express my deepest admiration and gratitude for my

parents, parents-in-law, all my family members and friends for their prayers, moral

support and encouragement. My special thanks to my beloved sons Muhammad

Abdullah Khan & Muhammad Abdur Rahman Khan for their lovely smiles and innocent

mischievous acts that always refresh my life.

Awais Gul Khan


ix

ABSTRACT


A new class of quasi variational inequalities involving three operators is considered and

investigated, which is called extended general quasi variational inequality. It is shown

that extended general quasi variational inequalities are equivalent to fixed point problems

and the implicit Wiener-Hopf equations. These equivalent formulations play an

important part to prove the existence of a solution of the extended general quasi

variational inequalities. These equivalent formulations are also used to develop new

iterative methods for solving quasi variational inequalities and their variant forms.

Convergence criteria of proposed methods is analyzed under some suitable conditions.

Dynamical systems related to the extended general quasi variational inequalities are

introduced. It is shown that the dynamical system can be used to discuss the existence of

the solution of extended general quasi variational inequalities, which requires only the

Lipschitz continuity. The stability of a solution of the extended general quasi variational

inequalities is investigated. The error bounds for a solution of the quasi variational

inequalities is considered and investigated using the merit function technique. A system

of nonlinear quasi variational inequalities is introduced. It is shown that this system of

quasi variational inequalities is equivalent to the system of fixed point problems. This

system of fixed point formulation is used in developing some iterative methods.

Convergence criteria of these iterative methods is investigated under some suitable

conditions. Several classes of quasi variational inequalities and related optimization

problems can be obtained as special cases of system of quasi variational inequalities. Our

results continue to hold for these cases and represent significant improvement and

refinement of the known results.

TABLE OF CONTENTS

1 Introduction 6

2 Preliminaries and Basic Concepts 12

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Relative Convex Sets and Relative Convex Functions . . . . . . . . . . . 14

2.3 Some Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Formulation and Existence Results 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Wiener-Hopf Equations Technique . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Iterative Methods 48

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Projection Operator Technique . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Wiener-Hopf Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Dynamical Systems 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1

5.2 Projected Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Wiener-Hopf Dynamical System . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Error Bounds 81

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.2 Normal Residue Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Regularized Merit Functions . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4 D-merit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7 System of Quasi Variational Inequalities 99

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 Conclusion 116

9 References 119

2

LIST OF FIGURES

________________________________________________

N. A.

3

LIST OF TABLES

________________________________________________

N. A.

4

LIST OF ABBREVIATIONS

________________________________________________

This is a list of symbols/abbreviations used in this thesis. Other symbols are de�ned

in the thesis, as needed.

h�; �i : inner product

k�k : norm

H : a real Hilbert space

: a closed and convex set

(u) : a closed and convex-valued set

T; g; h : nonlinear operators

I : identity operator

P : projection operator

P(u) : implicit projection operator

R (u) : residue vector

�T ; �g; �h : constants of strongly monotone of operators T; g; and h.

�T ; �g; �h : constants of Lipschitz continuous of operators T; g; and h.

2 : an element of

8 : for all

� : is equivalent to

5

Chapter 1

Introduction

6

The calculus of variations was developed by Euler and Lagrange. It essentially started

soon after the introduction of calculus by Newton and Leibnitz. Although some individ-

ual optimization problems had been investigated before that, notably the determination

of the paths of light by Fermat. Variational principles have played a signi�cant and

important role in the development of: the general theory of relativity, gauge �eld the-

ory in elementary particle physics, soliton theory, and optimization theory. One of the

most important developments in the calculus of variations over the last few decades has

been the emergence of the variational inequality theory. Variational inequalities were

introduced and studied by Fichera [35] and Stampacchia [121], in the �eld of potential

theory and mechanics. Stampacchia [121] proved that the minimum of the energy (vir-

tual work) function associated with the potential problems can be characterized by a

class of inequalities. This class of inequalities is called the variational inequality. This

result has played a signi�cant and crucial part in the studies of a wide class of problems

which arise in applied sciences and engineering. This theory provides us with a simple,

natural, general and uni�ed framework for studying a wide class of unrelated linear and

nonlinear problems arising in: elasticity, �uid �ow through porous media, economics,

transportation, oceanography, optimization, operations research, regional and applied

sciences [1-129].

It is important to emphasize that the variaional inequalities are de�ned on the convex

sets. A wide class of problems arise where the convex sets depend upon the solution

implicitly or explicitly. In this case, the convex set becomes convex-valued set. In such

cases, the variational inequality is called the quasi variational inequality. Bensoussan

and Lions [18] have shown that the problems of impulse control theory can be studied

via the framework of quasi variational inequalities. It has been shown that the quasi

variational inequalities give a natural and uni�ed framework for studying the problems

which arise in pure and applied sciences. Chan and Pang [21] introduced and studied the

generalized quasi variational inequality problems. It has been shown that the generalized

quasi variational inequality contains the quasi variational inequality problem as a special

7

case. Noor [67] has shown that the quasi variational inequality is equivalent to the �xed

point problem using the projection technique. Using this equivalence, he suggested some

iterative methods for solving the quasi variational inequality.

One of the most di¢ cult, interesting and important problems in variational inequal-

ity theory is the development of an e¢ cient and implementable algorithm. Projection

method represents an important computational tool for �nding approximate solution

of variational inequalities, which was developed in 1970 and 1980. This method has

been extended and modi�ed in various ways to other class of variational inequalities,

see, [67, 69, 70, 74, 75, 78, 83, 84, 90, 94, 95, 101, 102], for an account of the iterative

methods. Using essentially the projection technique, Shi [117] established the equiva-

lence between the variational inequality problems and system of equations, known as

Wiener-Hopf equations. This equivalence was used to suggest an iterative algorithm.

This technique was re�ned and developed by Noor [72, 79, 81, 84, 93, 95] to suggest

several iterative algorithms for di¤erent classes of variational inequalities.

In recent years, considerable interest has been shown in developing various classes

of variational inequalities, both for its own sake and for its applications. There are sig-

ni�cant recent developments of variational inequalities related to multivalued operators,

nonconvex optimization, iterative methods, Wiener-Hopf equations, dynamical systems

and merit functions. Noor [69] introduced and studied an important class of variational

inequalities using two operators, which is known as general variational inequality. Noor

[72] has shown that the general variational inequality are equivalent to the Wiener-Hopf

equations. Noor [72] has used this alternative equivalent formulation to suggest several

iterative methods for solving the general variational inequality and the related problems.

Noor [70] introduced and studied a new class of quasi variational inequality, which is

called a general quasi variational inequality involving two operators. It has proved that

the general quasi variational inequality is equivalent to a �xed point problem. Using this

�xed point equivalence, he proposed an iterative method for solving the general quasi

variational inequality and discussed the convergence of the proposed iterative method.

8

Motivated and inspired by the research going on in this area, Noor [73] introduced

and studied a new class of quasi variational inequality, which is known as generalized

multivalued quasi variational inequality. It has been shown that this class is most general

and includes many classes of quasi variational inequalities as a special cases. He has

considered the existence of a solution of problem using the projection technique and

proposed a number of iterative methods for solving the generalized multivalued quasi

variational inequality. He also established an equivalence relation between generalized

multivalued quasi variational inequality and the implicit Wiener-Hopf equations. He used

this equivalence to suggest a class of iterative methods for solving generalized multivalued

quasi variational inequality, see [73, 76] and references therein.

It is well known that the convexity plays an important role in the study of variational

inequality and its variant forms. The optimality of a di¤erentiable convex function on a

convex set can be characterized by the variational inequality. In recent years, the concept

of convexity has been generalized in many dimensions, see [23] and references therein.

Youness [128] introduced a concept of nonconvex sets and nonconvex functions. For

the properties and more detail of the nonconvex functions, see [23, 24, 25, 48, 83, 101].

Noor [90] has considered and studied the nonconvex function relative to two arbitrary

functions. It has shown [90] that the minimum of a di¤erentiable nonconvex function

involving three functions can be characterized by a class of variational inequalities, which

is called the extended general variational inequality. It also has been shown that many

classes of variational inequalities are the special cases of extended general variational

inequality. It has been proved that the extended general variational inequalities are

equivalent to the �xed point problems. This alternative �xed point formulation is used

to study the existence of a solution of extended general variational inequality as well as

to develop some iterative methods. Noor [87] has also studied the existence of a solution

of extended general variational inequality via auxiliary principle technique. Liu and Cao

[60] proposed a recurrent neural network based on the projection operator for solving the

extended general variational inequality.

9

Noor and Noor [100] introduced a new class of quasi variational inequality involving

three nonlinear operators. This class is called extended general quasi variational inequal-

ity. They have shown that extended general quasi variational inequalities are equivalent

to the �xed point problems and Wiener-Hopf equations. We again use this alternative

equivalent �xed point formulation to suggest several iterative methods quasi variational

inequalities. A brief thesis review is given below.

In chapter 2, we include some basic de�nitions and preliminary results. These results

will be useful in our main work. We shall also discuss the concept of relative convex sets

and relative convex functions. Some examples of relative convex sets and relative con-

vex functions are also given. In chapter 3, several special cases of extended general quasi

variational inequalities are discussed. Using the alternative �xed point formulation estab-

lished by Noor and Noor [100], we discuss the conditions for the existence of a solution of

extended general quasi variational inequalities. Implicit Wiener-Hopf equations related

to extended general quasi variational inequalities are also discussed. Since the extended

general quasi variational inequalities include variational inequalities, quasi variational in-

equalities, general variational inequalities, extended general variational inequalities and

quasi complementarity problems as special cases, one can deduce the similar results for

these problems under weaker conditions. In this respect, our results can be viewed as

re�nement of the previously known results for variational inequalities. Several special

cases are also investigated. In chapter 4, we suggest and analyze some one-step, two-

step and three-step iterative methods for solving the extended general quasi variational

inequalities and its variant forms.

In chapter 5, some dynamical systems associated with the extended general quasi

variational inequalities are investigated. These dynamical systems are called extended

general implicit projected dynamical system and extended general implicit Wiener-Hopf

dynamical system. It is shown that the solution of these dynamical systems converge

globally exponentially to a unique solution of the extended general quasi variational

inequalities under some suitable conditions. Some special cases are also discussed, which

10

can be obtained from our results. Results obtained in this chapter continue to hold for

these problems. In chapter 6, normal residue merit functions, regularized merit functions

and di¤erence merit functions are used to obtain error bounds for the solution of the

extended general quasi variational inequalities under some weaker conditions.

In chapter 7, we consider a new system of extended general quasi variational inequal-

ities involving four nonlinear operators. Using projection operator technique, we show

that system of extended general quasi variational inequalities is equivalent to a system

of �xed point problems. Using this alternative equivalent formulation, some Jacobi type

algorithms for solving a system of extended general quasi variational inequalities are sug-

gested and investigated. Convergence of these new methods is considered under some

suitable conditions. Several special cases are discussed.

11

Chapter 2

Preliminaries and Basic Concepts

12

In this chapter, we include some basic de�nitions and preliminary results. These results

will be useful in our main work. We shall also discuss the concept of relative convex sets

and relative convex functions. Some examples of relative convex sets and relative convex

functions are also given. It has been shown [90] that the minimum of a di¤erentiable

relative convex function involving three functions can be characterized by a class of

variational inequalities. For the sake of completeness, we shall give all the details of this

important result.

2.1 Preliminaries

In this section, de�nitions of inner product and norm are given. Some related de�nitions

are also included.

De�nition 2.1.1 Let V be a linear space over the �eld of real numbers R. An inner

product is a function h�; �i : V � V ! R with the following properties.

(i) hu; ui � 0; 8 u 2 V ;

(ii) hu; ui = 0; if and only if, u = 0;

(iii) hu; vi = hv; ui ; 8 u; v 2 V :

(iv) h�u+ �v; wi = � hu;wi+ � hv; wi ; 8 u; v; w 2 V and �; � 2 R:

The linear space V together with the inner product h�; �i is called an inner product

space.

De�nition 2.1.2 A complete inner product space is called a Hilbert space and it is de-

noted by H.

De�nition 2.1.3 Let V be a linear space. A norm k�k is a function from V to R with

the following properties.

13

(i) kuk � 0; 8 u 2 V ;

(ii) kuk = 0; if and only if, u = 0;

(iii) k�uk = j�j kuk ; 8 u 2 V and � 2 R;

(iv) ku+ vk � kuk+ kvk ; 8 u; v 2 V :

The linear space V equipped with the norm k�k ; (V ; k�k) ; is called a normed linear

space or simply a normed space.

We now de�ne a norm k�k associated with the inner product h�; �i as:

kuk2 = hu; ui ; 8u 2 H:

Lemma 2.1.1 Let H be a real Hilbert space. Then

1. ku+ vk2 = kuk2 + 2 hu; vi+ kvk2 ; 8u; v 2 H:

2. kuk2 + hu; vi � �14kvk2 ; 8u; v 2 H:

2.2 Relative Convex Sets and Relative Convex Func-

tions

In recent years, the concept of convexity has been generalized in many dimensions, see [23]

and references therein. Using the idea of a segmental type of non-connected convexity

for sets by taking into account only convex combinations of special types of points,

Youness [128] and Jian [48] introduced a concept of relative convex sets and relative

convex functions with one and two arbitrary functions, independently. For the properties

of relative convex functions, see [23, 48, 83, 88, 101, 128]. Noor [90] has considered and

studied the relative convex function related to two arbitrary functions. It has been

shown [90] that the minimum of a di¤erentiable relative convex function involving three

14

functions can be characterized by a class of variational inequalities. We now recall the

following well known concepts.

De�nition 2.2.1 [48] Let be any set in a Hilbert space H. The set is said to be

hg-convex (relative convex) set, if there exist two functions g; h : H! H such that

(1� t)h (u) + tg (v) 2 ; 8 u; v 2 H : h (u) ; g (v) 2 ; 0 � t � 1:

We now discuss some special cases of De�nition 2.2.1.

I. If h = g, then the De�nition 2.2.1, reduces to

De�nition 2.2.2 [128] Let be any set in H. The set is said to be g-convex (relative

convex) set, if there exists a function g : H! H such that

(1� t) g (u) + tg (v) 2 ; 8 u; v 2 H : g (u) ; g (v) 2 ; 0 � t � 1:

II. If h = I, where I is an identity operator, then the De�nition 2.2.1, reduces to

De�nition 2.2.3 [88] Let be any set in H. The set is said to be g-convex (relative

convex) set, if there exists a function g : H! H such that

(1� t)u+ tg (v) 2 ; 8 u; v 2 H : u; g (v) 2 ; 0 � t � 1:

III. If g = I, where I is an identity operator, then the De�nition 2.2.1, reduces to

De�nition 2.2.4 Let be any set in H. The set is said to be h-convex (relative

convex) set, if there exists a function h : H! H such that

(1� t)h (u) + tv 2 ; 8 u; v 2 H : h (u) ; v 2 ; 0 � t � 1:

IV. If h = g = I, where I is an identity operator, then the De�nition 2.2.1, reduces to

15

De�nition 2.2.5 Let be any set in H. The set is said to be convex set, if

(1� t)u+ tv 2 ; 8u; v 2 ; 0 � t � 1:

In other words, the set is a convex set if, for all u; v 2 , the line segment joining u

and v is also in the set :

We would like to point out that, De�nitions 2.2.2 and 2.2.3 are quit di¤erent from

each other. In both de�nitions a straight line segment joining two points of a given set

is displaced by a straight line segment. But in the De�nition 2.2.3, which is due to Noor

[88], the origin point of the straight line segment does not transform. For more detail, see

[24, 25]. It is also important to note that every convex set is a g-convex (relative convex)

set. This fact is straight forward by considering g : H! H as an identity function. But

the converse is not true, For this purpose we consider the following example.

Example 1 [30] Consider a set =��1; �1

2

�[ [0; 1] and a function g (u) = u2, for all

u 2 R. In this example it is clear that the set is g-convex but not a convex set.

De�nition 2.2.6 [48] The function F : ! H is said to be hg-convex (relative con-

vex) function, if there exist two functions g; h : H ! H such that for all u; v 2 H :

h (u) ; g (v) 2 ; and for 0 � t � 1, we have

F ((1� t)h (u) + tg (v)) � (1� t)F (h (u)) + tF (g (v)) :

We now discuss some special cases of De�nition 2.2.6.

I. If h = g, then the De�nition 2.2.6, reduces to

De�nition 2.2.7 [128] The function F : ! H is said to be g-convex (relative convex)

function, if there exists a function g : H! H such that for all u; v 2 H : g (u) ; g (v) 2 ;

and for 0 � t � 1, we have

F ((1� t) g (u) + tg (v)) � (1� t)F (g (u)) + tF (g (v)) :

16

II. If h = I, where I is an identity operator, then the De�nition 2.2.6, reduces to

De�nition 2.2.8 [88] The function F : ! H is said to be g-convex (relative convex)

function, if there exists a function g : H ! H such that for all u; v 2 H : u; g (v) 2 ;


F ((1� t)u+ tg (v)) � (1� t)F (u) + tF (g (v)) :

III. If g = I, where I is an identity operator, then the De�nition 2.2.6, reduces to

De�nition 2.2.9 The function F : ! H is said to be h-convex (relative convex)

function, if there exists a function h : H ! H such that for all u; v 2 H : h (u) ; v 2 ;


F ((1� t)h (u) + tv) � (1� t)F (h (u)) + tF (v) :

IV. If h = g = I, where I is an identity operator, then the De�nition 2.2.6, reduces to

De�nition 2.2.10 The function F : ! H is said to be convex function, if

F ((1� t)u+ tv) � (1� t)F (u) + tF (v) ; 8u; v 2 ; 0 � t � 1:

We now give an example of a g-convex function which is also a convex function.

Example 2 [29] Consider a set = [0; 1] � R and a function g (u) =pjuj; for all,

u 2 R. Let F : R! R be a function such that F (u) = u2; for all, u 2 R : g (u) 2 . The

function F (u) is g-convex on a g-convex set , because for all u; v 2 R : g (u) ; g (v) 2

and for all 0 � t � 1; we have

(1� t)F (g (u)) + tF (g (v))� F ((1� t) g (u) + tg (v))

= (1� t)F�p

juj�+ tF

�pjvj�� F

�(1� t)

pjuj+ t

pjvj�

17

= (1� t)�p

juj�2+ t�p

jvj�2��(1� t)

pjuj+ t

pjvj�2

= t (1� t)�p

juj �pjvj�2> 0:

The following example shows that a g-convex function on a g-convex set is not nec-

essarily a convex function.

Example 3 [128] Let F : R! R be a function such that

F (u) =

8<: 1; if u > 0;

�u; if u � 0:

Consider R; the set of real numbers, is a g-convex set and the function g : R ! R be

de�ned as g (u) = �u2. It is obvious that, the function F is not convex but it is g-convex.

In the next example, we shall show that if the function F is g-convex and h-convex

on a set , but F is not necessarily hg-convex.

Example 4 [48] Let = R; a set of real numbers. Consider functions F; g and h de�ned

as

F (u) =

8<: �u; if u � 0;

u; if u < 0;g (u) = u2; h (u) = � juj :

Clearly,

F ((1� t) g (u) + tg (v)) � (1� t)F (g (u)) + tF (g (v)) ; 8 u; v 2 and t 2 [0; 1] :

and

F ((1� t)h (u) + th (v)) � (1� t)F (h (u)) + tF (h (v)) ; 8 u; v 2 and t 2 [0; 1] :

Thus the function F is g-convex and h-convex on = R. But F is not hg-convex on

18

= R; because

F ((1� t)h (1) + tg (�1)) = 0 > �1 = (1� t)F (h (1)) + tF (g (�1)) ; for t = 1

2:

If F is hg-convex function on , then F is not necessarily convex, not g-convex and

not h-convex on ; see the following example.

