ITERATIVE METHODS FOR VARIATIONALINEQUALITIES OVER THE SET OF COMMONFIXED POINTS OF NONEXPANSIVE SEMIGROUPSON BANACH SPACES

Based on hybrid steepest descent method by Yamada, many authors havebeen considering methods for solving variational inequality over thefeasible set C with more complicated structure suc

THAI NGUYEN UNIVERSITY

PHAM THANH HIEU

ITERATIVE METHODS FOR VARIATIONAL

INEQUALITIES OVER THE SET OF COMMONFIXED POINTS OF NONEXPANSIVE SEMIGROUPS

College of Education - Thai Nguyen University (TNU)

Scientific supervisors:

1 Nguyen Thi Thu Thuy, PhD

2 Prof Nguyen Buong, PhD

Learning Resource Center - TNU;

Library of the College of Education - TNU

Variational inequality theory was introduced by Hartman andStampacchia (1966) as a tool for the study of partial differential equa-tions with applications principally drawn from mechanics Such vari-ational inequalities were infinitely dimensional rather than finitely di-mensional The breakthrough in finite-dimensional theory occurred in

1980 when Dafermos recognized that the traffic network equilibriumconditions as stated by Smith (1979) had a structure of a variationalinequality This unveiled the methodology for the study of problems

in economics, management science or operations research, and also inengineering, with a focus on transportation

To-date problems which have been formulated and studied as ational inequality problems include: traffic network equilibrium prob-lems, spatial price equilibrium problems, oligopolistic market equilib-rium problems, financial equilibrium problems, migration equilibriumproblems, as well as environmental network problems, and knowledgenetwork problems Variational inequality theory provides us with atool for formulating a variety of equilibrium problems; it also allows toanalyze qualitatively the problems in terms of existence and unique-ness of solutions, stability and sensitivity analysis, and it finally pro-vide us with algorithms and their convergence analysis for computa-tional purposes It contains, as special cases, such well-known prob-lems in mathematical programming as systems of nonlinear equations,optimization problems, complementarity problems, and fixed pointproblems

vari-Because of the important role of variational inequalities in matical theory as well as in many practical applications, it has alwaysbeen a topical subject which attracts numerous researchers Manymathematical methods and numerical algorithms for solving varia-tional inequalities have been developed such as projection method

mathe-by Lions (1977), auxiliary principle problem mathe-by Cohen (1980), mal point method by Martinet (1970) and Rockafellar (1976); inertialproximal point method proposed by Alvarez and Attouch (2001), andBrowder–Tikhonov regularization method (Browder, 1966; Tikhonov,1963), etc In Vietnam, in recent years the variational inequality prob-lem has become an interesting and important topic for many groups

proxi-of mathematical researchers major in Mathematical Analysis and plied Mathematics To name a few groups with publications on varia-tional inequalities, we can cite: Buong and Thuy (Buong, 2012; Thuy,2015), Yen (Lee et al., 2005; Tam et al., 2005), Muu and Anh (Anh

Ap-et al., 2005, 2012), Sach (Tuan and Sach, 2004; Sach Ap-et al., 2008) andKhanh (Bao and Khanh, 2005, 2006), In addition, variational in-equalities and some related problems such as fixed points and equilib-rium problems have also been the topic of many young researchers andPhD students, for instance, Tuyen (2011, 2012), Duong (2011), Lang(2011, 2012), Duong (2011), Thong (2011), Phuong (2013), Thanh(2015), Khanh (2015) and Ha (2015), and others

Let H be a Hilbert space with inner product h., i Let C be anonempty closed and convex subset of H and let F : H → H be amapping The classical variational inequality, CVI(F, C) for short, isstated as follows:

Find an element x∗ ∈ C such that hF (x∗), x − x∗i ≥ 0, ∀x ∈ C

(0.1)

It has been known that the classical variational inequality CVI(F, C)

is equivalent to the fixed point equation

x∗ = PC(I − µF )(x∗), (0.2)where PC is the metric projection from H onto C, and µ > 0 anarbitrary constant When F is η-strongly monotone and L-Lipschitzcontinuous, the mapping PC(I − µF ) in the right hand side of (0.2)

is a contraction Hence, the Banach contraction mapping principleguarantees that the Picard iteration based on (0.2) converges strongly

to the unique solution of (0.1) Such a method is called the projectionmethod We remark that the fixed-point formulation (0.2) involvesthe projection PC, which may not be easy to compute due to the

complexity of the convex set C In order to reduce the complexityprobably caused by the projection PC , Yamada (2001) introduced ahybrid steepest descent method for solving variational inequality (0.1)

in a Hilbert space His idea is using a nonexpansive mapping T whosefixed point set is the feasible set C, that is C = Fix(T ), instead ofthe metric projection PC, and a sequence {xn} is generated by thefollowing algorithm:

xn+1 = T xn − µλn+1F (T xn), n ≥ 0, (0.3)with µ ∈ (0, 2η/L2) and {λn}n≥1 ⊂ (0, 1] satisfying some controlconditions