Example 5 [48] Let = R; a set of real numbers. Let the function F : R ! R be

de�ned as

F (u) =

8<: �1; if u > 0;

�u; if u � 0:

Consider functions g and h as

g (u) = u3; h (u) = 0:

Then it is clear that, for all u; v 2 and for 0 < t < 1; we have

F ((1� t)h (u) + tg (v)) � (1� t)F (h (u)) + tF (g (v)) :

This shows that the function F is hg-convex function on = R. Now we show that F is

neither convex nor g-convex, since

F�1

2� (�2) +

�1� 1

2

�� 12

�= F

��34

�=3

4>1

2=1

2F (�2) + 1

2F�1

2

�;

and

F�1

2g (�2) + 1

2g

�1

2

��= F

��6316

�=63

16>56

16=1

2F (g (�2)) + 1

2F�g

�1

2

��:

Example 6 Every linear function is a convex function, but the converse is not true. For

example, F (u) = u2 is a convex function, but it is not linear.

19

De�nition 2.2.11 A function f is di¤erentiable at u in the direction of v, if

limt!0

f (u+ tv)� f (u)t

=Df0(u) ; v

E, 8u; v 2 H,

provided that the limit exists. If f is a linear function, then

limt!0

f (u+ tv)� f (u)t

= hf; vi , 8u; v 2 H.

Sometime, this derivative is called Frechet derivative of f at u 2 H in the direction

of v 2 H.

We now show that the minimum of a di¤erentiable hg-convex function on the hg-

convex set in H can be characterized by the variational inequality and this is the main

motivation of our next result. For the sake of completeness and to convey an idea of the

applications, we give all the details.

Theorem 2.2.1 [87, 90, 93] Let F : ! H be a di¤erentiable hg-convex function.

Then u 2 H : h(u) 2 is the minimum of hg-convex function F on , if and only if,

u 2 H : h(u) 2 satis�es the inequality

hF0 (h (u)) ; g (v)� h (u)i � 0; 8v 2 H : g (v) 2 ; (2.2.1)

where F0 (�) is the di¤erential of F at h (u) 2 :

Proof. Let u 2 H : h(u) 2 be a minimum of hg-convex function F on . Then, we

have

F (h (u)) � F (g (v)) ; 8v 2 H : g (v) 2 : (2.2.2)

Since is a hg-convex set, so for all u; v 2 H : h (u) ; g (v) 2 and 0 � t � 1, we have

g (vt) = (1� t)h (u) + tg (v) = h (u) + t (g (v)� h (u)) 2 : (2.2.3)

20

Setting g (v) = g (vt), in the relation (2:2:2), we have

F (h (u)) � F (g (vt)) = F (h (u) + t (g (v)� h (u))) ;

which can be written as:

F (h (u) + t (g (v)� h (u)))� F (h (u))t

� 0:

Since F is di¤erentiable function, therefore by taking the limit as t! 0, we have

hF0 (h (u)) ; g (v)� h (u)i � 0, 8v 2 H : g (v) 2 ;

is the desired result.

Conversely, let u 2 H : h (u) 2 satis�es (2:2:1). Now we have to show that

u 2 H : h (u) 2 is a minimum of F on . Since F is a hg-convex function, so we have

F ((1� t)h (u) + tg (v)) � (1� t)F (h (u)) + tF (g (v)) ; 8u; v 2 ; t 2 [0; 1]

= F (h (u)) + t (F (g (v))� F (h (u))) ;


F (g (v))� F (h (u)) � F (h (u) + t (g (v)� h (u)))� F (h (u))t

;

taking limit as t! 0, and then using (2:2:1) we have

F (g (v))� F (h (u)) � hF0 (h (u)) ; g (v)� h (u)i � 0;

which implies

F (h (u)) � F (g (v)) ; v 2 H : g (v) 2 :

21

Hence by the de�nition of minimum it follows that u 2 H : h (u) 2 is the minimum of

F on in H.

If h = g, then as a direct consequences of Theorem 2.2.1, we obtain the following

corollary.

Corollary 2.2.1 [78] Let F : ! H be a di¤erentiable g-convex function. Then u 2 H :

g(u) 2 is the minimum of g-convex function F on , if and only if, u 2 H : g(u) 2

satis�es the inequality

hF0 (g (u)) ; g (v)� g (u)i � 0; 8v 2 H : g (v) 2 ;

where F0 (�) is the di¤erential of F at g (u) 2 :

If h = g = I, where I is an identity operator, then Theorem 2.2.1 reduces to the

following result.

Corollary 2.2.2 [96, 97] Let F : ! H be a di¤erentiable convex function. Then u 2

is the minimum of convex function F on , if and only if, u 2 satis�es the inequality

hF0 (u) ; v � ui � 0; 8v 2 ;

where F0 (�) is the di¤erential of F at u 2 :

2.3 Some Operators

In this section, we de�ne some linear and nonlinear operators on a Hilbert space H.

De�nition 2.3.1 An operator T : H! H is said to be linear, if

T (�u+ �v) = �T (u) + �T (v) ; 8u; v 2 H and �; � 2 F:

Otherwise T : H! H will be nonlinear.

22

De�nition 2.3.2 [97] A nonlinear operator T : H ! H is called strongly monotone, if

there exists a constant � > 0, such that

hTu� Tv; u� vi � � ku� vk2 ; 8u; v 2 H.

De�nition 2.3.3 [97] A nonlinear operator T : H! H is called monotone, if

hTu� Tv; u� vi � 0; 8u; v 2 H.

De�nition 2.3.4 [85] A nonlinear operator T : H ! H is called strongly expanding, if

there exists a constant � > 0, such that

kTu� Tvk � � ku� vk ; 8u; v 2 H.

De�nition 2.3.5 [103] A nonlinear operator T : H! H is called strongly antimonotone,

if there exists a constant � > 0, such that

hTu� Tv; u� vi � �� ku� vk2 ; 8u; v 2 H.

De�nition 2.3.6 [68] A nonlinear operator T : H! H is called antimonotone, if

hTu� Tv; u� vi � 0; 8u; v 2 H.

De�nition 2.3.7 [97] A nonlinear operator T : H ! H is called Lipschitz continuous,

if there exists a constant � > 0, such that

kTu� Tvk � � ku� vk ; 8u; v 2 H.

Remark 2.3.1 From the above de�nitions, we have

(i) Strongly monotonicity implies monotonicity and strongly expanding.

23

(ii) If the operator T is both strongly monotone and Lipschitz continuous, then � � �.

(iii) If 0 < � < 1, then the nonlinear operator T : H! H is called contraction mapping.

(iv) If � = 1, then the nonlinear operator T : H! H is called nonexpansive mapping.

De�nition 2.3.8 [97] A nonlinear operator T : H! H is said to be cocoercive, if there

exists a constant > 0 such that

hTu� Tv; u� vi � kT (u)� T (v)k2 ; 8u; v 2 H:

De�nition 2.3.9 [85] A nonlinear operator T : H! H is said to be strongly g-monotone,

if there exists a mapping g : H! H and a constant �1 > 0 such that

hTu� Tv; g (u)� g (v)i � �1 kg (u)� g (v)k2 ; 8u; v 2 H:

De�nition 2.3.10 [85] A nonlinear operator T : H ! H is said to be strongly g-

expanding, if there exists a mapping g : H! H and a constant �1 > 0 such that

kTu� Tvk � �1 kg (u)� g (v)k ; 8u; v 2 H:

De�nition 2.3.11 [85] A nonlinear operator T : H ! H is said to be g-Lipschitz con-

tinuous, if there exists a mapping g : H! H and a constant �1 > 0 such that

kTu� Tvk � �1 kg (u)� g (v)k ; 8u; v 2 H:

From the De�nitions 2.3.9, 2.3.10 and 2.3.11, it is clear that if g = I; where I is an

identity operator, then above de�nitions reduce to the well known de�nitions of strongly

monotone, strongly expanding and Lipschitz continuous, respectively.

Lemma 2.3.2 [50] Let be a nonempty, closed and convex set in H. Then for a given

24

z 2 H, u 2 satis�es

hu� z; v � ui � 0; 8v 2 ; (2.3.1)

if and only if

u = P [z] ;

where P is the projection of H onto a closed and convex set in H.

We would like to mention that, the projection operator P has the following proper-

ties.

(i) The projection operator P is nonexpansive, that is,

kP [u]� P [v]k � ku� vk ; 8 u; v 2 H.

(ii) The projection operator P is cocoercive map with modulus 1 on H, that is,

hP [u]� P [v] ; u� vi � kP [u]� P [v]k2 ; 8 u; v 2 H. (2.3.2)

(iii) It follows from Lemma 2.3.2 that

hP [z]� z; v � P [z]i � 0; 8 z 2 H; v 2 : (2.3.3)

(iv) From (2:3:3), it is clear that

kP [z]� vk2 � kz � vk2 � kz � P [z]k2 ; 8 z 2 H; v 2 :

Lemma 2.3.3 [122] If f�ng1n=0 is a nonnegative sequence satisfying the following in-

equality:

�n+1 � (1� �n) �n + �n 8n � 0;

with 0 � �n � 1,1Pn=0

�n =1, and �n = o (�n), then limn!1

�n = 0.

25

Chapter 3

Formulation and Existence Results

26

3.1 Introduction

In this chapter we consider a new class of quasi variational inequalities. This new class is

called extended general quasi variational inequality. Some special cases of extended gen-

eral quasi variational inequalities are discussed. Using projection technique it is shown

that extended general quasi variational inequalities are equivalent to �xed point problems.

We discuss the existence of a solution of the extended general quasi variational inequali-

ties. Several special cases are also discussed. Some of the results obtained in this chapter

have been published in Afrika Matematika (2014), DOI 10.1007/s13370-014-0304-5, see

[107].

3.2 Problem Formulation

Let H be a real Hilbert space, whose norm and inner product are denoted by k�k and

h�; �i ; respectively. Let be a closed and convex set in H. Let : u ! (u) be a

point-to-set mapping which associates a closed and convex-valued set (u) of H with

any element u of H.

For given three nonlinear operators T; g; h : H ! H, consider a problem of �nding

u 2 H : h (u) 2 (u) such that

h�Tu+ h (u)� g (u) ; g (v)� h (u)i � 0; 8v 2 H : g (v) 2 (u) ; (3.2.1)

where � > 0 is a constant. The inequality of type (3:2:1) is called extended general quasi

variational inequality. This problem was introduced by Noor and Noor [100]. For the

�xed point formulation, existence of a solution, equivalence with Wiener-Hopf equation,

numerical methods and sensitivity analysis of the problem (3:2:1), see [100, 104, 108].

We now discuss some special cases of the problem (3:2:1).

27

3.3 Some Special Cases

I. If h = g, then the problem (3:2:1) is equivalent to �nding u 2 H : g (u) 2 (u) and

hTu; g (v)� g (u)i � 0; 8v 2 H : g (v) 2 (u) ; (3.3.1)

which is known as the general quasi variational inequality, introduced and studied

by Noor [70].

II. If h = I, where I is an identity operator, then the problem (3:2:1) is equivalent to

�nding u 2 (u) such that

h�Tu+ u� g (u) ; g (v)� ui � 0; 8v 2 H : g (v) 2 (u) ; (3.3.2)

is known as the general quasi variational inequality. This class is quite general and

uni�ed one. For more detail see, Noor et al [102].

III. If g = I, where I is an identity operator, then the problem (3:2:1) reduces to �nding

u 2 H : h (u) 2 (u) such that

h�Tu+ h (u)� u; v � h (u)i � 0; 8v 2 (u) ; (3.3.3)

is called the general quasi variational inequality.

IV. If g = h = I, where I is an identity operator, then the problem (3:2:1) is equivalent

to �nding u 2 (u) such that

hTu; v � ui � 0; 8v 2 (u) ; (3.3.4)

which is known as quasi variational inequality. This problem was introduced and

studied by Bensoussan and Lions [18] in the study of impulse control system. For

recent applications and for more detail, see [7, 10, 15, 16, 17, 33, 34, 52, 64, 65, 67,

28

70] and the references therein.

V. If (u) = , then the problem (3:2:1) is equivalent to �nding u 2 H : h (u) 2

and

h�Tu+ h (u)� g (u) ; g (v)� h (u)i � 0; 8v 2 H : g (v) 2 ; (3.3.5)

which is called the extended general variational inequality. This problem is pro-

posed and studied by Noor [90]. For the formulation, numerical algorithms and

recent applications, see [57, 60, 61, 93] and references therein.

VI. If (u) = and h = g, then the problem (3:2:1) is equivalent to the problem of

�nding u 2 H : g (u) 2 such that

hTu; g (v)� g (u)i � 0; 8v 2 H : g (v) 2 : (3.3.6)

This problem is known as general variational inequality, introduced and studied by

Noor [69]. It turned out that odd order and nonsymmetric obstacle, free, moving,

unilateral and equilibrium problems arising in various branches of pure and applied

sciences can be studied by the problem (3:3:6), see [57, 78, 84, 85, 89] and the

closely related references therein.

VII. If (u) = and h = I, where I is an identity operator, then the problem (3:2:1) is

equivalent to the problem of �nding u 2 such that

h�Tu+ u� g (u) ; g (v)� ui � 0; 8v 2 H : g (v) 2 ; (3.3.7)

is known as general variational inequality. This problem considered and studied by

Noor [88].

VIII. If (u) = and g = I, where I is an identity operator, then the problem (3:2:1) is

29

equivalent to the problem of �nding u 2 H : h (u) 2 such that

h�Tu+ h (u)� u; v � h (u)i � 0; 8v 2 ; (3.3.8)

which is also called the general variational inequality involving two nonlinear oper-

ators which was introduced and studied by Noor [69, 70].

XI. If (u) = and h = g = I, where I is an identity operator, then the problem

(3:2:1) is equivalent to the problem of �nding u 2 such that

hTu; v � ui � 0; 8v 2 ; (3.3.9)

which is the original variational inequality. It was introduced and studied by Stamp-

pachia [121]. For the recent applications, numerical algorithms, sensitivity analysis,

dynamical systems, generalizations and formulations of variational inequalities, see

[8, 9, 10, 11, 19, 20, 43, 49, 50, 83, 84, 97] and the references therein.

X. If � (u) = fu 2 H : hu; vi � 0;8v 2 (u)g is a polar (dual) cone of a closed convex-

valued set (u) in H, then problem (3:3:1) is equivalent to �nding u 2 H such that

g (u) 2 (u) ; Tu 2 � (u) ; hTu; g (u)i = 0; (3.3.10)

which is known as the general quasi complementarity problem, see [41, 52, 69, 84].

If g = I, the identity operator, then problem (3:3:10) is called the generalized

quasi complementarity problem. For g (u) = u � m (u), where m is a point-to-

point mapping, then problem (3:3:10) is called the quasi (implicit) complementarity

problem, see [77, 84] and the references therein.

From the above discussion, it is clear that the extended general quasi variational

inequality (3:2:1) is most general and includes several previously known classes of vari-

ational inequalities and related problems as special cases. These variational inequalities

30

have important applications in mathematical programming and engineering sciences. For

the recent applications, numerical methods, sensitivity analysis, dynamical systems and

formulation of quasi variational inequalities and related �elds, see [1-129] and the refer-

ences therein.

Lemma 3.3.1 [67] Let (u) be a closed convex-valued set in H. Then, for a given

z 2 H, u 2 (u) satis�es the inequality

hu� z; v � ui � 0, 8v 2 (u) ;

if and only if,

u = P(u) [z] ;

where P(u) is the projection of H onto the closed convex-valued set (u) in H.

We would like to point out that the implicit projection operator P(u) is not nonexpan-

sive. We shall assume that the implicit projection operator P(u) satis�es the Lipschitz

type continuity, which plays an important and fundamental role in the existence the-

ory and in developing numerical methods for solving extended general quasi variational

inequality (3:2:1) and its variant forms.

Assumption 3.1 [67] The implicit projection operator P(u) satis�es the condition

P(u) [w]� P(v) [w] � � ku� vk , 8u; v; w 2 H, (3.3.11)

where � > 0 is a positive constant.

In many important applications [41, 64, 65, 67] the convex-valued set (u) can be

written as

(u) = m (u) + ; (3.3.12)

31

where m (u) is a point-to-point mapping and is a convex set. In this case, we have

P(u) [w] = Pm(u)+ [w] = m (u) + P [w �m (u)] , 8u; v 2 H. (3.3.13)

We note that if (u) is as, de�ned by (3:3:12) ; and m (u) is a Lipschitz continuous

mapping with constant > 0, then using the relation (3:3:13), we have

P(u) [w]� P(v) [w] = km (u) + P [w �m (u)]�m (v)� P [w �m (v)]k

� km (u)�m (v)k+ kP [w �m (u)]� P [w �m (v)]k

� 2 km (u)�m (v)k

� 2 ku� vk , 8u; v; w 2 H;

which shows that Assumption 3.1 holds with � = 2 > 0.

One can show that the extended general quasi variational inequality (3:2:1) is equiv-

alent to the �xed point problem by invoking Lemma 3.3.1.

Lemma 3.3.2 [101] The function u 2 H : h (u) 2 (u) is a solution of the extended

general quasi variational inequality (3:2:1) ; if and only if, u 2 H : h (u) 2 (u) satis�es

the relation

h (u) = P(u) [g (u)� �Tu] ; (3.3.14)

where P(u) is the implicit projection operator and � > 0 is a constant.

Lemma 3.3.2 implies that the extended general quasi variational inequality (3:2:1)

is equivalent to the implicit �xed point problem (3:3:14). This alternative equivalent

formulation is very useful from the numerical and theoretical point of view.

From (3:3:14), we can write

u = u� h (u) + P(u) [g (u)� �Tu] ; (3.3.15)

32

this enables us to de�ne the mapping F (u) as:

F (u) = u� h (u) + P(u) [g (u)� �Tu] : (3.3.16)

3.4 Existence

We now study those conditions under which the extended general quasi variational in-

equalities (3:2:1) has a solution and this is the main motivation for obtaining our next

result.

Theorem 3.4.1 Let the operators T; g; h : H! H be strongly monotone with constants

�T > 0; �g > 0; �h > 0 and Lipschitz continuous with constants �T > 0; �g > 0; �h > 0;

respectively. If Assumption 3.1 holds and � > 0 satis�es the condition

�� T�2T�� <

q�2T � k (2� k) �2T

�2T; �T > �T

pk (2� k); k < 1; (3.4.1)

k = � +q1� 2�g + �2g +

q1� 2�h + �2h; (3.4.2)

then there exists a solution u 2 H : h (u) 2 (u) satisfying the problem (3:2:1).

Proof. Let u 2 H : h (u) 2 (u) be a solution of (3:2:1). To prove the existence of a

solution of problem (3:2:1) ; it is enough to show that the problem (3:3:16) has a unique

�xed point. Thus, for u1 6= u2 2 H : h (u1) 6= h (u2) 2 (u), consider

kF (u1)� F (u2)k

= u1 � h (u1) + P(u1) [g (u1)� �Tu1]� u2 + h (u2)� P(u2) [g (u2)� �Tu2]

� ku1 � u2 � (h (u1)� h (u2))k+ P(u1) [g (u1)� �Tu1]� P(u2) [g (u2)� �Tu2]

� ku1 � u2 � (h (u1)� h (u2))k+ P(u1) [g (u1)� �Tu1]� P(u2) [g (u1)� �Tu1]

+ P(u2) [g (u1)� �Tu1]� P(u2) [g (u2)� �Tu2]

33

� ku1 � u2 � (h (u1)� h (u2))k+ � ku1 � u2k+ kg (u1)� g (u2)� � (Tu1 � Tu2)k

� � ku1 � u2k+ ku1 � u2 � (g (u1)� g (u2))k

+ ku1 � u2 � (h (u1)� h (u2))k+ ku1 � u2 � � (Tu1 � Tu2)k ; (3.4.3)

where we have used Assumption 3.1.

Since operator T is strongly monotone with constant �T > 0 and Lipschitz continuous

with constant �T > 0, so

ku1 � u2 � � (Tu1 � Tu2)k2

= ku1 � u2k2 � 2� hTu1 � Tu2; u1 � u2i+ �2 kTu1 � Tu2k2

� ku1 � u2k2 � 2��T ku1 � u2k2 + �2�2T ku1 � u2k2

=�1� 2��T + �2�2T

�ku1 � u2k2 : (3.4.4)

In a similar way, we have

ku1 � u2 � (g (u1)� g (u2))k2 ��1� 2�g + �2g

�ku1 � u2k2 ; (3.4.5)

and

ku1 � u2 � (h (u1)� h (u2))k2 ��1� 2�h + �2h

�ku1 � u2k2 ; (3.4.6)

where we have used strongly monotonicity and Lipschitz continuity of operators g and h

with constants �g > 0, �g > 0 and �h > 0, �h > 0, respectively.

Combining (3:4:3)� (3:4:6), we have

kF (u1)� F (u2) k � f� +q1� 2�g + �2g +

q1� 2�h + �2h

+

q1� 2��T + �2�2Tgku1 � u2k

= fk + t (�)g ku1 � u2k

= �ku1 � u2k; (3.4.7)

34

where

� = k + t (�) ;

t (�) =

q1� 2��T + �2�2T :

From the condition (3:4:1), it follows that � < 1: This shows that the mapping F (u)

de�ned by (3:3:16) is a contraction mapping and consequently it has a unique �xed point

u 2 H : h (u) 2 (u) satisfying the problem (3:2:1). This is the desired result.

If (u) = , then Theorem 3.4.1 reduces to the following result for the problem

(3:3:5).

Theorem 3.4.2 [90] Let the operators T; g; h : H ! H be strongly monotone with

constants �T > 0; �g > 0; �h > 0 and Lipschitz continuous with constants �T > 0;

�g > 0; �h > 0; respectively. If � > 0 satis�es the condition

�� T�2T�� <

q�2T � k (2� k) �2T

�2T; �T > �T

pk (2� k); k < 1; (3.4.8)

k =q1� 2�g + �2g +

q1� 2�h + �2h;

then there exists a unique solution u 2 H : h (u) 2 satisfying the problem (3:3:5).

3.5 Wiener-Hopf Equations Technique

In this Section, we consider the problem of solving the extended general implicit Wiener-

Hopf equations. These problems are related with the extended general quasi variational

inequality (3:2:1). To be more precise, let Q(u) = I� gh�1P(u), where I is an identity

operator and h�1 exists. For given nonlinear operators T; g; h, we consider the problem

35

of �nding z 2 H such that

Th�1P(u) [z] + ��1Q(u) [z] = 0; (3.5.1)

which is called the extended general implicit Wiener-Hopf equation.

We now discuss some special cases of the problem (3:5:1).

I. If h = g, then the problem (3:5:1) is equivalent to �nding z 2 H such that

Tg�1P(u) [z] + ��1Q(u) [z] = 0; (3.5.2)

which is known as the general implicit Wiener-Hopf equation related to the problem

(3:3:1) :

II. If h = I, where I is an identity operator, then the problem (3:5:1) is equivalent to

�nding z 2 H such that

TP(u) [z] + ��1Q(u) [z] = 0; (3.5.3)

which is known as the general implicit Wiener-Hopf equation related to the problem

(3:3:2), introduced and studied by Noor et al. [102].

III. If g = I, where I is an identity operator, then the problem (3:5:1) reduces to �nding

z 2 H such that

Th�1P(u) [z] + ��1Q(u) [z] = 0; (3.5.4)

is called the general implicit Wiener-Hopf equation related to the problem (3:3:3) :

IV. If g = h = I, where I is an identity operator, then the problem (3:5:1) is equivalent

to �nding z 2 H such that

TP(u) [z] + ��1Q(u) [z] = 0; (3.5.5)

36

which is known as the implicit Wiener-Hopf equation related to the problem (3:3:4) :

V. If (u) = , then the problem (3:5:1) is equivalent to �nding z 2 H such that

Th�1P [z] + ��1Q [z] = 0; (3.5.6)

which is called the extended general Wiener-Hopf equation associated with the

problem (3:3:5) : This problem was introduced and studied by Noor [91].

VI. If (u) = and h = g, then the problem (3:5:1) is equivalent to the problem of

�nding z 2 H such that

Tg�1P [z] + ��1Q [z] = 0; (3.5.7)

which is known as the general Wiener-Hopf equation related to the problem (3:3:6) :

This problem was introduced and studied by Noor [72].

VII. If (u) = and h = g = I, where I is an identity operator, then the problem

(3:5:1) is equivalent to the problem of �nding u 2 such that

TP [z] + ��1Q [z] = 0; (3.5.8)

which is the original Wiener-Hopf equations associated with the problem (3:3:9).

This problem was mainly due to Shi [117]. It has been shown that the Wiener-

Hopf equations have played an important and signi�cant role in developing several

numerical techniques for solving variational inequalities and related optimization

problems.

It has been shown in [101] that the problems (3:2:1) and (3:5:1) are equivalent. This

equivalence has been used to study the sensitivity analysis of the extended general quasi

variational inequality (3:2:1). For the sake of completeness and to convey an idea, we

include its proof.

37

Lemma 3.5.3 [101] The solution u 2 H : h(u) 2 (u) satis�es the extended general

quasi variational inequalities (3:2:1), if and only if, z 2 H is a solution of the extended

general implicit Wiener-Hopf equations (3:5:1), where

h (u) = P(u) [z] (3.5.9)

z = g (u)� �Tu; (3.5.10)

where � > 0 is a constant.

Proof. Let u 2 H : h(u) 2 (u) be a solution of (3:2:1). Then, from Lemma 3.3.2,

we have

h (u) = P(u) [g (u)� �Tu] . (3.5.11)

Let

z = g (u)� �Tu. (3.5.12)

Then

h (u) = P(u) [z] : (3.5.13)

Combining (3:5:12) and (3:5:13), and using the fact that h�1 exists, we have

z = g (u)� �Tu

= g�h�1

�P(u) [z]

�� T

�h�1

�P(u) [z]

��;


0 =�I� gh�1P(u)

�[z] + �Th�1P(u) [z]

= Q(u) [z] + �Th�1P(u) [z] ;

from which it follows that z 2 H is a solution of the extended general implicit Wiener-

Hopf equations (3:5:1).

38

Conversely, let z 2 H be a solution of (3:5:1), then

0 = �Th�1P(u) [z] +Q(u) [z]

= �Th�1P(u) [z] + z � gh�1P(u) [z]

= �Tu+ z � g (u) ; using (3:5:9) ,

which implies

h (u) = P(u) [g (u)� �Tu] ;

which shows u 2 H : h (u) 2 (u) is the solution of problem (3:2:1). This completes the

proof.

Lemma 3.5.3 implies that the extended general quasi variational inequalities (3:2:1)

and the extended general implicit Wiener-Hopf equations (3:5:1) are equivalent. We use

this equivalent formulation to study the existence of a solution of the problem (3:5:1).

This is the main motivation of our next result.

From (3:5:10) ; we can de�ne a mapping F (z) as

F (z) = g (u)� �Tu; (3.5.14)

and

h (u) = P(u) [z] : (3.5.15)

The following theorem provides the conditions under which the existence of the solution

to extended general implicit Wiener-Hopf equations (3:5:1) are guaranteed.



respectively. If Assumption 3.1 holds and � > 0 satis�es the condition (3:4:1) ; then there

exists a solution z 2 H satisfying the problem (3:5:1).

Proof. Let z 2 H be a solution of (3:5:1). To prove the existence of a solution of

39

problem (3:5:1) ; it is enough to show that the mapping (3:5:14) has a unique �xed point.

Thus, for z1 6= z2 2 H, consider

kF (z1)� F (z2)k = kg (u1)� g (u2)� � (Tu1 � Tu2)k

� ku1 � u2 � (g (u1)� g (u2))k+ ku1 � u2 � � (Tu1 � Tu2)k

��q

1� 2�g + �2g +q1� 2��T + �2�2T

�ku1 � u2k ; (3.5.16)

where we have used the results from (3:4:4) and (3:4:5).

From (3:5:15) ; using Assumption 3.1 and relation (3:4:6), we have

ku1 � u2k � ku1 � u2 � (h (u1)� h (u2))k+ P(u1) [z1]� P(u2) [z2]

� ku1 � u2 � (h (u1)� h (u2))k+ P(u1) [z1]� P(u2) [z1]

+ P(u2) [z1]� P(u2) [z2]

�� +

q1� 2�h + �2h

�ku1 � u2k+ kz1 � z2k ;

which can be simpli�ed as:

ku1 � u2k �1

1� � �p1� 2�h + �2h

kz1 � z2k : (3.5.17)

Combining (3:5:16) and (3:5:17), we have

kF (z1)� F (z2)k �

q1� 2�g + �2g +

p1� 2��T + �2�2T

1� � �p1� 2�h + �2h

kz1 � z2k

= �1 kz1 � z2k ;

where

�1 =

q1� 2�g + �2g +

p1� 2��T + �2�2T

1� � �p1� 2�h + �2h

:

From the condition (3:4:1), it follows that �1 < 1: This shows that the mapping F (z)

40


z 2 H satisfying the problem (3:5:1). This is the desired result.

If (u) = , then the Theorem 3.5.1 reduces to the following result.



respectively. If � > 0 satis�es the condition (3:4:8) ; then there exists a solution z 2 H

satisfying the problem (3:5:6).

We now de�ne the residue vector R (u) as:

R (u) = h (u)� P(u) [g (u)� �Tu] : (3.5.18)

It is clear from Lemma 3:3:2 that problem (3:2:1) has a solution u 2 H : h (u) 2 (u), if

and only if, u 2 H : h (u) 2 (u) is a zero of the equation

R (u) = 0: (3.5.19)

Lemma 3.5.4 The mapping Qu : H! H; de�ned as:

Qu (w) = w � P(u) [w] ;

is a cocoercive on H with modulus 1.

Proof. From the relation (2:3:2), we have

P(u) [w]� P(u) [z] ; w � z

�� P(u) [w]� P(u) [z] 2 ;

41


0 �P(u) [w]� P(u) [z] ; w � z � P(u) [w] + P(u) [z]

�=

�w � P(u) [w]

��z � P(u) [z]

�; P(u) [w]� P(u) [z]

�= hQu (w)�Qu (z) ; w �Qu (w)� z +Qu (z)i

= hQu (w)�Qu (z) ; w � zi � hQu (w)�Qu (z) ; Qu (w)�Qu (z)i ;

form which have

hQu (w)�Qu (z) ; w � zi � kQu (w)�Qu (z)k2 ;

which is the required result.

We now show that the residue vectorR (u), de�ned by (3:5:18), is strongly g-monotone

and Lipschitz continuous on H.

Lemma 3.5.5 Let T is strongly g-monotone and g-Lipschitz continuous with constant

�T > 0 and �T > 0, respectively. Let h is strongly g-monotone with constant �h > 0. If

g is strongly expanding with constant �g > 0 and Assumption 3.1 holds, then the residue

vector R (u), de�ned by (3:5:18), is strongly g-monotone on H:

Proof. For all u; v 2 H; consider

hR (u)�R (v) ; g (u)� g (v)i

=h (u)� P(u) [g (u)� �Tu]� h (v) + P(v) [g (v)� �Tv] ; g (u)� g (v)

�= hh (u)� h (v) ; g (u)� g (v)i+ hg (u)� P(u) [g (u)� �Tu]� g (v)

+P(v) [g (v)� �Tv]� (g (u)� g (v)) ; g (u)� g (v)i

= hh (u)� h (v) ; g (u)� g (v)i � hg (u)� g (v) ; g (u)� g (v)i

+g (u)� P(u) [g (u)� �Tu]� g (v) + P(v) [g (v)� �Tv] ; g (u)� g (v)

�

42

� �h kg (u)� g (v)k2 � kg (u)� g (v)k2

+h(g (u)� �Tu)� P(u) [g (u)� �Tu]� (g (v)� �Tv) + P(v) [g (v)� �Tv]

+ (�Tu� �Tv) ; g (u)� g (v)i

= �h kg (u)� g (v)k2 � kg (u)� g (v)k2 + � hTu� Tv; g (u)� g (v)i

+h(g (u)� �Tu)� P(u) [g (u)� �Tu]

� (g (v)� �Tv) + P(v) [g (v)� �Tv] ; g (u)� g (v)i; (3.5.20)

where we have used the strongly g-monotonicity of operator h with constant �h > 0.

Let w = g (u) � �Tu and z = g (v) � �Tv, then from (3:5:20) using the strongly

g-monotonicity of operator T with constant �T > 0, we have

hR (u)�R (v) ; g (u)� g (v)i

� �h kg (u)� g (v)k2 � kg (u)� g (v)k2 + ��T kg (u)� g (v)k2

+�w � P(u) [w]

��z � P(v) [z]

�; g (u)� g (v)

�= (�h � 1 + ��T ) kg (u)� g (v)k2

+�w � P(u) [w]

��z � P(u) [z]

�; g (u)� g (v)

��P(u) [z]� P(v) [z] ; g (u)� g (v)

�; (3.5.21)

Let w � P(u) [w] = Qu (w) and z � P(u) [z] = Qu (z), then form (3:5:21) using the

Assumption 3.1, Lemma 2.1.1, Lemma 3.5.4, g-Lipschitz continuity of T with constant

�T > 0 and strongly expandicity of g with constant �g > 0, we get

hR (u)�R (v) ; g (u)� g (v)i

= (�h � 1 + ��T ) kg (u)� g (v)k2 � P(u) [z]� P(v) [z] kg (u)� g (v)k

+ hQu (w)�Qu (z) ; g (u)� g (v)i

� (�h + ��T � 1) kg (u)� g (v)k2 � � ku� vk kg (u)� g (v)k

+ hQu (w)�Qu (z) ; w + �Tu� z � �Tvi

43

� (�h + ��T � 1) kg (u)� g (v)k2 ��

�gkg (u)� g (v)k2

+ hQu (w)�Qu (z) ; w � zi+ hQu (w)�Qu (z) ; � (Tu� Tv)i

��h + ��T � 1�

�

�g

�kg (u)� g (v)k2 + kQu (w)�Qu (z)k2

+ hQu (w)�Qu (z) ; � (Tu� Tv)i

��h + ��T � 1�

�

�g

�kg (u)� g (v)k2 � �

2

4kTu� Tvk2

��h + ��T � 1�

�

�g

�kg (u)� g (v)k2 � �

2�2T4

kg (u)� g (v)k2

= �1 kg (u)� g (v)k2 ;

which implies that

hR (u)�R (v) ; g (u)� g (v)i � �1 kg (u)� g (v)k2 ;

where

�1 = �h + ��T � 1��

�g� �

2�2T4

> 0:

This implies that the residue vector R (u), de�ned by (3:5:18), is strongly g-monotone

with constant �1 > 0. This is the desired result.

Lemma 3.5.6 Let operators T and g are strongly monotone and Lipschitz continuous

with constants �T > 0; �g > 0 and �T > 0; �g > 0, respectively. If h is Lipschitz

continuity with constant �h > 0 and Assumption 3.1 holds, then the residue vector R (u),

de�ned by (3:5:18), is Lipschitz continuous on H:

Proof. For all u; v 2 H, consider

kR (u)�R (v)k � kh (u)� h (v)k

+ P(u) [g (u)� �Tu]� P(v) [g (v)� �Tv]

� �h ku� vk+ P(u) [g (u)� �Tu]� P(v) [g (u)� �Tu]

+ P(v) [g (u)� �Tu]� P(v) [g (v)� �Tv]

44

� �h ku� vk+ � ku� vk+ kg (u)� g (v)� � (Tu� Tv)k

� (� + �h) ku� vk+ ku� v � (g (u)� g (v))k+ ku� v � � (Tu� Tv)k

�� + �h +

q1� 2�g + �2g +

q1� 2��T + �2�2T

�ku� vk

= �2 ku� vk ;

where

�2 = � + �h +q1� 2�g + �2g +

q1� 2��T + �2�2T > 0:

In the above, we have used the strongly monotonicity and Lipschitz continuity of

operators T and g with constants �T > 0; �g > 0 and �T > 0; �g > 0, respectively. We

have also used the Lipschitz continuity of h with constant �h > 0 and Assumption 3.1.

Thus the residue vector R (u), de�ned by (3:5:18), is Lipschitz continuous with con-

stant �2 > 0. This completes the proof.

For a constant > 0; the relation (3:5:19) can be written as:

u = u� R (u) : (3.5.22)

This �xed point formulation enables us to de�ne the mapping G (u) as:

G (u) = u� R (u) : (3.5.23)

We now discuss the conditions under which the �xed point mapping G (u), de�ned by

(3:5:23) ; has a unique �xed point. Consequently, the �xed point of the problem (3:5:23)

is the zero of the problem (3:5:19) :

Theorem 3.5.3 Let the operator R : H! H be strongly monotone with constant �R >

0 and Lipschitz continuous with constant �R > 0: If the constant > 0 satis�es the

condition

0 < <2�R

�2R; (3.5.24)

then there exists a solution u 2 H satisfying the problem (3:5:19).

45

Proof. Let u 2 H be a solution of (3:5:19). To prove the existence of a solution

of problem (3:5:19) ; it is enough to show that the problem (3:5:23) has a unique �xed

point. Thus, for u1 6= u2 2 H, consider

kG (u1)�G (u2)k2 = ku1 � u2 � (R (u1)�R (u2))k2

= hu1 � u2 � (R (u1)�R (u2)) ; u1 � u2 � (R (u1)�R (u2))i

= ku1 � u2k2 � 2 hR (u1)�R (u2) ; u1 � u2i

+ 2 kR (u1)�R (u2)k2

��1� 2 �R + 2�2R

�ku1 � u2k2 ; (3.5.25)

where we have used strongly monotonicity and Lipschitz continuity of operator R with

constants �R > 0 and �R > 0, respectively.

From the relation (3:5:25), we have

kG (u1)�G (u2)k � �R ku1 � u2k ;

where

�R =

q1� 2 �R + 2�2R:

From the condition (3:5:24), it follows that �R < 1: This shows that the mapping G (u)


u 2 H satisfying the problem (3:5:23). This is the desired result.

3.6 Conclusion

In this chapter, we have introduced and considered a class of quasi variational inequalities

involving three nonlinear operators. It has been shown that the extended general quasi

variational inequalities are equivalent to the �xed point problems and implicit Wiener-

Hopf equations. These alternative equivalent formulations are very useful from numerical

46

and theoretical point of view. Several special cases are also discussed, which can be

obtained from our results. Results obtained in this chapter continue to hold for these

problems.