In this work, Yamada also considered the case when C : = ∩N

i=1Fix(Ti),the set of common fixed points of a finite family of nonexpansive map-pings (Ti)Ni=1, and proposed a cyclic iterative algorithm for solvingvariational inequality (0.1) over the feasible set C := ∩Ni=1Fix(Ti).The strong convergence of the method is proved under an additionalcondition, namely an invariance property of the set of common fixedpoints of combinations of nonexpansive mappings in the family Based

on hybrid steepest descent method by Yamada, many authors havebeen considering methods for solving variational inequality over thefeasible set C with more complicated structure such as the commonfixed point set of countably infinite family of nonexpansive mappings(Yao et al., 2010; Wang, 2011) or nonexpansive semigroups which isthe uncountably infinite family of nonexpansive mappings (Yang etal., 2012) These research works are important because they containmany applications arising from the theory of signal recovery problems,power control problems, bandwidth allocation problems and optimalcontrol problems In this thesis, we are interested in methods forsolving variational inequalities over the set of common fixed points

of nonexpansive semigroups {T (s) : s ≥ 0} This problem is linkedwith the evolution equation in the field of partial differential equations.Consider the differential equation dudt + Au(t) = 0 which describes anevolution system where A is an accretive map from a Banach space Einto itself In Hilbert spaces, accretive operators are called monotone

At equilibrium state, dudt = 0, and so a solution of Au = 0 describesthe equilibrium or stable state of the system This is very desirable

in many applications, for example, in ecology, economics, physics, toname a few Many studies showed that the solutions of an evolutionequation with a m-accretive mapping A : E → E in a Banach spaceconstitute a nonexpansive semigroup generated by operator A, andfurther, the set of common fixed points of {T (s) : s ≥ 0} is the set ofzero points of A, that is F := ∩s≥0Fix(T (s)) = A−1(0)

Along with the results achieved on different methods for solvingvariational inequality (0.1) in a Hilbert space H, many authors haverecently studied solution methods for variational inequalities in Ba-nach spaces We know that, among Banach spaces, Hilbert space H

is a space with very nice geometrical properties such as the ogram identity, or the existence of an inner product, the uniqueness

parallel-of the projection onto a nonempty, closed and convex subset parallel-of H,etc These properties make the study of the problem in Hilbert spacesmuch simpler than studying the problem in general Banach spaces

On the other hand, some methods for solving the problem converges

in a Hilbert space but not necessarily in a general Banach space Thisexplains an important number of research works on extensions andgeneralizations recently appeared in the literature in the framework ofBanach spaces For some recent published results on solution methodsfor variational inequalities in Banach spaces, one needs to assume, inorder to ensure their strong convergence, the weakly continuity of thenormalized duality mapping Until now it has been shown that the

lp, 1 < p < ∞, satisfies this weakly continuity property while the

Lp[a, b], 1 < p < ∞, does not A natural question arising here iswhether it is possible to develop methods for solving variational in-equalities in Banach spaces without requiring the weakly continuity ofthe normalized duality mapping If the answer is affirmative, then thescope of applications of the algorithms in question can be expandedtowards more general Banach spaces such as Lp[a, b], rather applicableonly for lp

Another aspect of variational inequalities is that it is an ill-posedproblem To solve the class of these problems, we have to use sta-ble methods, the so-called regularization methods In practice, theinput data are usually collected by observations or direct measure-

ments This means that there are errors on the input data, andthe results received from the problem will not reliable enough; so

it can lead to a wrong decision based on what we have considered

as the solutions of the problem These known facts yielded manyinteresting research publications for ill-posed problems including vari-ational inequalities based on the Browder–Tikhonov regularization In

2012, Buong and Phuong proposed a Browder–Tikhonov tion method for problem of accretive variational inequalities over theset of common fixed points of countably infinite family of nonexpan-sive mappings {Ti}∞

regulariza-i=1 in Banach spaces E using V -mapping as animprovement of W -mapping in some results of other authors

Therefore, we can say that the variational inequality problem tracted numerous mathematicians, not only in Vietnam but also inthe international community of researchers, to develop effective so-lution methods for solving this problem The investigation of theproblem in the framework of Banach spaces is a natural and neces-sary research topic to understand the problem in infinite dimension.For these reasons we chose a subject for this dissertation whose title is

at-“Iterative methods for variational inequalities over the set of mon fixed points of nonexpansive semigroups on Banach spaces”.The main goal of this thesis is to study hybrid steepest methods andregularization methods for solving variational inequalities over the set

com-of common fixed points com-of nonexpansive semigroups in Banach spaces.Specifically, the dissertation will address the following issues:

1 Devise implicit iterations based on hybrid steepest descent ods for accretive variational inequalities in uniformly convex Banachspaces without the use of sequentially weakly continuity property ofthe normalized duality mapping of Banach spaces

meth-2 Propose and analyze the corresponding explicit iterations of theseimplicit iterative methods for the same problem

3 Suggest Browder–Tikhonov regularization methods for accretivevariational inequalities and combine with inertial proximal point method

to construct inertial proximal point regularization method for tional inequalities in uniformly convex and smooth Banach spaces;present another combination of the Browder–Tikhonov regularization

varia-method with an explicit algorithm for variational inequalities in formly convex and q-uniformly smooth Banach spaces.

uni-Besides the introduction, conclusion and references, the contents

of the dissertation are presented in three chapters In Chapter 1,

we present some important preliminaries to prepare the presentation

of the main results in the next chapters, specifically as some metrical characteristics of Banach spaces, monotone type mappings,Lipschitz continuous mappings and variational inequalities in Banachspaces, like classical variational inequalities and some related prob-lems, monotone variational inequalities and accretive variational in-equalities In Chapter 2, we introduce and analyze implicit iterativemethods for accretive variational inequalities based on hybrid steep-est descent methods in uniformly convex Banach spaces whose norm

geo-is uniformly Gˆateaux differentiable Also in this chapter we give theexplicit versions of the corresponding implicit iterations for the sameproblem In Chapter 3, we combine the Browder–Tikhonov regular-ization method with the inertial proximal point method to obtain theinertial proximal point regularization method for variational inequal-ities We also combine the Browder–Tikhonov regularization methodwith an explicit iterative technique to devise an iterative regulariza-tion method for variational inequalities in uniformly smooth Banachspaces We finally present some numerical results to illustrate theproposed methods at the end of Chapter 2 and Chapter 3

Chapter 1

Preliminaries

Chapter 1 of the dissertation is devoted to introduce some basicpreliminaries serving for the presentation of research results in thenext chapters Specifically, this chapter consists of 4 sections:

Section 1.1 is set up for the presentation of some geometrical teristics of Banach spaces, definitions and some properties of monotoneand accretive mappings, and Lipschitz continuous mapping

charac-In Section 1.2 we introduce nonexpansive semigroups and an cation of nonexpansiveness for the Cauchy problem

appli-In Section 1.3, we give the statement of the problem of classicalvariational inequalities and some related problems such as system ofequations, complementarity problem, optimization problem and fixedpoint problem

In Section 1.4 we describe the problem of monotone and accretiveinequalities in general Banach spaces Also in this section we presentthe hybrid steepest descent method proposed by Yamada for solving avariational inequality over the set of common fixed points of a family

of nonexpansive mappings

Section 1.5 gives the statement of the problem of accretive tional inequalities over the feasible set that is the set of common fixedpoints of nonexpansive semigroups in Banach spaces This problem isdenoted VI∗(F, F ) which will be considered throughout this disserta-tion

varia-Let F : E → E be an η-accretive and γ-pseudocontractive mappingwith η + γ > 1 Let {T (t) : t ≥ 0} be a nonexpansive semigroup

on E such that F := ∩s≥0Fix(T (s)) 6= ∅, where F denotes the set ofcommon fixed points of the nonexpansive semigroup {T (t) : t ≥ 0}

We consider the problem:

Find p∗ ∈ F such that hF p∗, j(x − p∗)i ≥ 0 ∀x ∈ F (1.1)

Proposition 1.1 Let E be a real uniformly convex Banach spacewith a uniformly Gˆateaux differentiable norm Let F : E → E

be an η-strongly accretive and γ-pseudocontractive mapping with

η, γ ∈ (0, 1) satisfying η + γ > 1 Let {T (s) : s ≥ 0} be a pansive semigroup on E such that F := ∩s≥0Fix(T (s)) 6= ∅ Then,the problem (1.1) has one and only one solution p∗ ∈ F

nonex-In the next chapters we will propose some methods for solving cretive variational inequalities based on hybrid steepest descent ap-proach in uniformly convex Banach spaces having Gˆateaux differen-tiable norm

ac-Chapter 2

Hybrid Steepest Descent Methods for Variational Inequalities over the Set of Common Fixed Points of