47

Chapter 4

Iterative Methods

48

4.1 Introduction

In the previous chapter, we have considered a class of quasi variational inequalities. This

class is called extended general quasi variational inequalities (3:2:1) : Several special cases

of this class are also discussed, which show that the extended general quasi variational

inequalities (3:2:1) are quite general. In this chapter, we suggest and analyze some one-

step, two-step and three-step iterative methods for solving the problem (3:2:1) and its

variant forms. We would like to mention that the contents of this chapter have been

published in Applied Mathematics & Information Sciences, 7(3), 917-925, see [101] and

Afrika Matematika, DOI 10.1007/s13370-014-0304-5, see [107].

4.2 Projection Operator Technique

The �xed point formulation (3:3:14) can be written as:

u = (1� �n)u+ �n�u� h (u) + P(u) [g (u)� �Tu]

, (4.2.1)

where 0 � �n � 1 is a controlling parameter. This �xed point formulation enables us

to suggest and analyze the following iterative method for solving the extended general

quasi variational inequalities (3:2:1).

Algorithm 4.1 For a given u0 2 H, �nd the approximate solution un+1 by the iterative

schemes

un+1 = (1� �n)un + �n�un � h (un) + P(un) [g (un)� �Tun]

, n = 0; 1; 2; ::: (4.2.2)

where 0 � �n � 1 for all n � 0.

The Algorithm 4.1 is known as the Mann iteration process for solving the extended

general quasi variational inequalities (3:2:1).

We now discuss some special cases of Algorithm 4.1.

49

I. If h = g, then Algorithm 4.1 reduces to the following iterative method for solving

the general quasi variational inequalities (3:3:1).

Algorithm 4.2 [94] For a given u0 2 H, �nd the approximate solution un+1 by the

iterative schemes

un+1 = (1� �n)un + �n�un � g (un) + P(un) [g (un)� �Tun]

, n = 0; 1; 2; :::


II. If h = I, I is an identity operator, then Algorithm 4.1 reduces to the following

iterative scheme for solving problem (3:3:2).


iterative schemes

un+1 = (1� �n)un + �nP(un) [g (un)� �Tun] , n = 0; 1; 2; :::


III. If g = I, I is an identity operator, then Algorithm 4.1 reduces to the following

algorithm for solving problem (3:3:3).


schemes

un+1 = (1� �n)un + �n�un � h (un) + P(un) [un � �Tun]

, n = 0; 1; 2; :::


IV. If h = g = I, I is an identity operator, then Algorithm 4.1 collapse to the following

algorithm for solving the quasi variational inequalities (3:3:4).

50


iterative schemes

un+1 = (1� �n)un + �nP(un) [un � �Tun] , n = 0; 1; 2; :::


V. If (�) = , a closed and convex set in H, then Algorithm 4.1 collapse to the

following algorithm for solving the extended general variational inequalities (3:3:5).


iterative schemes

un+1 = (1� �n)un + �n fun � h (un) + P [g (un)� �Tun]g , n = 0; 1; 2; :::


For di¤erent and appropriate choice of the operators T; g; h; set and the space

H, one can obtain several known and new iterative methods for solving a wide class of

variational inequalities and related complementarity problems. This clearly shows that

the iterative methods, which we considered here are more general and unifying ones.

We now consider the convergence analysis of Algorithm 4.1 and this is the main

motivation of our next result.



respectively. Let Assumption 3.1 and condition (3:4:1) holds. If 0 � �n � 1, for all n � 0

and1Pn=0

�n =1, then the approximate solution un obtained from Algorithm 4.1 converges

to a solution u 2 H : h (u) 2 (u) satisfying the extended general quasi variational

inequality (3:2:1).

51

Proof. Let u 2 H : h (u) 2 (u) be a solution of the extended general quasi

variational inequality (3:2:1). Then, using (4:2:1) and (4:2:2), we have

kun+1 � uk = (1� �n)un + �n �un � h (un) + P(un) [g (un)� �Tun]� (1� �n)u� �n

�u� h (u) + P(u) [g (u)� �Tu]

� (1� �n) kun � uk+ �n kun � u� (h (un)� h (u))k

+�n P(un) [g (un)� �Tun]� P(u) [g (u)� �Tu]


+�n P(un) [g (un)� �Tun]� P(u) [g (un)� �Tun]

+�n P(u) [g (un)� �Tun]� P(u) [g (u)� �Tu]


+��n kun � uk+ �n kg (un)� g (u)� (�Tun � �Tu)k


+��n kun � uk+ �n kun � u� (g (un)� g (u))k

+�n kun � u� � (Tun � Tu)k ; (4.2.3)

where we have used Assumption 3.1. Since the operator T is strongly monotone with

constant �T > 0 and Lipschitz continuous with constant �T > 0, so it follows that

kun � u� � (Tun � Tu)k2 = kun � uk2 � 2� hTun � Tu; un � ui+ �2 kTun � Tuk2

��1� 2��T + �2�2T

�kun � uk2 : (4.2.4)


kun � u� (g (un)� g (u))k2 ��1� 2�g + �2g

�kun � uk2 ; (4.2.5)

kun � u� (h (un)� h (u))k2 ��1� 2�h + �2h

�kun � uk2 ; (4.2.6)

52

using the strongly monotonicity and Lipschitz continuity of operators g and h with con-

stants �g > 0, �g > 0 and �h > 0, �h > 0, respectively.

From (4:2:3)� (4:2:6), we have

kun+1 � uk � (1� �n) kun � uk+ ��n kun � uk

+�n

q1� 2�g + �2g kun � uk+ �n

q1� 2�h + �2h kun � uk

+�n

q1� 2��T + �2�2T kun � uk

= (1� �n) kun � uk+ �n (k + t (�)) kun � uk

= (1� �n) kun � uk+ �n� kun � uk ;

where

t (�) =

q1� 2��T + �2�2T ; (4.2.7)

and

� = k + t (�) < 1; using (3:4:1) . (4.2.8)

Thus, we have

kun+1 � uk � (1� �n) kun � uk+ �n� kun � uk

= [1� (1� �)�n] kun � uk

�nQi=0

[1� (1� �)�i] kui � uk :

Since1Pn=0

�n diverges and 1�� > 0, we have limn!1

�nQi=0

[1� (1� �)�i]�= 0. Consequently

the sequence fung converges strongly to u 2 H : h (u) 2 (u) satisfying (3:2:1). This

completes the proof.

It is worth mentioning that for h = g, we obtain the following result from the Theorem

4.2.1 for solving the general quasi variational inequality (3:3:1).

Corollary 4.2.1 [94] Let the operators T; g : H ! H be both strongly monotone with

53

constants �T > 0; �g > 0 and Lipschitz continuous with constants �T > 0; �g > 0;

respectively. Let Assumption 3.1 holds and � > 0 satis�es the conditions

�� T�2T�� <

q�2T � k (2� k) �2T

�2T; �T > �T

pk (2� k); k < 1;

k = � + 2q1� 2�g + �2g:

If 0 � �n � 1, for all n � 0 and1Pn=0

�n =1, then the approximate solution un obtained

from Algorithm 4.2 converges to a solution u 2 H : g (u) 2 (u) satisfying the general

quasi variational inequality (3:3:1).

We would like to emphasize that if h = g = I, then we recover the following result

from the Theorem 4.2.1 for solving the quasi variational inequality (3:3:4).

Corollary 4.2.2 [102] Let the operator T : H! H be strongly monotone with constant

�T > 0 and Lipschitz continuous with constant �T > 0; respectively. Let Assumption 3.1

holds and � > 0 satis�es the conditions

�� T�2T�� <

q�2T � � (2� �) �2T

�2T; �T > �T

p� (2� �); � < 1:

If 0 � �n � 1, for all n � 0 and1Pn=0

�n =1, then the approximate solution un obtained

from Algorithm 4.3 converges to a solution u 2 (u) satisfying the quasi variational

inequality (3:3:4).

We again use the �xed point formulation (3:3:14) to suggest the following implicit

methods for solving the extended general quasi variational inequality (3:2:1).

Algorithm 4.7 For a given u0 2 H, compute the approximate solution un+1 by the

iterative scheme

h (un+1) = P(un) [g (un)� �Tun+1] , n = 0; 1; 2; ::: .

54


iterative scheme

h (un+1) = P(un) [g (un+1)� �Tun+1] , n = 0; 1; 2; ::: .

To implement these implicit methods, one usually uses the predictor-corrector tech-

nique. Consequently, Algorithm 4.7 and Algorithm 4.8 can be written in the following

form.


iterative schemes

h (wn) = P(un) [g (un)� �Tun]

h (un+1) = P(un) [g (un)� �Twn] , n = 0; 1; 2; ::: .


iterative schemes

h (wn) = P(un) [g (un)� �Tun]

h (un+1) = P(un) [g (wn)� �Twn] , n = 0; 1; 2; ::: .

Algorithm 4.9 is called the extragradient method. The implementation and compar-

ison of Algorithm 4.9 is an open problem. Algorithm 4.10 is also known as the modi�ed

extragradient method. Such type of the modi�ed projection methods for solving the

variational inequalities and their variant forms are due to Noor [84]. We remark that

Algorithm 4.9 and Algorithm 4.10 are quite di¤erent. The problem of comparing these

methods is an open problem.

55

Using the technique of updating the solution, we can rewrite (3:3:14) in the following

form as:

h (x) = P(u) [g (u)� �Tu] ;

h (y) = P(x) [g (x)� �Tx] ;

h (u) = P(y) [g (y)� �Ty] .

These modi�ed �xed point formulations help us to propose and analyze the following

three-step iterative method for solving the problem (3:2:1).


schemes

h (xn) = P(un) [g (un)� �Tun] ;

h (yn) = P(xn) [g (xn)� �Txn] ;

h (un+1) = P(yn) [g (yn)� �Tyn] ; n = 0; 1; 2; : : : .

Invoking Algorithm 4.11, we now suggest another three-step scheme for solving the

extended general quasi variational inequalities (3:2:1) :


schemes

xn = (1� n)un + n�un � h (un) + P(un) [g (un)� �Tun]

; (4.2.9)

yn = (1� �n)un + �n�xn � h (xn) + P(xn) [g (xn)� �Txn]

; (4.2.10)

un+1 = (1� �n)un + �n�yn � h (yn) + P(yn) [g (yn)� �Tyn]

; (4.2.11)

where 0 � �n; �n; n � 1, for all n � 0.

For di¤erent and appropriate choice of the operators and spaces, one can obtain several

known and new iterative methods for solving a wide class of variational inequalities and

56

related problems.

We now consider the convergence analysis of Algorithm 4.12 and this is the main

motivation of our next result. In a similar way, one can analyze the convergence criteria

of other algorithms.


�T > 0; �g > 0; �h > 0 and Lipschitz continuous with constants �T > 0; �g > 0; �h > 0,

respectively. Let 0 � �n; �n; n � 1, for all n � 0 and1Pn=0

�n = 1. If Assumption 3.1

and condition (3:4:1) holds, then the approximate solution un obtained from Algorithm

4.12 converges to a solution u 2 H : h (u) 2 (u) satisfying the problem (3:2:1).

Proof. Let u 2 H : h (u) 2 (u) be a solution of the problem (3:2:1). Then, using

Lemma 3.3.2, we have

u = (1� n)u+ n�u� h (u) + P(u) [g (u)� �Tu]

; (4.2.12)

= (1� �n)u+ �n�u� h (u) + P(u) [g (u)� �Tu]

; (4.2.13)

= (1� �n)u+ �n�u� h (u) + P(u) [g (u)� �Tu]

; (4.2.14)

where 0 � �n; �n n � 1 are constants.

Using Assumption 3.1, from (4:2:11) and (4:2:14), we have

kun+1 � uk � (1� �n) kun � uk+ �n kyn � u� (h (yn)� h (u))k

+�n P(yn) [g (yn)� �Tyn]� P(u) [g (u)� �Tu]

� (1� �n) kun � uk+ �n kyn � u� (h (yn)� h (u))k

+�n P(yn) [g (yn)� �Tyn]� P(u) [g (yn)� �Tyn]

+�n P(u) [g (yn)� �Tyn]� P(u) [g (u)� �Tu]

� (1� �n) kun � uk+ �n kyn � u� (h (yn)� h (u))k

+�n� kyn � uk+ �n kg (yn)� g (u)� � (Tyn � Tu)k

57

� (1� �n) kun � uk+ �n� kyn � uk+ �n kyn � u� (g (yn)� g (u))k

+�n kyn � u� (h (yn)� h (u))k+ �n kyn � u� � (Tyn � Tu)k . (4.2.15)

Since the operator T is strongly monotone with constant �T > 0 and Lipschitz continuous

with constant �T > 0, then it follows that

kyn � u� � (Tyn � Tu)k2 = kyn � uk2 � 2� hTyn � Tu; yn � ui+ kTyn � Tuk2

��1� 2��T + �2�2T

�kyn � uk2 . (4.2.16)


kyn � u� (g (yn)� g (u))k2 ��1� 2�g + �2g

�kyn � uk2 , (4.2.17)

and

kyn � u� (h (yn)� h (u))k2 ��1� 2�h + �2h

�kyn � uk2 , (4.2.18)

where we have used the strongly monotonicity and Lipschitz continuity of operators g, h

with constants �g > 0, �g > 0 and �h > 0, �h > 0, respectively.


kun+1 � uk � (1� �n) kun � uk+ �nf� +q1� 2�g + �2g +

q1� 2�h + �2h

+

q1� 2��T + �2�2Tg kyn � uk

= (1� �n) kun � uk+ �n fk + t (�)g kyn � uk

= (1� �n) kun � uk+ �n� kyn � uk , (4.2.19)

where

t (�) =

q1� 2��T + �2�2T , (4.2.20)

and

� = k + t (�) < 1.

58

Similarly, from (4:2:10) and (4:2:13), we have

kyn � uk = k (1� �n)un + �n�xn � h (xn) + P(xn) [g (xn)� �Txn]

� (1� �n)u� �n

�u� h (u) + P(u) [g (u)� �Tu]

k

� (1� �n) kun � uk+ ��n kxn � uk . (4.2.21)

In a similar way, from (4:2:9) and (4:2:12), we have

kxn � uk = k (1� n)un + n�un � h (un) + P(un) [g (un)� �Tun]

� (1� n)u� n

�u� h (u) + P(u) [g (u)� �Tu]

k

� (1� n) kun � uk+ � n kun � uk

= (1� n (1� �)) kun � uk � kun � uk . (4.2.22)

From (4:2:21) and (4:2:22), we have

kyn � uk � (1� �n) kun � uk+ ��n kun � uk

= (1� �n (1� �)) kun � uk � kun � uk . (4.2.23)

Using (4:2:19) and (4:2:23), we have

kun+1 � uk � (1� �n) kun � uk+ ��n kun � uk

= (1� �n (1� �)) kun � uk

�nQi=0

[1� (1� �)�i] kui � uk .

Since1Pn=0

�n diverges and 1�� > 0, we have limn!1

�nQi=0

[1� (1� �)�i]�= 0. Consequently

the sequence fung converges strongly to u 2 H : h (u) 2 (u) satisfying (3:2:1). This is

the desired result.

59

4.3 Wiener-Hopf Technique

In this section we introduce another class of iterative schemes for solving the extended

general quasi variational inequality (3:2:1) and its related forms. Lemma 3.5.3 implies

that the extended general quasi variational inequality (3:2:1) and the extended general

implicit Wiener-Hopf equation (3:5:1) are equivalent. We use this equivalent formulation

to suggest a number of iterative methods for solving the extended general quasi variational

inequalities (3:2:1).

I. Using (3:5:9), the extended general implicit Wiener-Hopf equation (3:5:1) can be

rewritten in the form as:

Q(u) [z] = ��Th�1P(u) [z] ;

which implies that

z = gh�1P(u) [z]� �Th�1P(u) [z] = g (u)� �Tu:

This �xed point formulation enables to suggest the following iterative method for

solving problem (3:5:1).

Algorithm 4.13 For a given z0 2 H, compute the approximate solution zn+1 by the

iterative schemes

h (un) = P(un) [zn] (4.3.1)

zn+1 = (1� �n) zn + �n fg (un)� �Tung ; n = 0; 1; 2; : : : (4.3.2)

where 0 � �n � 1; for all n � 0.

II. By an appropriate and suitable rearrangement of the terms and using (3:5:9), the

60

Wiener-Hopf equations (3:5:1) can be written as:

0 = Th�1P(u) [z] + ��1Q(u) [z]�Q(u) [z] +Q(u) [z]

= Th�1P(u) [z]��1� ��1

�Q(u) [z] +

�I� gh�1P(u)

�[z]

= Th�1P(u) [z]��1� ��1

�Q(u) [z] + z � gh�1P(u) [z]

= Tu��1� ��1

�Q(u) [z] + z � g (u) ;


z = g (u)� Tu+�1� ��1

�Q(u) [z] ;

which is another �xed point formulation. Using this �xed point formulation, we

can suggest the following iterative method.


iterative schemes

h (un) = P(un) [zn]

zn+1 = g (un)� Tun +�1� ��1

�Q(un) [zn] ; n = 0; 1; 2; : : : .

III. If T is linear and T�1 exists, then the implicit Wiener-Hopf equations (3:5:1) can

be written as:

Q(u) [z] = z � gh�1P(u) [z] ;

which implies

Tg�1�z �Q(u) [z]

�= Th�1P(u) [z]

= ��1Q(u) [z] ; using (3:5:1)

61

which implies

z = Q(u) [z]� ��1gT�1Q(u) [z]

=�I� ��1gT�1

�Q(u) [z] :

This �xed point formulation allows us to suggest the following iterative method for

solving the extended general quasi variational inequality (3:2:1).


iterative schemes

zn+1 =�I� ��1gT�1

�Q(un) [zn] ; n = 0; 1; 2; : : : .

For g = h; (u) � , Algorithm 4.13- Algorithm 4.15 are due to Noor [72]. In

brief, by an appropriate and suitable rearrangements of the terms of the extended general

implicit Wiener-Hopf equations (3:5:1), one can suggest and analyze a number of iterative

methods for solving the extended general quasi variational inequality (3:2:1) and related

optimization problems. The investigation of such type of projection iterative methods

and the veri�cation of their numerical e¢ ciency, further research e¤orts are needed.

We now consider the convergence analysis of Algorithm 4.13. In a similar way, one

can study the convergence analysis of Algorithm 4.14 and Algorithm 4.15.

Theorem 4.3.1 Let the operators T; g; h satisfy all the assumptions of Theorem 3.4.1. If

the Assumption 3.1 holds and � > 0 satis�es the condition (3:4:1), then the approximate

solution fzng obtained from Algorithm 4.13 converges to a solution z 2 H satisfying the

extended general implicit Wiener-Hopf equation (3:5:1) strongly in H.

Proof. Let u 2 H be a solution of (3:2:1). Then, using Lemma 3.5.3, we have

z = (1� �n) z + �n fg (u)� �Tug ; (4.3.3)

62

where 0 � �n � 1; and1Pn=0

�n =1.

From (3:4:4) ; (3:4:5) ; (4:3:2) and (4:3:3) ; we have

kzn+1 � zk � (1� �n) kzn � zk+ �n kg (un)� g (u)� � (Tun � Tu)k

� (1� �n) kzn � zk+ �n kun � u� (g (un)� g (u))k

+�n kun � u� � (Tun � Tu)k

� (1� �n) kzn � zk

+�n

�q1� 2�g + �2g +

q1� 2��T + �2T

�kun � uk : (4.3.4)

Also from (3:4:6) ; (3:5:9) ; (4:3:1) and Assumption 3.1, we have

kun � uk � kun � u� (h (un)� h (u))k+ P(un) [zn]� P(u) [z]

� kun � u� (h (un)� h (u))k+ P(un) [zn]� P(u) [zn]

+ P(u) [zn]� P(u) [z]

�� +

q1� 2�h + �2h

�kun � uk+ kzn � zk ;

which implies that

kun � uk �1

1� � �p1� 2�h + �2h

kzn � zk : (4.3.5)

Combining (4:3:4) and (4:3:5) ; we have

kzn+1 � zk � (1� �n) kzn � zk+ �n�1 kzn � zk ; (4.3.6)

where

�1 =

q1� 2�g + �2g +

p1� 2��T + �2T

1� � �p1� 2�h + �2h

: (4.3.7)

Using (3:4:1) and (3:4:2), we see that �1 < 1.

63

Consequently, from (4:3:6), we have

kzn+1 � zk � (1� �n) kzn � zk+ �n�1 kzn � zk

= (1� �n (1� �1)) kzn � zk

�nQi=0

(1� �i (1� �1)) kzi � zk :

Since1Pn=0

�n diverges and 1 � �1 > 0, we have limn!1

�nQi=0

(1� (1� �1)�i)�= 0. Conse-

quently the sequence fzng converges strongly to z 2 H satisfying (3:5:1), the required

result.

4.4 Conclusion

In this chapter, alternative equivalent �xed point formulations have been used to sug-

gest and analyze several iterative methods for solving the quasi variational inequalities.

Convergence of these new algorithms is also investigated under some mild conditions.

Several special cases are also discussed. Results obtained in this chapter continue to hold

for these problems. We expect that these ideas and techniques will motivate and inspire

the interested readers to explore its applications in various �elds.

64

Chapter 5

Dynamical Systems

65

5.1 Introduction

Dynamics is a concise term referring to the study of time evolving processes, and the

corresponding system of equations, which describes this evolution, is called a dynamical

system. Nonlinear systems are widely used as models to describe complex physical phe-

nomena in various �eld of sciences, such as �uid dynamics, solid state physics, plasma

physics, mathematical biology and chemical kinetics, vibrations, heat transfer and so on.

It is well known that these problems can be studied via the quasi variational inequalities.

Dynamical systems can be solved by using some analytical techniques such as Homo-

topy Perturbation Method, Variational Iteration Method, Neural Network techniques

and their variant forms, see [60, 61, 62] and references therein.

In recent years, several dynamical systems associated with variational inequalities are

being investigated using the projection operator methods and Wiener-Hopf equations.

This can be traced back to Dupuis and Nagurney [31], Friesz et al [36] and Noor [81].

The dynamical systems method is more attractive due to its wide applicability, �exibility

and numerical e¢ ciency. In this method, variational inequality problem is reformulated

as an initial value problem. It enables us to study the stability properties of the unique

solution of the variational inequality problem. There are two types of the projected

dynamical systems. The �rst type is due to Dupuis and Nagurney [31], which is known

as local projected dynamical systems where as, the second one which is due to Friesz

et al [36] is called global projected dynamical systems. In projected dynamical systems

the right-hand side of the ordinary di¤erential equation is a projection operator and is

discontinuous. The discontinuity arises from the constraints governing the applications

in question. The novel and innovative feature of the projected dynamical systems is that

the set of the stationary/equilibrium points of the solutions of the dynamical systems

correspond to the set of the solutions of the variational inequalities. Consequently, the

equilibrium problems which can be formulated in the setting of variational inequalities can

now be studied in the more general setting of the dynamical systems. These dynamical

systems enable us to describe the trajectories of real economics and physical process prior

66

to reaching steady states. Noor [81] has introduced Wiener-Hopf dynamical systems for

variational inequalities, using �xed point formulation. Xia and Wang [123] have shown

that the projected dynamical systems can be used e¤ectively in designing neural network

for solving variational inequalities, see [60, 61]. The neural network methods are used

to �nd the approximate solutions of the global projected dynamical systems. Neural

network methods are robust and e¢ cient one.

In this chapter, using the projection operator technique we introduce some new dy-

namical systems for extended general quasi variational inequalities (3:2:1). These dy-

namical systems are called extended general implicit projected dynamical system and

extended general implicit Wiener-Hopf dynamical system. We prove that solution of

these new dynamical systems converge globally exponentially to a solution of the ex-

tended general quasi variational inequalities under some suitable conditions. Some spe-

cial cases are also discussed, which can be obtained from our results. Results obtained

in this chapter continue to hold for these problems. This work has been accepted for

publication in Annals of Functional Analysis 6(1), 193-209, see [108].

5.2 Projected Dynamical System

We use the residue vector, de�ned in (3:5:18) ; to consider the following dynamical system

du

dt= ��R (u)

= ��P(u) [g (u)� �Tu]� h (u)

; u (t0) = u0 2 H; (5.2.1)

associated with problem (3:2:1), where � > 0 is a constant. The dynamical system (5:2:1)

is called the extended general implicit projected dynamical system. Since right hand side

is related to the projection operator, and thus, is discontinuous on the boundary of (u).

It is clear from the de�nition that the solution of dynamical system (5:2:1) belongs to

the constraint set (u). From this it is clear that the results such as the existence,

uniqueness and continuous dependence on the given data can be studied.

67

De�nition 5.2.1 An element u 2 H : h (u) 2 (u) is an equilibrium point of the

dynamical system (5:2:1) ; if dudt= 0, that is,

P(u) [g (u)� �Tu]� h (u) = 0:

Thus it is clear that u 2 H : h (u) 2 (u) is a solution of the problem (3:2:1), if and

only if, u 2 H : h (u) 2 (u) is an equilibrium point of the dynamical system (5:2:1).

In a similar way, one can de�ne the concept of equilibrium points for other dynamical

systems.

De�nition 5.2.2 [81] The dynamical system (5:2:1) is said to converge to the solution

set � of problem (3:2:1) if, irrespective of the initial point, the trajectory of the dynamical

system satis�es

limt!1

dist (u (t) ;�) = 0;

where

dist (u;�) = infv2�

ku� vk :

Clearly, if the set � has a unique point u�, then we have

limt!1

u (t) = u�:

The stability of the dynamical system at u� in the Lyapunov sense, con�rms that the

dynamical system is also globally asymptotically stable at u�.

De�nition 5.2.3 [81] The dynamical system is said to be globally exponentially stable

with degree �1 at u� if, irrespective of the initial point, the trajectory of the system u (t)

satis�es

ku (t)� u�k � �1 ku (t0)� u�k e��1(t�t0); 8t � t0;

where �1 > 0 and �1 > 0 are constants independent of the initial point. It is clear that

globally exponential stability is necessarily globally asymptotically stable and the dynamical

68

system converges arbitrarily fast.

Lemma 5.2.1 (Gronwall�s Lemma [81]) Let u and v be real valued non-negative contin-

uous functions with domain ft : t � t0g and let � (t) = �0 jt� t0j, where �0 is a monotone

increasing function. If, for t � t0,

u (t) � � (t) +tRt0

u (s) v (s) ds;

then

u (t) � � (t) � exp

tRt0

v (s) ds

!:

Theorem 5.2.1 Let the operators T; g and h be Lipschitz continuous with constants

�T > 0, �g > 0 and �h > 0; respectively. If the constant � > 0 and Assumption

3.1 holds, then for each u0 2 H, there exists a unique continuous solution u (t) of the

dynamical system (5:2:1) with u (t0) = u0 over [t0;1).

Proof. Let

G (u) = ��P(u) [g (u)� �Tu]� h (u)

:

To prove that G (u) is Lipschitz continuous for all u1 6= u2 2 H, we have to consider

kG (u1)� G (u2)k

= � P(u1) [g (u1)� �Tu1]� h (u1)� �P(u2) [g (u2)� �Tu2]� h (u2)

� � P(u1) [g (u1)� �Tu1]� P(u2) [g (u1)� �Tu1] +� P(u2) [g (u1)� �Tu1]� P(u2) [g (u2)� �Tu2] + � kh (u1)� h (u2)k

� �� ku1 � u2k+ � kg (u1)� g (u2)k+ � kh (u1)� h (u2)k+ �� kTu1 � Tu2k

� �� + �g + �h + ��T

�ku1 � u2k ;

where we have used the Assumption 3.1 and Lipschitz continuity of the operators T; g

and h with constants �T > 0; �g > 0 and �h > 0; respectively.

69

This implies that the operator G (u) is a Lipschitz continuous in H. So for each

u0 2 H , there exists a unique and continuous solution u (t) of the dynamical system

(5:2:1), de�ned in an interval t0 � t < T1 with the initial condition u (t0) = u0. Let

[t0; T1) be its maximal interval of existence. Now we have to show that T1 =1.

Consider, dudt = kG (u)k

= � P(u) [g (u)� �Tu]� h (u)

= � P(u) [g (u)� �Tu]� P(u) [g (u)] + P(u) [g (u)]� P(u�) [g (u�)] + P(u�) [g (u�)]� h (u)

� �

P(u) [g (u)� �Tu]� P(u) [g (u)] + � P(u) [g (u)]� P(u�) [g (u)] +� P(u�) [g (u)]� P(u�) [g (u�)] + � P(u�) [g (u�)] + � kh (u)k

� � kg (u)� �Tu� g (u)k+ �� ku� u�k+ � kg (u)� g (u�)k

+� P(u�) [g (u�)] + � kh (u)k

� ��T kuk+ �� + �g

�ku� u�k+ �

P(u�) [g (u�)] + ��h kuk= �

�� + �g + �h + ��T

�kuk+ �

�� + �g

�ku�k+

P(u�) [g (u�)] ; (5.2.2)where we have used the Assumption 3.1, and Lipschitz continuity of operators T; g and

h with constants �T > 0; �g > 0 and �h > 0; respectively.

Now, integrating (5:2:2) over the interval [t0; t], we have

ku (t)k � ku (t0)k � k1 (t� t0) + k2tRt0

ku (s)k ds;

from which by using Lemma 5.2.1, we have

ku (t)k � ku (t0)k+ k1 (t� t0) + k2tRt0

ku (s)k ds

= fku (t0)k+ k1 (t� t0)g exp (k2 (t� t0)) ; t 2 [t0; T1) ;

70

where

k1 = �� + �g

�ku�k+

P(u�) [g (u�)] k2 = �

�� + �g + �h + ��T

�> 0:

This shows that the solution is bounded on [t0;1).

We now show that the trajectory of the solution of the dynamical system (5:2:1)

converges globally exponentially to the unique solution of problem (3:2:1).

Theorem 5.2.2 Let the operators T; g and h : H ! H be Lipschitz continuous with

constants �T > 0, �g > 0 and �h > 0; respectively. Let the operator h : H ! H

be strongly monotone with contant �h > 0 and Assumption 3.1 holds. If the constant

�h > � + �g + ��T , then the dynamical system (5:2:1) converges globally exponentially to

the unique solution of problem (3:2:1).

Proof. Since the operators T; g and h are Lipschitz continuous, then it follows from

the Theorem 5.2.1 that the dynamical system (5:2:1) has a unique solution u (t) over

[t0; T1) for any �xed u0 2 H.

Let u (t) = u (t; t0;u0) be a solution of problem (5:2:1). For a given u� 2 H : h (u�) 2

(u�), satisfying problem (3:2:1), consider the following Lyapunov function:

L (u) = 1

2ku (t)� u�k2 ; u 2 H: (5.2.3)

Thus

dLdt

=

�u (t)� u�; du

dt

�= ��

u� u�; h (u)� P(u) [g (u)� �Tu]

�= �� hh (u)� h (u�) ; u� u�i+ �

P(u) [g (u)� �Tu]� h (u�) ; u� u�

�� h ku� u�k2 + �

P(u) [g (u)� �Tu]� h (u�) ku� u�k ; (5.2.4)

71

where we have used the strongly monotonicity of the operator h with constant �h > 0.

Since u� 2 H : h (u�) 2 (u�) is the solution of problem (3:2:1), therefore using

Lemma 3.3.2,we have

h (u�) = P(u�) [g (u�)� �Tu�] : (5.2.5)

Using the relation (5:2:5), Assumption 3.1 and Lipschitz continuity of operators T and

g with constants �T > 0 and �g > 0; respectively, we have

P(u) [g (u)� �Tu]� h (u�) = P(u) [g (u)� �Tu]� P(u�) [g (u�)� �Tu�]

� P(u) [g (u)� �Tu]� P(u�) [g (u)� �Tu] + P(u�) [g (u)� �Tu]� P(u�) [g (u�)� �Tu�]

� � ku� u�k+ kg (u)� g (u�)k+ � kTu� Tu�k

�� + �g + ��T

�ku� u�k : (5.2.6)


dLdt

� ��h � � � �g � ��T

�ku� u�k2

= �k3 ku� u�k2 ;

where

k3 = ��h � � � �g � ��T

�> 0;

which implies that

ku (t)� u�k � ku (t0)� u�k exp (�k3 (t� t0)) :

This shows that the trajectory of the solution of the dynamical system (5:2:1) converges

globally exponentially to the unique solution of problem (3:2:1).

Corollary 5.2.1 Let the operators T; g and h : H ! H be Lipschitz continuous with

72

constants �T > 0, �g > 0 and �h > 0; respectively. Let the operator h : H ! H be

monotone and Assumption 3.1 holds. If the constant � > 0, then the dynamical system

(5:2:1) converges globally exponentially to the unique solution of problem (3:2:1).

For suitable and appropriate choice of the operators and spaces one can obtain the

results of Noor [81, 82] and others as special cases from our results.

5.3 Wiener-Hopf Dynamical System

In this section, we introduce and study a new dynamical system related to the extended

general quasi variational inequalities (3:2:1) : This dynamical system is called extended

general implicit Wiener-Hopf dynamical system.

Using Lemma 3.5.3; the implicit Wiener-Hopf equation (3:5:1) can be written as:

g (u)� �Tu� gh�1P(u) [g (u)� �Tu] + �Th�1P(u) [g (u)� �Tu] = 0; (5.3.1)

from, which we have

g (u) = �Tu+ gh�1P(u) [g (u)� �Tu]� �Th�1P(u) [g (u)� �Tu] : (5.3.2)

Thus it is clear from Lemma 3.5.3 that u 2 H : h (u) 2 (u) is a solution of problem

(3:2:1) ; if and only if, u 2 H : h (u) 2 (u) satis�es the equation (5:3:1).

For a constant � > 0 and using the relation (5:3:1), we suggest a new dynamical

system:du

dt= �

�gh�1P(u) [g (u)� �Tu]� �Th�1P(u) [g (u)� �Tu]

+�Tu� g (u)g ; u (t0) = u0 2 (u) : (5.3.3)

This problem is called the extended general implicit Wiener-Hopf dynamical system

related to the extended general quasi variational inequalities (3:2:1). Here the right-hand

73

side is associated with the implicit projection and hence is discontinuous on the boundary

of a closed and convex-valued set (u). It is clear from the de�nition that the solution

to the dynamical system (5:3:3) belongs to the constraint set (u). This implies that

the results, such as existence, uniqueness and continuous dependence on the data of the

solution of (5:3:3) can be studied.

We now study the main properties of the proposed Wiener-Hopf dynamical system

and analyze the global stability of the systems. First of all, we discuss the existence and

uniqueness of the dynamical system (5:3:3) and this is the main motivation of our next

result.

Theorem 5.3.1 Let the non-linear operators T; g and h�1 be the Lipschitz continuous

with constants �T > 0; �g > 0 and �h�1 > 0; respectively. If the constant � > 0 and

Assumption 3.1 holds, then for each u0 2 H, there exists a unique continuous solution

u (t) of the dynamical system (5:3:3) with u (t0) = u0 over [t0;1).

Proof. Let

G (u) = ��gh�1P(u) [g (u)� �Tu]� �Th�1P(u) [g (u)� �Tu] + �Tu� g (u)

:

To prove that G (u) is Lipschitz continuous for all u1 6= u2 2 H, we consider

kG (u1)� G (u2)k = �kgh�1P(u1) [g (u1)� �Tu1]

��Th�1P(u1) [g (u1)� �Tu1] + �Tu1 � g (u1)

��gh�1P(u2) [g (u2)� �Tu2]

��Th�1P(u2) [g (u2)� �Tu2] + �Tu2 � g (u2)k

� �kgh�1P(u1) [g (u1)� �Tu1]� gh�1P(u2) [g (u2)� �Tu2] k

+�� Th�1P(u1) [g (u1)� �Tu1]� Th�1P(u2) [g (u2)� �Tu2]

+�� kTu1 � Tu2k+ � kg (u1)� g (u2)k

74

� ��g + ��T

� � h�1P(u1) [g (u1)� �Tu1]�h�1P(u2) [g (u2)� �Tu2]

+ ku1 � u2k ; (5.3.4)

where we have used the Lipschitz continuity of the operators T and g with constants

�T > 0 and �g > 0, respectively. Now, h�1P(u1) [g (u1)� �Tu1]� h�1P(u2) [g (u2)� �Tu2] � �h�1

P(u1) [g (u1)� �Tu1]� P(u2) [g (u2)� �Tu2] � �h�1

P(u1) [g (u1)� �Tu1]� P(u2) [g (u1)� �Tu1] +�h�1

P(u2) [g (u1)� �Tu1]� P(u2) [g (u2)� �Tu2] � �h�1� ku1 � u2k+ �h�1 kg (u1)� g (u2)� � (Tu1 � Tu2)k

� �h�1� ku1 � u2k+ �h�1 kg (u1)� g (u2)k+ �h�1� kTu1 � Tu2k

� �h�1�� + �g + ��T

�ku1 � u2k ; (5.3.5)

where we have used the Assumption 3.1 and Lipschitz continuity of the operators T , g

and h�1 with constants �T > 0, �g > 0 and �h�1 > 0, respectively.


kG (u1)� G (u2)k � ��g + ��T

� ��h�1

�� + �g + ��T

�ku1 � u2k+ ku1 � u2k

= �

��g + ��T

� ��h�1

�� + �g + ��T

�+ 1ku1 � u2k :

This implies that the operator G (u) is Lipschitz continuous in H. So for each u0 2

H , there exists a unique and continuous solution u (t) of the dynamical system (5:3:3),

de�ned in an interval t0 � t < T1 with the initial condition u (t0) = u0. Let [t0; T1) be its

maximal interval of existence. Now we have to show that T1 =1.