Nonexpansive Semigroups

This chapter consists of three sections In Section 2.1, we pose three implicit iterative schemes based on hybrid steepest descentmethod for variational inequalities VI∗(F, F ) and in Section 2.2 wegive the explicit versions of the methods studied in Section 2.1 A nu-merical example illustrating the proposed methods is presented anddiscussed in Section 2.3 Results of this chapter is taken from thearticles (1) and (2) of the list of research papers published related tothe dissertation

pro-2.1 Implicit Hybrid Steepest Descent Methods

2.1.1 State the Method

In this section we propose three implicit iterative methods based

on the hybrid steepest descent method by Yamada for variational equalities (1.1) in uniformly convex Banach spaces having uniformly

in-Gˆateaux differentiable norm The first method is a convex nation of two mappings Fk and Tk defined, respectively, by Fkx =(I − λkF )x and Tkx = t1

combi-k

R tk

0 T (s)xds, x ∈ E

Method 2.1 Start from an arbitrary point x1 ∈ E, define {xk}

by the following equation:

xk = γkFkxk + (1 − γk)Tkxk, k ≥ 1, (2.1)where γk ∈ (0, 1), λk ∈ (0, 1] and tk > 0 satisfy that λk → 0, tk →

∞ as k → ∞

In the second methods, we do not use Bochner integral Tk butnonexpansive mapping T (tk) instead.

Method 2.2 Start from an arbitrary point x1 ∈ E, define {yk}

by the following equation:

yk = γkFkyk + (1 − γk)T (tk)yk, k ≥ 1, (2.2)where λk ∈ (0, 1], γk ∈ (0, 1) and tk > 0 satisfy that limk→∞tk =limk→∞ γk

tk = 0

One might see that the structure of the two implicit iterative ods (2.1) and (2.2) is similar to each other but in the method (2.2),using direct mappings T (tk) with tk → 0, k → ∞ without usingBochner integral, the method (2.2) is considered simpler to implementthan the method (2.1) With the third method, by taking the com-posite of two mappings Tk and Fk, we construct an iterative sequenceimplicitly for variational inequalities VI∗(F, F ) as follows

meth-Method 2.3 Start from an arbitrary point x1 ∈ E, define {wk}

by the following equation:

wk = TkFkwk, k ≥ 1, (2.3)where λk ∈ (0, 1] and tk > 0 such that λk → 0 and tk → ∞, as

k → ∞

2.1.2 The Strong Convergence

Theorem 2.1 Let E be a real uniformly convex Banach spacewith a uniformly Gˆateaux differentiable norm Let F : E → E

be an η-strongly accretive and γ-pseudocontractive mapping with

η, γ ∈ (0, 1) satisfying η + γ > 1 Let {T (s) : s ≥ 0} be a pansive semigroup on E such that F := ∩s≥0Fix(T (s)) 6= ∅ Then,sequence {xk} defined by (2.1) converges strongly to p∗, the uniquesolution of variational inequality (1.1) as k → ∞

nonex-Theorem 2.2 Let E, F , {T (s) : s ≥ 0} and F be as in nonex-Theorem2.1 Then, sequence {yk} defined by (2.2) converges strongly to

p∗, the unique solution of variational inequality (1.1) as k → ∞

Theorem 2.3 Let E, F , {T (s) : s ≥ 0} and F be as in Theorem2.1 Then, sequence {wk} defined by (2.3) converges strongly to

p∗, the unique solution of variational inequality (1.1) as k → ∞.Remark 2.1

(a) The proofs of convergence of the method (2.1) in Theorem 2.1, ofthe method (2.2) in Theorem 2.2 and of the method (2.3) in Theorem2.3 do not require weakly continuity property of the normalized dualitymapping of Banach spaces E

(b) When C = F := ∩∞i=1Fix(Ti) is the set of common fixed points ofcountably infinite family of nonexpansive mappings, in 2013, Buongand Phuong proposed two implicit methods for solving (1.1) in a realuniformly convex Banach space with a uniformly Gˆateaux differen-tiable norm The first method has the same structure as (2.1) whilethe mapping Tk of (2.1) is replaced by Vk mapping

(c) For some research results on the implicit iterative methods for thevariational inequalities over the set of common fixed points of a family

of nonexpansive mappings, we would like to mention those of Ceng et

al (2008), Chen and He (2007), Shioji and Takahashi (1998), Suzuki(2003), and Xu (2005) Ceng et al (2008) also used Banach limit toprove the strong convergence of their methods

2.2 Explicit Hybrid Steepest Descent Methods

2.2.1 State the Method

When constructing implicit iterative schemes in Section 2.2, a sible difficulty encountered by those methods in practice is the cal-culation of xk at each iteration k Indeed, we have to solve at eachstep an equation to find approximately xk, and after a finite number

pos-of iterations we hope to obtain xk closed to the exact solution of theinterested problem Stemming from the idea to overcome this issue

of implicit iterative methods, we devise two explicit iterative methodsbased on (2.1) and (2.3)

Method 2.4 Start from an initial guess x1 ∈ E arbitrarily, we