75

Consider,

kG (u)k =

dudt

= � gh�1P(u) [g (u)� �Tu] ��Th�1P(u) [g (u)� �Tu] + �Tu� g (u)

� � gh�1P(u) [g (u)� �Tu]� g (u) + �� Th�1P(u) [g (u)� �Tu]� Tu

� ��g + ��T

� h�1P(u) [g (u)� �Tu]� u ; (5.3.6)

where we have used the Lipschitz continuity of T and g with constants �T > 0 and

�g > 0, respectively. Now,

h�1P(u) [g (u)� �Tu]� u =

h�1P(u) [g (u)� �Tu]� h�1P(u) [g (u)] + h�1P(u) [g (u)]�h�1P(u�) [g (u�)] + h�1P(u�) [g (u�)]� u

� �h�1

P(u) [g (u)� �Tu]� P(u) [g (u)] + �h�1 P(u) [g (u)]� P(u�) [g (u)] +�h�1

P(u�) [g (u)]� P(u�) [g (u�)] + �h�1 P(u�) [g (u�)] + kuk� �h�1

��T kuk+ � ku� u�k+ �g ku� u�k+

P(u�) [g (u�)] + kuk�

��h�1

�� + �g + ��T

�+ 1kuk

+�h�1�� + �g

�ku�k+

P(u�) [g (u�)] ; (5.3.7)

where we have used the Lipschitz continuity of operators T; g and h�1 with constants

�T > 0; �g > 0 and �h�1 > 0; respectively and the Assumption 3.1.


kG (u)k =

dudt

� ��g + ��T

� ��h�1

��T + � + �g

�+ 1kuk

+�h�1��g + ��T

� �� + �g

�ku�k+

P(u�) [g (u�)] : (5.3.8)

76

Integrating (5:3:8) over the interval [t0; t], we have

ku (t)k � ku (t0)k � k5 (t� t0) + k6tRt0

ku (s)k ds;

from which by using the Lemma 5.2.1, we have

ku (t)k � ku (t0)k+ k4 (t� t0) + k5tRt0

ku (s)k ds

= fku (t0)k+ k5 (t� t0)g exp (k6 (t� t0)) ; t 2 [t0; T1) ;

where

k4 = �h�1��g + ��T

� �� + �g

�ku�k+

P(u�) [g (u�)] ;k5 = �

��g + ��T

� ��h�1(� + �g + ��T ) + 1

> 0:

This shows that the solution is bounded on [t0;1).

We now show that the trajectory of the solution of the dynamical system (5:3:3)

converges globally exponentially to the unique solution of problem (3:2:1).

Theorem 5.3.2 Let the operators T; g and h�1 be Lipschitz continuous with constants

�T > 0; �g > 0 and �h�1 > 0; respectively. Let the operator g be strongly monotone with

constant �g > 0 and Assumption 3.1 holds. If the constant

�g > ��T + �h�1��g + ��T

� �� + �g + ��T

�;

then the dynamical system (5:3:3) converges globally exponentially to the unique solution

of problem (3:2:1).

Proof. Since the operators T; g and h�1 satis�es all the conditions of Theorem 5.3.1,

therefore it follows from Theorem 5.3.1 that the dynamical system (5:3:3) has a unique

solution u (t) over [t0; T1) for any �xed point u0 2 H.

77

Let u (t) = u (t; t0;u0) be a solution of (5:3:3). For a given u� 2 H : h2 (u�) 2 (u),

satisfying problem (3:2:1), consider the following Lyapunov function:

L (u) = 1

2ku (t)� u�k2 ; u 2 H:

Thus by using (5:3:3), we have

dLdt

=

�u (t)� u�; du

dt

�= ��hu� u�; g (u)� gh�1P(u) [g (u)� �Tu] + �Th�1P(u) [g (u)� �Tu]� �Tui

= �� hg (u)� g (u�) ; u� u�i+ �hgh�1P(u) [g (u)� �Tu]

��Th�1P(u) [g (u)� �Tu] + �Tu� g (u�) ; u� u�i

� ��g ku� u�k2 + � ku� u�k kgh�1P(u) [g (u)� �Tu]

��Th�1P(u) [g (u)� �Tu] + �Tu� g (u�) k; (5.3.9)

where we have used the strongly monotonicity of the operator g with constant �g > 0.

Since u� 2 H : h2 (u�) 2 (u) is the solution of problem (3:2:1), therefore using

(5:3:2), we have

gh�1P(u) [g (u)� �Tu]� �Th�1P(u) [g (u)� �Tu] + �Tu� g (u�) = kgh�1P(u) [g (u)� �Tu]� �Th�1P(u) [g (u)� �Tu] + �Tu

��gh�1P(u�) [g (u

�)� �Tu�]� �Th�1P(u�) [g (u�)� �Tu�] + �Tu�k

� gh�1P(u) [g (u)� �Tu]� gh�1P(u�) [g (u�)� �Tu�] +� Th�1P(u) [g (u)� �Tu]� Th�1P(u�) [g (u�)� �Tu�]

+� kTu� Tu�k

78

� �h�1��g + ��T

� P(u) [g (u)� �Tu]� P(u�) [g (u�)� �Tu�] + ��T ku� u�k� �h�1

��g + ��T

�f� ku� u�k+ kg (u)� g (u�)k+ � kTu� Tu�kg+ ��T ku� u�k

��h�1

��g + ��T

� �� + �g + ��T

�+ ��T

ku� u�k ; (5.3.10)

where we have used the Assumption 3.1 and the Lipschitz continuity of the operators T;

g and h�1 with constants �T > 0; �g > 0 and �h�1 > 0; respectively.


dLdt

� ��g � ��T � �h�1

��g + ��T

� �� + �g + ��T

�ku� u�k2

= �k6 ku� u�k2 ;

where

k6 = ��g � ��T � �h�1

��g + ��T

� �� + �g + ��T

�> 0;

which implies that

ku (t)� u�k � ku (t0)� u�k exp (�k6 (t� t0)) ;

This shows that the trajectory of the solution of the dynamical system (5:3:3) converges


Corollary 5.3.2 Let the operators T; g and h�1 be Lipschitz continuous with constants

�T > 0; �g > 0 and �h�1 > 0; respectively. Let the operator g be monotone and As-

sumption 3.1 holds. If the constant � > 0, then the dynamical system (5:3:3) converges


5.4 Conclusion

In this chapter, we have introduced two new dynamical systems associated with the

extended general quasi variational inequalities. These dynamical systems have been used

79

to investigate the existence of the solutions of the quasi variational inequalities. Several

special cases are discussed. Results obtained in this chapter can be extended for quasi

split feasibility problems, which is another direction for future research. The ideas on

the technique of the work presented in this chapter may inspire the interested readers to

discover novel and innovative applications of quasi variational inequalities in pure and

applied sciences.

80

Chapter 6

Error Bounds

81

6.1 Introduction

In recent years, much attention has been given to reformulate the variational inequality

as an optimization problem. A function which can constitute an equivalent optimization

problem is called a merit (gap) function. Merit functions turn out to be very useful in

designing new globally convergent algorithms and in analyzing the rate of convergence

of some iterative methods. Various merit (gap) functions for variational inequalities and

complementarity problems have been suggested and proposed by many authors, see [38,

39, 42, 44, 85, 86, 111, 112, 113, 114, 115, 118, 120, 124, 126, 127, 129] and the references

therein. Error bounds are functions which provide a measure of the distance between a

solution set and an arbitrary point. Therefore, error bounds play an important role in

the analysis of global or local convergence analysis of algorithms for solving variational

inequalities. To the best of our knowledge, very few merit functions have been considered

for general variational inequalities and quasivariational inequalities.

In this chapter, we consider normal residue merit functions, regularized merit func-

tions and di¤erence merit functions for extended general quasi variational inequalities

(3:2:1) and some other related aspects. We also obtain error bounds for the solution of

the extended general quasi variational inequalities (3:2:1) under some weaker conditions.

Since the extended general quasi variational inequalities include variational inequalities,

quasi variational inequalities, general variational inequalities, extended general varia-

tional inequalities and quasi complementarity problems as special cases, one can deduce

the similar results for these problems under weaker conditions. In this respect, our results

can be viewed as re�nement of the previously known results for variational inequalities.

Several special cases are also investigated.

De�nition 6.1.1 [85] A functionM : H! R [ f+1g is called a merit (gap) function

for the inequality (3:2:1), if and only if

(i) M (u) � 0; 8u 2 H : h (u) 2 (u) :

(ii) M (u) = 0; if and only if, u 2 H : h (u) 2 (u) solves (3:2:1) :

82

We now introduce some merit functions associated with the problem (3:2:1). Using

these merit functions, we obtain some error bounds for the solution of the problem (3:2:1).

6.2 Normal Residue Vector

It is well known that the normal residue vector kR (u)k is a merit function for extended

general quasi variational inequality (3:2:1) : We use the relation (3:5:18) to derive the

error bound for the solution of the problem (3:2:1).

Theorem 6.2.1 Let u 2 H : h (u) 2 (u) be a solution of the problem (3:2:1). Let

the operator T be strongly h-monotone with constant �T > 0 and h-Lipschitz continuous

with constant �T > 0. Let the operators g and h be Lipschitz continuous with constants

�g > 0 and �h > 0; respectively, and h is also strongly expanding with constant �h > 0.

If � >�g�h�T �

2hand Assumption 3.1 holds, then

1

k1kR (u)k � ku� uk � k2 kR (u)k ; 8u 2 H;

where k1 and k2 are generic constants.

Proof. Let u 2 H : h (u) 2 (u) be a solution of the problem (3:2:1). Then

h�Tu+ h (u)� g (u) ; g (v)� h (u)i � 0; 8v 2 H : g (v) 2 (u) : (6.2.1)

Setting g (v) = P(u) [g (u)� �Tu] in (6:2:1), we have

�Tu+ h (u)� g (u) ; P(u) [g (u)� �Tu]� h (u)

�� 0: (6.2.2)

Setting u = P(u) [g (u)� �Tu] ; z = g (u)� �Tu and v = h (u) in Lemma 2.3.2, we have

P(u) [g (u)� �Tu]� g (u) + �Tu; h (u)� P(u) [g (u)� �Tu]

�� 0;

83

which implies that

��Tu+ g (u)� P(u) [g (u)� �Tu] ; P(u) [g (u)� �Tu]� h (u)

�� 0: (6.2.3)

By adding (6:2:2) and (6:2:3), we have

0 � h� (Tu� Tu) + (h (u)� h (u))� (g (u)� g (u))

+�h (u)� P(u) [g (u)� �Tu]

�; P(u) [g (u)� �Tu]� h (u)i;

using (3:5:18), we haveTu� Tu; h (u)� P(u) [g (u)� �Tu]

�� 1

�

h (u)� h (u)� (g (u)� g (u)) +R (u) ; P(u) [g (u)� �Tu]� h (u)

�: (6.2.4)

Since h is strongly expanding with constant �h and T is strongly h-monotone with con-

stant �T > 0, such that

�T �2h ku� uk

2 � �T kh (u)� h (u)k2

� hTu� Tu; h (u)� h (u)i

=Tu� Tu; h (u)� P(u) [g (u)� �Tu]

�+Tu� Tu; P(u) [g (u)� �Tu]� h (u)

�;

using (3:5:18) and (6:2:4), we have

�T �2h ku� uk

2 � 1

�

h (u)� h (u)� (g (u)� g (u)) +R (u) ; P(u) [g (u)� �Tu]� h (u)

�+ hTu� Tu;�R (u)i ;

84

using the h-Lipschitz continuity of T with constant �T > 0, Lipschitz continuity of g and

h with constants �g > 0 and �h > 0, respectively, we have

��T �2h ku� uk

2 � hh (u)� h (u)� (g (u)� g (u)) +R (u) ;�R (u)� (h (u)� h (u))i

+� hTu� Tu;�R (u)i

= hh (u)� h (u) ;�R (u)i � hh (u)� h (u) ; h (u)� h (u)i

+ hg (u)� g (u) ; R (u)i+ hg (u)� g (u) ; h (u)� h (u)i

� hR (u) ; R (u)i+ hR (u) ;� (h (u)� h (u))i+ � hTu� Tu;�R (u)i

� 2 kh (u)� h (u)k kR (u)k � kh (u)� h (u)k2

+ kg (u)� g (u)k kR (u)k+ kg (u)� g (u)k kh (u)� h (u)k

�kR (u)k2 + � kTu� Tuk kR (u)k

� 2 kh (u)� h (u)k kR (u)k+ kg (u)� g (u)k kR (u)k

+ kg (u)� g (u)k kh (u)� h (u)k+ � kTu� Tuk kR (u)k

��2�h + �g

�ku� uk kR (u)k

+��T kh (u)� h (u)k kR (u)k+ �g�h ku� uk2

��2�h + �g + ��T�h

�ku� uk kR (u)k+ �g�h ku� uk

2 ;

which implies that

ku� uk �2�h + �g + ��T�h

��T �2h � �g�h

kR (u)k = k2 kR (u)k ; (6.2.5)

where

k2 =2�h + �g + ��T�h

��T �2h � �g�h

:

85

Using the relation (3:5:18), we have

kR (u)k = h (u)� P(u) [g (u)� �Tu]

� kh (u)� h (u)k+ P(u) [g (u)� �Tu]� P(u) [g (u)� �Tu]

� �h ku� uk+ P(u) [g (u)� �Tu]� P(u) [g (u)� �Tu]

+ P(u) [g (u)� �Tu]� P(u) [g (u)� �Tu]

� �h ku� uk+ � ku� uk+ kg (u)� g (u)� � (Tu� Tu)k

� (� + �h) ku� uk+ kg (u)� g (u)k+ � kTu� Tuk

� (� + �h) ku� uk+ �g ku� uk+ ��T kh (u)� h (u)k

�� + �g + �h + ��T�h

�ku� uk

= k1 ku� uk ;

where we have used the Assumption 3.1, Lipschitz continuity of g and h with constants

�g > 0 and �h > 0, respectively and h-Lipschitz continuity of T with constant �T > 0,

and

k1 = � + �g + �h + ��T�h:

Thus, we have1

k1kR (u)k � ku� uk : (6.2.6)


1

k1kR (u)k � ku� uk � k2 kR (u)k ; 8u 2 H; (6.2.7)


Letting u = 0 in (6:2:7), we have

1

k1kR (0)k � kuk � k2 kR (0)k : (6.2.8)

86

Combining (6:2:7) and (6:2:8), we obtain a relative error bound for any point u 2 H :

h (u) 2 (u).

Theorem 6.2.2 Assume that all the assumptions of Theorem 6.2.1 hold. If 0 6= u 2 H :

h (u) 2 (u) is a solution of (3:2:1), then

c1 kR (u)k = kR (0)k � ku� uk = kuk � c2 kR (u)k = kR (0)k :

If (u) = , then Theorem 6.2.1 reduces to the following result.

Corollary 6.2.1 Let u 2 H : h (u) 2 be a solution of the problem (3:3:5). Let the

operator T be strongly h-monotone with constant �T > 0 and h-Lipschitz continuous

with constant �T > 0. Let the operators g and h be Lipschitz continuous with constants

�g > 0 and �h > 0; respectively, and h is also strongly expanding with constant �h > 0.

If � >�g�h�T �

2h, then

1


where

k1 = �h + �g + ��T�h and k2 =2�h + �g + ��T�h

��T �2h � �g�h

:

If h = g = I, where I is an identity operator, then Theorem 6.2.1 reduces to the

following result of Noor [86].

Corollary 6.2.2 Let u 2 (u) be a solution of the problem (3:3:4). Let T be strongly

monotone with constant �T > 0 and Lipschitz continuous with constant �T > 0. If

Assumption 3.1 holds, then

1


where

k1 = 2 + � + ��T and k2 =1 + ��T��T

:

87

If (u) = and h = g, then Theorem 6.2.1 reduces to the following result of Noor

[85].

Corollary 6.2.3 Let u 2 H : g (u) 2 be a solution of the problem (3:3:6). Let the

operator T be strongly g-monotone with constant �T > 0 and g-Lipschitz continuous with

constant �T > 0. Let the operator g be Lipschitz continuous with constant �g > 0 and g

is also strongly expanding with constant �g > 0. Then

1


where

k1 = (2 + ��T ) �g and k2 =�g (1 + ��T )

��T �2h

:

6.3 Regularized Merit Functions

Note that the normal residue vector (merit function) R (u) ; de�ned by (3:5:18) ; is non-

di¤erentiable. To over come the nondi¤erentiability, which is a serious drawback of the

residue merit function, we consider an other merit function associated with problem

(3:2:1). This merit function can be viewed as a regularized merit function, see [38]. We

consider the function for all u 2 H : h (u) 2 (u), such that

M� (u) =Tu; h (u)� P(u) [g (u)� �Tu]

�� 1

2�

h (u)� P(u) [g (u)� �Tu] 2 ; (6.3.1)from which it follows thatM� (u) � 0; for all u 2 H : h (u) 2 (u).

We now discuss some special cases of the regularized merit functionM� (u), de�ned

by (6:3:1) :

I. If (u) = , then the merit function M� (u), de�ned by (6:3:1), reduces to the

following merit function.

M� (u) = hTu; h (u)� P [g (u)� �Tu]i�1

2�kh (u)� P [g (u)� �Tu]k2 ; (6.3.2)

88

for all u 2 H : h (u) 2 : The merit function M� (u) ; de�ned by (6:3:2), is

associated with the problem (3:3:5).

II. If h = g, then the merit functionM� (u), de�ned by (6:3:1), reduces to the following

merit function for the general quasi variational inequalities (3:3:1).

M� (u) =Tu; g (u)� P(u) [g (u)� �Tu]

�� 1

2�

g (u)� P(u) [g (u)� �Tu] 2 ;(6.3.3)

for all u 2 H : g (u) 2 (u) :

III. If h = g = I, where I is an identity operator, then the merit function M� (u),

de�ned by (6:3:1), reduces to the following merit function.

M� (u) =Tu; u� P(u) [u� �Tu]

�� 1

2�

u� P(u) [u� �Tu] 2 ; (6.3.4)

for all u 2 (u) : The merit function M� (u) ; de�ned by (6:3:4), introduced and

studied by Noor [86] in 2007, for the quasi variational inequalities (3:3:4).

IV. If (u) = and h = g, then the merit function M� (u), de�ned by (6:3:1), is

equivalent to:

M� (u) = hTu; g (u)� P [g (u)� �Tu]i �1

2�kg (u)� P [g (u)� �Tu]k2 ; (6.3.5)

for all u 2 H : g (u) 2 : Noor [85] introduced and studied the merit function

M� (u) ; de�ned by (6:3:5), for the general variational inequalities (3:3:6).

V. If (u) = and h = g = I, where I is an identity operator, then the merit function

M� (u), de�ned by (6:3:1), reduces to the well-known regularized merit function:

M� (u) = hTu; u� P [u� �Tu]i �1

2�ku� P [u� �Tu]k2 8u 2 ; (6.3.6)

which is a mainly due to Fukushima [38], for the classical variational inequalities

89

(3:3:9).

We now show that the functionM� (u) ; de�ned by (6:3:1) ; is a merit function and

this is the main motivation of our next result.

Theorem 6.3.1 For all u 2 H : h (u) 2 (u), we have

M� (u) �1

4�kR� (u)k2 �

1

�kg (u)� h (u)k2 :

In Particular, we haveM� (u) = 0, if and only if, u 2 H : h (u) 2 (u) is a solution of

the problem (3:2:1).

Proof. Setting u = P(u) [g (u)� �Tu], z = g (u) � �Tu and v = h (u) in Lemma

2.3.2, we have

�Tu+ P(u) [g (u)� �Tu]� g (u) ; h (u)� P(u) [g (u)� �Tu]

�� 0;

from which using the relation (6:3:1) and Lemma 2.1.1, we have

0 � h�Tu�R� (u) + h (u)� g (u) ; R� (u)i

= hTu;R� (u)i �1

�hR� (u) ; R� (u)i �

1

�hg (u)� h (u) ; R� (u)i

� M� (u) +1

2�kR� (u)k2 �

1

�kR� (u)k2 �

1

�hg (u)� h (u) ; R� (u)i

= M� (u)�1

2�kR� (u)k2 +

1

�kg (u)� h (u)k2 + 1

4�kR� (u)k2

= M� (u)�1

4�kR� (u)k2 +

1

�kg (u)� h (u)k2 ;

which implies that

M� (u) �1

4�kR� (u)k2 �

1

�kg (u)� h (u)k2 ;

which is the required result. Clearly we haveM� (u) � 0; 8u 2 H : h (u) 2 (u) : Now if

M� (u) = 0; then clearly R� (u) = 0. Hence by Lemma 3.3.2, we see that u 2 H : h (u) 2

90

(u) is the solution of problem (3:2:1). Conversely, if u 2 H : h (u) 2 (u) is the solution

of problem (3:2:1), then h (u) = P(u) [g (u)� �Tu] by Lemma 3.3.2. Consequently from

(6:3:1), we see thatM� (u) = 0, the required result.

If (u) = , then the Theorem 6.3.1 reduces to the following result for the function

M� (u) ; de�ned by (6:3:2).

Corollary 6.3.4 For all u 2 H : h (u) 2 , we have

M� (u) �1

4�kR� (u)k2 �

1

�kg (u)� h (u)k2 :

In Particular, we haveM� (u) = 0, if and only if, u 2 H : h (u) 2 is a solution of the

problem (3:3:5).

If h = g = I, where I is an identity operator, then the Theorem 6.3.1 reduces to the

following result of Noor [86] for the functionM� (u) ; de�ned by (6:3:4).

Corollary 6.3.5 For all u 2 (u), we have

M� (u) �1

2�kR� (u)k2 :

In Particular, we haveM� (u) = 0, if and only if, u 2 (u) is a solution of the problem

(3:3:4).

If (u) = and h = g, then the Theorem 6.3.1 reduces to the following result of

Noor [85] for the functionM� (u) ; de�ned by (6:3:5).

Corollary 6.3.6 For all u 2 H : g (u) 2 , we have

M� (u) �1

2�kR� (u)k2 :

In Particular, we haveM� (u) = 0, if and only if, u 2 H : g (u) 2 is a solution of the

problem (3:3:6).

91

From Theorem 6.3.1, we see that the function M� (u) de�ned by (6:3:1) is a merit

function for the extended general quasi variational inequality (3:2:1). It is clear that the

regularized merit function is di¤erentiable whenever T; g and h are di¤erentiable. We

now derive the error bounds without using the Lipschitz continuity of the operator T .

Theorem 6.3.2 Let u 2 H : h (u) 2 (u) be a solution of the problem (3:2:1). Let the

operator T be a strongly h-monotone with constant �T > 0. If the operator h is strongly

expanding with a constant �h > 0, then

ku� uk2 � 4�

�2h (4��T � 3)

�M� (u) +

1

�kg (u)� h (u)k2

�; 8u 2 H: (6.3.7)

Proof. Since u 2 H : h (u) 2 (u) be a solution of the problem (3:2:1), and setting

g (v) = h (u), we have

h�Tu+ h (u)� g (u) ; h (u)� h (u)i � 0;

form which, using Lemma 2.1.1, we have

hTu; h (u)� h (u)i � 1

�hg (u)� h (u) ; h (u)� h (u)i

� �1�kg (u)� h (u)k2 � 1

4�kh (u)� h (u)k2 : (6.3.8)

From the relation (6:3:1) ; using strongly h-monotonicity of operator T with constant

�T > 0; strongly expandicity of operator h with constant �h > 0; and the relation

(6:3:8) ; we have

M� (u) � hTu; h (u)� h (u)i � 1

2�kh (u)� h (u)k2

= hTu� Tu; h (u)� h (u)i+ hTu; h (u)� h (u)i � 1

2�kh (u)� h (u)k2

��T �

1

2�

�kh (u)� h (u)k2 + hTu; h (u)� h (u)i

92

��T �

3

4�

�kh (u)� h (u)k2 � 1

�kg (u)� h (u)k2

� �2h4�(4��T � 3) ku� uk2 �

1

�kg (u)� h (u)k2 ;

which implies that

ku� uk2 � 4�

�2h (4��T � 3)

�M� (u) +

1

�kg (u)� h (u)k2

�:

This is the required result.


Corollary 6.3.7 Let u 2 H : h (u) 2 be a solution of the problem (3:3:5). Let T be

a strongly h-monotone with constant �T > 0. If h is strongly expanding with a constant

�h > 0, then

ku� uk2 � 4�

�2h (4��T � 3)

�M� (u) +

1

�kg (u)� h (u)k2

�; 8u 2 H;

where the functionM� (u) ; de�ned by (6:3:2).



Corollary 6.3.8 Let u 2 (u) be a solution of the problem (3:3:4). Let the operator T

be a strongly monotone with constant �T > 0. Then

ku� uk2 � 2�

2��T � 1M� (u) ; 8u 2 H;



Noor [85].

93

Corollary 6.3.9 Let u 2 H : g (u) 2 be a solution of the problem (3:3:6). Let the

operator T be a strongly g-monotone with constant �T > 0. If the operator g is strongly

expanding with a constant �g > 0, then

ku� uk2 � 2�

�2g (2��T � 1)M� (u) ; 8u 2 H;


6.4 D-merit Functions

We now consider another merit function associated with the problem (3:2:1), which can

be viewed as a di¤erence of two regularized merit functions and known as D-gap functions

or D-merit functions. Such type of merit functions were �rst introduced by Peng [114] and

later studied by many authors for solving variational inequalities and complementarity

problems, see [39, 58, 85, 86, 113, 115, 118, 119]. Here we de�ne the D-merit function by

a formal di¤erences of the regularized merit function de�ned by (6:3:1). To this end, we

consider the following function

D�;� (u) = M� (u)�M� (u)

= hTu;R� (u)�R� (u)i �1

2�kR� (u)k2 +

1

2�kR� (u)k2 ; 8u 2 H:(6.4.1)

It is clear that the D�;� (u) is everywhere �nite. We now show that the function D�;� (u)

de�ned by (6:4:1) is indeed a merit function for the extended general quasi variational

inequalities (3:2:1) and this is the main motivation of our next result.

Theorem 6.4.1 For all u 2 H and � > � > 0; we have

(�� ) kR� (u)k2 � 2��D�;� (u) � (�� ) kR� (u)k2 : (6.4.2)

In particular, D�;� (u) = 0; if and only if, u 2 H solves the problem (3:2:1).

94

Proof. Setting u = P(u) [g (u)� �Tu] ; v = P(u) [g (u)� �Tu] and z = h (u)� �Tu

in Lemma 2.3.2, we have

P(u) [g (u)� �Tu]� h (u) + �Tu; P(u) [g (u)� �Tu]� P(u) [g (u)� �Tu]

�� 0

which implies that

hTu;R� (u)�R� (u)i �1

�hR� (u) ; R� (u)�R� (u)i : (6.4.3)

From (6:4:1) and (6:4:3), we have

D�;� (u) � � 1

2�kR� (u)k2 +

1

2�kR� (u)k2 +

1

�hR� (u) ; R� (u)�R� (u)i

=1

2�kR� (u)k2 �

1

2�kR� (u)k2 +

1

�kR� (u)k2 �

1

�hR� (u) ; R� (u)i

=1

2

�1

�� 1�

�kR� (u)k2 +

1

2�kR� (u)k2 +

1

2�kR� (u)k2 �

1

�hR� (u) ; R� (u)i

=1

2

�1

�� 1�

�kR� (u)k2 +

1

2�kR� (u)�R� (u)k2

� 1

2

�1

�� 1�

�kR� (u)k2 ;

which implies the right most inequality of the required result

2��D�;� (u) � (�� ) kR� (u)k2 : (6.4.4)

In a similar way, by setting u = P(u) [g (u)� �Tu] ; v = P(u) [g (u)� �Tu] and z =

h (u)� �Tu in Lemma 2.3.2, we have

P(u) [g (u)� �Tu]� h (u) + �Tu; P(u) [g (u)� �Tu]� P(u) [g (u)� �Tu]

�� 0

95

which implies that

hTu;R� (u)�R� (u)i �1

�hR� (u) ; R� (u)�R� (u)i : (6.4.5)

From (6:4:1) and (6:4:5), we have

D�;� (u) � � 1

2�kR� (u)k2 +

1

2�kR� (u)k2 +

1

�hR� (u) ; R� (u)�R� (u)i

=1

2�kR� (u)k2 �

1

2�kR� (u)k2 �

1

�kR� (u)k2 +

1

�hR� (u) ; R� (u)i

=1

2

�1

�� 1�

�kR� (u)k2 �

1

2�kR� (u)k2 �

1

2�kR� (u)k2 +

1

�hR� (u) ; R� (u)i

=1

2

�1

�� 1�

�kR� (u)k2 �

1

2�kR� (u)�R� (u)k2

� 1

2

�1

�� 1�

�kR� (u)k2 ;

which implies the left most inequality of the required result

(�� ) kR� (u)k2 � 2��D�;� (u) : (6.4.6)


(�� ) kR� (u)k2 � 2��D�;� (u) � (�� ) kR� (u)k2 ;


Theorem 6.4.2 Let u 2 H : h (u) 2 (u) be a solution of (3:2:1). If the operator T

is strongly h-monotone with constant �T > 0 and h is strongly expanding with constant

�h > 0; then

ku� uk2 � 4��

�2h (4��T � 3�+ 2�)

�D�;� (u) +

1

�kg (u)� h (u)k2

�; 8u 2 H (6.4.7)

96

Proof. Since u 2 H : h (u) 2 (u) be a solution of (3:2:1) and setting g (v) = h (u)

in (3:2:1), we have

h�Tu+ h (u)� g (u) ; h (u)� h (u)i � 0;

form which, using Lemma 2.1.1, we have

hTu; h (u)� h (u)i � 1

�hg (u)� h (u) ; h (u)� h (u)i

� �1�kg (u)� h (u)k2 � 1

4�kh (u)� h (u)k2 : (6.4.8)

From (6:4:1) ; using the strongly h-monotonicity of operator T with constant �T > 0;

strongly expandicity of operator h with constant �h > 0; and the relation (6:4:8) ; we

have

D�;� (u) � hTu; h (u)� h (u)i � 1

2�kh (u)� h (u)k2 + 1

2�kh (u)� h (u)k2

= hTu� Tu; h (u)� h (u)i+ hTu; h (u)� h (u)i ��1

2�� 1

2�

�kh (u)� h (u)k2

��T �

1

2�+1

2�

�kh (u)� h (u)k2 + hTu; h (u)� h (u)i

��T �

3

4�+1

2�

�kh (u)� h (u)k2 � 1

�kg (u)� h (u)k2

� �2h4��

(4��T � 3�+ 2�) ku� uk2 �1

�kg (u)� h (u)k2 ;

which implies that

ku� uk2 � 4��

�2h (4��T � 3�+ 2�)

�D�;� (u) +

1

�kg (u)� h (u)k2

�:



Corollary 6.4.10 Let u 2 H : h (u) 2 be a solution of the problem (3:3:5). Let the

operator T be a strongly h-monotone with constant �T > 0. If the operator h is strongly

97

expanding with a constant �h > 0, then

ku� uk2 � 4��

�2h (4��T � 3�+ 2�)

�D�;� (u) +

1

�kg (u)� h (u)k2

�; 8u 2 H:



Corollary 6.4.11 Let u 2 (u) be a solution of the problem (3:3:4). Let the operator T

be a strongly monotone with constant �T > 0. Then

ku� uk2 � 2��

2��T � �+ �D�;� (u) ; 8u 2 H:


Noor [85].

Corollary 6.4.12 Let u 2 H : g (u) 2 be a solution of the problem (3:3:6). Let T be a

strongly g-monotone with constant �T > 0. If the operator g is strongly expanding with

a constant �g > 0, then

ku� uk2 � 2��

�2h (2��T � �+ �)D�;� (u) ; 8u 2 H:

6.5 Conclusion

In this chapter, we have introduced some new classes of merit functions for extended

general quasi variational inequalities. Using these functions, we have found the error

bounds for the solution of extended general quasi variational inequalities. Several special

cases are also investigated. We would like to mention that the contents of this chapter

have been submitted for publication in an international journal.

98

Chapter 7

System of Quasi Variational Inequalities

99

7.1 Introduction

In this chapter, we consider a new system of extended general quasi variational inequal-

ities involving four nonlinear operators. Using projection operator technique, we show

that system of extended general quasi variational inequalities is equivalent to a system

of �xed point problems. Using this alternative equivalent formulation, some Jacobi type

algorithms for solving a system of extended general quasi variational inequalities are sug-

gested and investigated. Convergence of these new methods is considered under some

suitable conditions. Several special cases are discussed. Results obtained in this chap-

ter continue to hold for these problems. The ideas and techniques of this chapter may

stimulate further research in this �eld. The results of this chapter have been published

in Applied Mathematics and Computation, 245, 566-574, see [105].

7.2 Problem Formulation

Let H be a real Hilbert space, whose norm and inner product are denoted by k�k and

h�; �i, respectively. Let 1; 2 be two nonempty, closed and convex sets in H.

For given nonlinear operators T1; T2; g; h : H ! H and point-to-set mappings 1 :

y ! 1(y) and 2 : x ! 2(x), which associate closed and convex-valued sets 1(y)

and 2(x) with elements x; y 2 H, consider a problem of �nding x; y 2 H : h (y) 2

1 (y) ; h (x) 2 2 (x) such that

h�1T1x+ h (y)� g (x) ; g (v)� h (y)i � 0; 8v 2 H : g (v) 2 1 (y)

h�2T2y + h (x)� g (y) ; g (v)� h (x)i � 0; 8v 2 H : g (v) 2 2 (x)

9=; (7.2.1)

where �1 > 0 and �2 > 0 are constants. The system (7:2:1) is called a system of extended

general quasi variational inequalities involving four operators. Recently, Noor et al [105]

introduced and studied system (7:2:1) :

100

We now discuss some special cases of system (7:2:1).

I. If 1 (y) = (y) and 2 (x) = (x), then problem (7:2:1) reduces to �nd x; y 2 H :

h (y) 2 (y) ; h (x) 2 (x) such that

h�1T1x+ h (y)� g (x) ; g (v)� h (y)i � 0; 8v 2 H : g (v) 2 (y)

h�2T2y + h (x)� g (y) ; g (v)� h (x)i � 0; 8v 2 H : g (v) 2 (x)

9=; (7.2.2)

The system (7:2:2) is called a system of extended general quasi variational inequal-

ities with four nonlinear operators.

II. If 1 (y) = 1 and 2 (x) = 2, then problem (7:2:1) collapse to �nd x; y 2 H :

h1 (y) 2 1; h2 (x) 2 2 such that

h�1T1x+ h (y)� g (x) ; g (v)� h (y)i � 0; 8v 2 H : g (v) 2 1h�2T2y + h (x)� g (y) ; g (v)� h (x)i � 0; 8v 2 H : g (v) 2 2

9=; ; (7.2.3)

is a system of extended general variational inequalities involving four nonlinear

operators.

III. If h = g, then problem (7:2:2) reduces to �nd x; y 2 H : g (y) 2 (y) ; g (x) 2 (x)

and

h�1T1x+ g (y)� g (x) ; g (v)� g (y)i � 0; 8v 2 H : g (v) 2 (y)

h�2T2y + g (x)� g (y) ; g (v)� g (x)i � 0; 8v 2 H : g (v) 2 (x)

9=; (7.2.4)

which is called a system of general quasi variational inequalities with three nonlinear

operators.

IV. If g = h = I; where I is an identity operator, then system (7:2:1) reduces to �nd

101

y 2 1 (y) and x 2 2 (x) such that

h�1T1x+ y � x; v � yi � 0; 8v 2 1 (y)

h�2T2y + x� y; v � xi � 0; 8v 2 2 (x)

9=; (7.2.5)

is called a system of quasi variational inequalities, introduced by Noor and Noor

[101].

V. If 1 (y) = (y) and 2 (x) = (x), then system (7:2:5) collapse to �nd x 2 (x)

and y 2 (y) such that

h�1T1x+ y � x; v � yi � 0; 8v 2 (y)

h�2T2y + x� y; v � xi � 0; 8v 2 (x)

9=; (7.2.6)

which is also, introduced by Noor and Noor [101].

VI. If T1 = T2 = T and �1 = �2 = �, then problem (7:2:2) reduces to �nd x 2 H :

h (x) 2 (x) and

h�Tx+ h (x)� g (x) ; g (y)� h (x)i � 0; 8y 2 H : g (y) 2 (x) (7.2.7)

which is called extended general quasi variational inequality, introduced and studied

by Noor and Noor [100] and Noor et al [104].

VII. If (x) = , then problem (7:2:7) reduces to �nd x 2 H : h (x) 2 such that

h�Tx+ h (x)� g (x) ; g (y)� h (x)i � 0; 8y 2 H : g (y) 2 (7.2.8)

which is called extended general variational inequality, introduced and studied by

Noor [90]. For the formulation, applications, numerical methods and other as-

pects of extended general variational inequalities, see [87, 90, 93, 95] and references

therein.

102

VIII. If h = g and (x) = , then problem (7:2:7) reduces to the problem of �nding

x 2 H : g (x) 2 such that

hTx; g (y)� g (x)i � 0; 8y 2 H : g (y) 2 (7.2.9)

which is called general variational inequality. This problem introduced and studied

by Noor [69], in 1988. For the formulation, applications, numerical methods and

other aspects of general variational inequalities, see [69, 84, 85, 89] and references

therein.

For suitable and appropriate choice of operators and spaces, one can obtain several

new and known classes of variational inequalities.

We now show that the nonconvex minimax problem can be characterized by a system

of variational inequalities of the type (7:2:1). This is the main motivation of the next

example.

Example 7 Consider the following nonconvex minimax problem as

minx2H:h(x)22

�max

y2H:h(y)21f (h (x) ; h (y))

�; (7.2.10)

where f is twice di¤erentiable in H � H. The solution of (7:2:10) is equivalent to the

saddle point of f (h (x) ; h (y)), that is, a point x�; y� 2 H : h (y�) 2 1; h (x�) 2 2

satis�es

f (h (x�) ; h (y)) � f (h (x�) ; h (y�)) � f (h (x) ; h (y�)) ;

for all x; y 2 H : h (y) 2 1; h (x) 2 2.

Using the technique of Bazaraa et al [14], one can show that x�; y� is a saddle point

of f (h (x) ; h (y)) ; if and only if, it satis�es

h�1rxf (x�; y�) ; g (x)� h (y�)i � 0; 8x 2 H : g (x) 2 1

h�2ryf (x�; y�) ; g (x)� h (x�)i � 0; 8x 2 H : g (x) 2 2

9=; ; (7.2.11)

103

where �1 > 0 and �2 > 0 are constants.

Clearly problem (7:2:11) is a special case of (7:2:1) with rxf (x�; y�) = T1x and

ryf (x�; y�) = T2y:

7.3 Main Results

In this section, we show that the system of extended general quasi variational inequalities

(7:2:1) is equivalent to a system of �xed point problems. This alternative equivalent

formulation is used to suggest some parallel algorithms for solving (7:2:1), using the

technique of Noor and Noor [101].

Lemma 7.3.1 The system of extended general quasi variational inequalities (7:2:1) has

a solution, x; y 2 H : h (y) 2 1 (y) ; h (x) 2 2 (x), if and only if, x; y 2 H : h (y) 2

1 (y) ; h (x) 2 2 (x) satis�es the relations

h (y) = P1(y) [g (x)� �1T1x] (7.3.1)

h (x) = P2(x) [g (y)� �2T2y] ; (7.3.2)

where �1 > 0 and �2 > 0 are constants.

Lemma 7.3.1 implies that the system (7:2:1) is equivalent to the �xed point problems

(7:3:1) and (7:3:2). This alternative equivalent formulation is very useful from numerical

and theoretical point of view. Using this �xed point formulation, we suggest and analyze

some iterative algorithms.

We can rewrite (7:3:1) and (7:3:2) in the following equivalent forms:

y = (1� �n) y + �n�y � h (y) + P1(y) [g (x)� �1T1x]

(7.3.3)

x = (1� �n)x+ �n�x� h (x) + P2(x) [g (y)� �2T2y]

; (7.3.4)

where 0 � �n; �n � 1 for all n � 0.

104

This alternative equivalent formulation is used to suggest the following parallel algo-

rithms for solving system of extended general quasi variational inequalities (7:2:1) and

its variant forms.

Algorithm 7.1 For given x0; y0 2 H : h (y0) 2 1 (y) and h (x0) 2 2 (x) ; �nd xn+1and yn+1 by the iterative schemes

yn+1 = (1� �n) yn + �n�yn � h (yn) + P1(yn) [g (xn)� �1T1xn]

(7.3.5)

xn+1 = (1� �n)xn + �n�xn � h (xn) + P2(xn) [g (yn)� �2T2yn]

; (7.3.6)


Algorithm 7.1 is called the parallel algorithm, which is suitable for implementation

on two di¤erent processor computers. It is well known that parallel algorithms are better

than the sequential iterative methods. Algorithm 7.1 can be regarded as a Jacobi method

type iterative method.


I. If 1 (y) = (y) and 2 (x) = (x), then Algorithm 7.1 reduces to following

parallel projection algorithm for solving the system (7:2:2).

Algorithm 7.2 For given x0; y0 2 H : h (x0) 2 (x) ; h (y0) 2 (y) ; �nd xn+1 and yn+1by the iterative schemes

yn+1 = (1� �n) yn + �n�yn � h (yn) + P(yn) [g (xn)� �1T1xn]

xn+1 = (1� �n)xn + �n

�xn � h (xn) + P(xn) [g (yn)� �2T2yn]

;


II. If h = g, then Algorithm 7.2 reduces to the following parallel algorithm for solving

system (7:2:4).

105

Algorithm 7.3 For given x0; y0 2 H : g (x0) 2 (x) ; g (y0) 2 (y), compute sequences

fxng and fyng by the iterative schemes

yn+1 = (1� �n) yn + �n�yn � g (yn) + P(yn) [g (xn)� �1T1xn]

xn+1 = (1� �n)xn + �n

�xn � g (xn) + P(xn) [g (yn)� �2T2yn]

;


III. If 1 (y) = 1 and 2 (x) = 2, then Algorithm 7.1 collapse to the following

iterative method for solving system (7:2:3).

Algorithm 7.4 For arbitrary chosen initial points x0; y0 2 H : h (y0) 2 1; h (x0) 2 2;

sequences fxng and fyng are computed by

yn+1 = (1� �n) yn + �n fyn � h (yn) + P1 [g (xn)� �1T1xn]g

xn+1 = (1� �n)xn + �n fxn � h (xn) + P2 [g (yn+1)� �2T2yn]g ;


IV. If T1 = T2 = T; g = h = I; where I is an identity operator, then Algorithm 7.1

reduces to the following algorithm.

Algorithm 7.5 For given x0 2 2 (x) ; y0 2 1 (y) �nd the approximate solution xn; ynby the iterative schemes

yn+1 = (1� �n) yn + �nP1(yn) [xn � �1Txn]

xn+1 = (1� �n)xn + �nP2(xn) [yn � �2Tyn] ;


V. If g = h = I; where I is an identity operator, then Algorithm 7.1 reduces to:

106

Algorithm 7.6 For given x0 2 2 (x) ; y0 2 1 (y) �nd the approximate solution xn; ynby the iterative schemes

yn+1 = (1� �n) yn + �nP1(yn) [xn � �1T1xn]

xn+1 = (1� �n)xn + �nP2(xn) [yn � �2T2yn] ;


For suitable and appropriate choice of operators and spaces, one can obtain several

new and known iterative methods for solving system of extended general quasi variational

inequalities and related problems. It has been shown [96] that the problem (7:2:1) has a

solution under some suitable conditions.

We now investigate the convergence analysis of Algorithm 7.1. This is the main

motivation of our next result.

Theorem 7.3.1 Let operators T1; T2; g; h : H! H be strongly monotone with constants

�T1 > 0; �T2 > 0; �g > 0; �h > 0 and Lipschitz continuous with constants �T1 > 0;

�T2 > 0; �g > 0; �h > 0; respectively. If Assumption 3.1 and following conditions hold:

(i) �T1 =q1� 2�1�T1 + �21�2T1 < 1.

(ii) �T2 =q1� 2�2�T2 + �22�2T2 < 1.

(iii) 0 � �n; �n � 1; 8 n � 0,

�n (1� � � �h)� �n (�g + �T1) � 0

�n (1� � � �h)� �n (�g + �T2) � 0;

such that1Pn=0

(�n (1� � � �h)� �n (�g + �T1)) =1

107

1Pn=0

(�n (1� � � �h)� �n (�g + �T2)) =1;

where

�g =q1� 2�g + �2g; and �h =

q1� 2�h + �2h;

then sequences fxng and fyng obtained from Algorithm 7.1 converge to x and y;

respectively.

Proof. Let x; y 2 H : h (y) 2 1 (y) ; h (x) 2 2 (x) be a solution of (7:2:1). Then

from (7:3:4) ; (7:3:6) and using Assumption 3.1, we have

kxn+1 � xk = k (1� �n)xn + �n�xn � h (xn) + P2(xn) [g (yn)� �2T2yn]

� (1� �n)x� �n

�x� h (x) + P2(x) [g (y)� �2T2y]

k

� (1� �n) kxn � xk+ �n kxn � x� (h (xn)� h (x))k

+�n P2(xn) [g (yn)� �2T2yn]� P2(x) [g (y)� �2T2y]


+�n P2(xn) [g (yn)� �2T2yn]� P2(x) [g (yn)� �2T2yn]

+�n P2(x) [g (yn)� �2T2yn]� P2(x) [g (y)� �2T2y]

� (1� �n) kxn � xk+ ��n kxn � xk+ �n kxn � x� (h (xn)� h (x))k

+�n kyn � y � (g (yn)� g (y))k+ �n kyn � y � �2 (T2yn � T2y)k :(7.3.7)

Since operator T2 is strongly monotone and Lipschitz continuous with constants �T2 > 0

and �T2 > 0, respectively. Then it follows that

kyn � y � �2 (T2yn � T2y)k2 = kyn � yk2 � 2�2 hT2yn � T2y; yn � yi+ kT2yn � T2yk

2

��1� 2�2�T2 + �22�2T2

�kyn � yk2 . (7.3.8)


kxn � x� (h (xn)� h (x))k2 ��1� 2�h + �2h

�kxn � xk2 ; (7.3.9)

108

and

kyn � y � (g (yn)� g (y))k2 ��1� 2�g + �2g

�kyn � yk2 , (7.3.10)

where we have used the strongly monotonicity and Lipschitz continuity of operators g, h

with constants �g > 0, �h > 0 and �g > 0, �h > 0; respectively.


kxn+1 � xk � (1� �n) kxn � xk+ ��n kxn � xk+ �nq1� 2�h + �2h kxn � xk

+�n

q1� 2�g + �2g kyn � yk+ �n

q1� 2�2�T2 + �22�2T2 kyn � yk

= (1� �n (1� � � �h)) kxn � xk+ �n (�g + �T2) kyn � yk : (7.3.11)

Similarly, from (7:3:3) ; (7:3:5) ; using strongly monotonicity and Lipschitz continuity of

operators T1; g; h with constants �T1 > 0; �g > 0; �h > 0 and �T1 > 0; �g > 0; �h > 0,

respectively and using Assumption 3.1, we have

kyn+1 � yk = (1� �n) yn + �n �yn � h (yn) + P1(yn) [g (xn)� �1T1xn]� (1� �n) y � �n

�y � h (y) + P1(y) [g (x)� �1T1x]

� (1� �n) kyn � yk+ �n kyn � y � (h (yn)� h (y))k

+�n P1(yn) [g (xn)� �1T1xn]� P1(y) [g (x)� �1T1x]


+�n P1(yn) [g (xn)� �1T1xn]� P1(y) [g (xn)� �1T1xn]

+�n P1(y) [g (xn)� �1T1xn]� P1(y) [g (x)� �1T1x]

� (1� �n) kyn � yk+ ��n kyn � yk+ �n kyn � y � (h (yn)� h (y))k

+�n kxn � x� (g (xn)� g (x))k+ �n kxn � x� �1 (T1xn � T1x)k

� (1� �n) kyn � yk+ ��n kyn � yk+ �n�h kyn � yk

+�n�g kxn � xk+ �n�T1 kxn � xk

= (1� �n (1� � � �h)) kyn � yk+ �n (�g + �T1) kxn � xk : (7.3.12)

109

Adding (7:3:11) and (7:3:12), we have

kxn+1 � xk+ kyn+1 � yk � (1� �n (1� � � �h)) kxn � xk+ �n (�g + �T2) kyn � yk

+(1� �n (1� � � �h)) kyn � yk+ �n (�g + �T1) kxn � xk

= (1� �n (1� � � �h) + �n (�g + �T1)) kxn � xk

+(1� �n (1� � � �h) + �n (�g + �T2)) kyn � yk

� max (�1; �2) (kxn � xk+ kyn � yk)

= � (kxn � xk+ kyn � yk) ; (7.3.13)

where

� = max (�1; �2)

�1 = 1� (�n (1� � � �h)� �n (�g + �T1))

�2 = 1� (�n (1� � � �h)� �n (�g + �T2)) :

Using assumption (iii), we have � < 1. Thus, using Lemma 2.3.3, it follows from (7:3:13)

that

limn!1

[kxn+1 � xk+ kyn+1 � yk] = 0:

This implies that

limn!1

kxn+1 � xk = 0,

and

limn!1

kyn+1 � yk = 0:

This is the desired result.

We now suggest some other iterative methods for solving the system of extended

general quasi variational inequalities (7:2:1) by using some suitable substitutions.

110

Using Lemma 7.3.1, one can easily show that x; y 2 H : h (y) 2 1 (y) ; h (x) 2 2 (x)

is a solution of (7:2:1) if and only if, x; y 2 H : h (y) 2 1 (y) ; h (x) 2 2 (x) satis�es

h (y) = P1(y) [z] (7.3.14)

h (x) = P2(x) [w] (7.3.15)

z = g (x)� �1T1x (7.3.16)

w = g (y)� �2T2y: (7.3.17)

This alternative formulation can be used to suggest and analyze the following iterative

methods for solving the system (7:2:1).

Algorithm 7.7 For given x0; y0 2 H : h (y0) 2 1 (y) ; h (x0) 2 2 (x) �nd xn+1 and

yn+1 by the iterative schemes

yn+1 = (1� �n) yn + �n�yn � h (yn) + P1(yn) [zn]

(7.3.18)

xn+1 = (1� �n)xn + �n�xn � h (xn) + P2(xn) [wn]

(7.3.19)

zn = g (xn)� �1T1xn (7.3.20)

wn = g (yn)� �2T2yn; (7.3.21)



I. If 1 (y) = (y) and 2 (x) = (x), then Algorithm 7.7 reduces to:

Algorithm 7.8 For given x0; y0 2 H : h (x0) 2 (x) ; h (y0) 2 (y) �nd the approximate

solutions xn+1 and yn+1 by the iterative schemes

yn+1 = (1� �n) yn + �n�yn � h (yn) + P(yn) [zn]

xn+1 = (1� �n)xn + �n

�xn � h (xn) + P(xn) [wn]

111

zn = g (xn)� �1T1xn

wn = g (yn)� �2T2yn;


II. If g = h = I, where I is an identity operator, and �n = �n = 1, then Algorithm 7.7

reduces to:

Algorithm 7.9 [101] For given x0; y0, �nd the approximate solutions xn+1 and yn+1 by

the iterative schemes

yn+1 = P1(yn) [zn]

xn+1 = P2(xn) [wn]

zn = xn � �1T1xn

wn = yn � �2T2yn; 8 n � 0:

For appropriate and suitable choice of operators and spaces, one can obtain several

new and known iterative methods for solving system of extended general quasi variational

inequalities and related optimization problems.

We now consider the convergence analysis of Algorithm 7.7, using the technique of

Theorem 7.3.1. For the sake of completeness and to convey an idea, we include all the

details.

Theorem 7.3.2 Let operators T1; T2; g; h : H! H be strongly monotone with constants

�T1 > 0; �T2 > 0; �g > 0; �h > 0 and Lipschitz continuous with constants �T1 > 0;

�T2 > 0; �g > 0; �h > 0; respectively. If Assumption 3.1 and all the conditions of

Theorem 7.3.1 hold, then sequences fxng and fyng obtained from Algorithm 7.7 converge

to x and y; respectively.

112

Proof. Let x; y 2 H : h (y) 2 1 (y) ; h (x) 2 2 (x) be a solution of (7:2:1). Then

from (7:3:9) ; (7:3:15) ; (7:3:19) and using Assumption 3.1, we have

kxn+1 � xk � (1� �n) kxn � xk+ �n kxn � x� (h (xn)� h (x))k

+�n P2(xn) [wn]� P2(x) [w]


+�n P2(xn) [wn]� P2(x) [wn] + �n P2(x) [wn]� P2(x) [w]

� (1� �n) kxn � xk+ �n�h kxn � xk+ �n� kxn � xk+ �n kwn � wk

= (1� �n (1� � � �h)) kxn � xk+ �n kwn � wk : (7.3.22)

Similarly, from (7:3:12) ; (7:3:14) ; (7:3:18) and using Assumption 3.1, we have

kyn+1 � yk � (1� �n) kyn � yk+ �n kyn � y � (h (yn)� h (y))k

+�n P1(yn) [zn]� P1(y) [z]


+�n P1(yn) [zn]� P1(y) [zn] + �n P1(y) [zn]� P1(y) [z]

� (1� �n) kyn � yk+ �n�h kyn � yk+ �n� kyn � yk+ �n kzn � zk

= (1� �n (1� � � �h)) kyn � yk+ �n kzn � zk : (7.3.23)

From (7:3:8) ; (7:3:10) ; (7:3:17) and (7:3:21), we have

kwn � wk = kg (yn)� �2T2yn � g (y) + �2T2yk

� kyn � y � (g (yn)� g (y))k+ kyn � y � �2 (T2yn � T2y)k

� (�g + �T2) kyn � yk : (7.3.24)

113

Similarly, from (7:3:12) ; (7:3:16) and (7:3:20), we have

kzn � zk = kg (xn)� �1T1xn � g (x) + �1T1xk

� kxn � x� (g (xn)� g (x))k+ kxn � x� �1 (T1xn � T1x)k

� (�g + �T1) kxn � xk : (7.3.25)

Combining (7:3:22) ; (7:3:24) and (7:3:23) ; (7:3:25), we have

kxn+1 � xk � (1� �n (1� � � �h)) kxn � xk+ �n (�g + �T2) kyn � yk ; (7.3.26)

and

kyn+1 � yk � (1� �n (1� � � �h)) kyn � yk+ �n (�g + �T1) kxn � xk : (7.3.27)

Adding (7:3:26) and (7:3:27), we have

kxn+1 � xk+ kyn+1 � yk � (1� �n (1� � � �h)) kxn � xk+ �n (�g + �T2) kyn � yk

+(1� �n (1� � � �h)) kyn � yk+ �n (�g + �T1) kxn � xk

= (1� �n (1� � � �h) + �n (�g + �T1)) kxn � xk

+(1� �n (1� � � �h) + �n (�g + �T2)) kyn � yk

� max (�1; �2) (kxn � xk+ kyn � yk)

= � (kxn � xk+ kyn � yk) ; (7.3.28)

where

� = max (�1; �2)

�1 = 1� (�n (1� � � �h)� �n (�g + �T1))

�2 = 1� (�n (1� � � �h)� �n (�g + �T2)) :

114

Using assumption (iii), we have � < 1. Thus, using Lemma 2.3.3, it follows from (7:3:28)

that

limn!1

[kxn+1 � xk+ kyn+1 � yk] = 0:

This implies that

limn!1

kxn+1 � xk = 0,

and

limn!1

kyn+1 � yk = 0:


7.4 Conclusion

In this chapter, we have considered a new system of extended general quasi variational

inequalities. We have established that the system of extended general variational inequal-

ities is equivalent to systems of �xed point problems. These equivalent formulations have

been used to propose and analyze several Jacobi type algorithms for solving system of

extended general quasi variational inequalities. Convergence of these new Jacobi type

algorithms is investigated under some suitable conditions. Several special cases are also

discussed. The implementation of these algorithms and their comparison with other tech-

niques need further research. This problem is itself an interesting problem for the future

research. The idea and technique introduced in this chapter may motivate for further

research in this area. The researchers are encouraged to explore the novel and innovative

applications of the system of extended general quasi variational inequalities and their

variant forms in pure and applied sciences.

115

Chapter 8

Conclusion

116

In this work, we have

� Considered and investigated a new class of quasi variational inequalities involving

three di¤erent nonlinear operators. Several special cases are also discussed.

� It has been shown that the extended general quasi variational inequalities are equiv-

alent to the �xed point problems and implicit Wiener-Hopf equations.

� These equivalent formulations have been used to suggest and analyze several iter-

ative methods for solving the quasi variational inequalities and related problems.

� Two projected dynamical systems related to the extended general quasi variational

inequalities are proposed. It has been shown that the solution of the proposed dy-

namical system converges globally exponentially to the solution of extended general

quasi variational inequalities. Some special cases are also discussed, which can be

obtained from our results. Results obtained for the proposed dynamical systems

continue to hold for these problems.

� Several merit functions related to the extended general quasi variational inequalities

are proposed. Using these merit functions, we have investigated error bounds for

the solution of extended general quasi variational inequalities. Several special cases

are also discussed.

� A new system of extended general quasi variational inequalities is considered and

investigated. It has been shown that the system of extended general quasi vari-

ational inequalities are equivalent to the system of �xed point problems. These

alternative formulations have been used to proposed several Jacobi type algorithms

for solving system of extended general quasi variational inequalities.

� We would like to emphasize that the results obtained in this thesis can be ex-

tended and generalized for multi-valued variational and variational like inequalities

involving bifunction and related optimization problems.

117

It is worthy to mention that following research papers have been published/accepted:

i. Noor, M. A., Noor, K. I., & Khan, A. G. (2013). Some iterative schemes for

solving extended general quasi variational inequalities. Applied Mathematics &

Information Sciences. 7(3), 917-925.

ii. Noor, M. A., Noor, K. I., & Khan, A. G. (2014). Parallel schemes for solving a

system of extended general quasi variational inequalities. Applied Mathematics and

Computation. 245, 566-574.

iii. Noor, M. A., Noor, K. I., & Khan, A. G. (2014). Three step algorithms for solv-

ing extended general variational inequalities. Journal of Advanced Mathematical

Studies. 7(2), 38-48.

iv. Noor, M. A., Noor, K. I., & Khan, A. G. (2015). Dynamical systems for quasi

variational inequalities. Annals of Functional Analysis. 6(1), 193-209.

v. Noor, M. A., Kamal, R., Noor, K. I., & Khan, A. G. (2015). Sensitivity analysis for

general variational inclusions involving di¤erence of operators. Journal of Advance

Mathematical Studies. 8(1), 1-8.

vi. Noor, M. A., Noor, K. I., & Khan, A. G. (2014). Three step iterative algo-

rithms for solving a class of quasi variational inequalities. Afrika Matematika.

DOI 10.1007/s13370-014-0304-5.

118

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