júlia salamon: parametric vector equilibrium problems

S C I E N T I A

Publishing HouseC ·luj-Napoca 2011

PARAMETRIC VECTOREQUILIBRIUM PROBLEMS

JÚLIA SALAMON

JÚLIA SALAMON


SAPIENTIA

ERDÉLYI MAGYAR

TUDOMÁNYEGYETEM

SAPIENTIA

ALAPÍTVÁNY

PARTIUMIKERESZTÉNY

EGYETEM

S A P I E N T I A B O O K S

JÚLIA SALAMON


S C I E N T I APublishing HouseCluj-Napoca·2011

SAPIENTIA BOOKS 72.Natural Sciences

This book was published with the support of the Sapientia Foundation.

Published byScientia Publishing House400112 Cluj-Napoca, str. Matei Corvin 4.Tel./Fax: +40-264-593694E-Mail: [email protected]

Publisher in Chief:Zoltán Kása

The present volume is the PhD dissertation of the author who takesprofessional responsibility for it.

Members of the PhD commission:Dr. József Kolumbán (thesis supervisor)Dr. Petru JebeleanDr. Adrian PetruşelDr. Daniela InoanDr. Wolfgang Breckner

First English edition: 2011©Scientia, 2011All rights reserved, including the rights for photocopying, public lecturing, radioand television broadcast and translation of the whole work and of chapters aswell.

Descrierea CIP a Bibliotecii Naţionale a RomânieiJÚLIA SALAMON

Parametric vector equilibrium problems / Júlia Salamon. – Cluj-Napoca:Scientia, 2011Bibliogr.ISBN 978-973-1970-46-2

571

CONTENTS

Preface 11

1 Preliminaries 17

1.1 Cones 17

1.2 Limit points of the nets 18

1.3 Vector topological pseudomonotonicity 22

1.4 Mosco convergence 31

1.5 Hadamard well-posedness 32

1.6 Existence results 33

1.6.1 Equilibrium problems 33

1.6.2 Weak vector equilibrium problems 33

1.6.3 Vector equilibrium problems 34

1.6.4 Strong vector equilibrium problems 36

1.6.5 Generalized vector equilibrium problems with trifunctions 36

1.7 Parametric equilibrium problems 38

2 Sensitivity analysis of vector equilibrium problems 42

2.1 Semicontinuity of the solution mapping 42

2.1.1 Upper semicontinuity of the solution mapping 44

2.1.2 Lower semicontinuity and continuity of solution mapping 47

2.2 Hölder continuity of the solution mapping 55

3 Closedness of the solution mapping 60

3.1 Parametric weak vector equilibrium problems 62

3.2 Parametric vector equilibrium problems 75

3.3 Parametric strong vector equilibrium problems 79

3.4 Parametric generalized vector equilibrium 88

3.4.1 Strong solutions 88

3.4.2 Weak solutions 96

6 CONTENTS

4 Parametric operator equilibrium problems 103

4.1 Existence results 103

4.2 Closedness of the solution mapping 105

5 Applications 110

5.1 Parametric vector optimization problems 110

5.2 Parametric vector variational inequalities 112

5.3 Parametric vector complementarity problems 117

Bibliography 124

Index 139

Kivonat 141

Rezumat 142

About the author 143

TARTALOM

Előszó 11

1. Bevezető fogalmak 17

1.1. Kúpok 17

1.2. A hálók határértékpontjai 18

1.3. Vektor topológikus pszeudomonotonitás 22

1.4. Mosco konvergencia 31

1.5. Hadamard jól értelmezettség (well-posedness) 32

1.6. Létezési tételek 33

1.6.1. Egyensúlyi feladatok 33

1.6.2. Gyenge vektor egyensúlyi feladatok 33

1.6.3. Vektor egyensúlyi feladatok 34

1.6.4. Erős vektor egyensúlyi feladatok 36

1.6.5. Vektor egyensúlyi feladatok háromváltozós függvényekkel 36

1.7. Paraméteres egyensúlyi feladatok 38

2. Vektor egyensúlyi feladatok érzékenység vizsgálata 42

2.1. Megoldás függvény féligfolytonossága 42

2.1.1. Megoldás függvény felülről féligfolytonossága 44

2.1.2. Megoldás függvény alulról féligfolytonossága 47

2.2. Megoldás függvény Hölder folytonossága 55

3. Megoldás függvény grafikonjának zártsága 60

3.1. Paraméteres gyenge vektor egyensúlyi feladatok 62

3.2. Paraméteres vektor egyensúlyi feladatok 75

3.3. Paraméteres erős vektor egyensúlyi feladatok 79

3.4. Paraméteres vektor egyensúlyi feladatok háromváltozósfüggvényekkel 88

3.4.1. Erős megoldások 88

3.4.2. Gyenge megoldások 96

8 TARTALOM

4. Paraméteres operátor egyensúlyi feladatok 103

4.1. Létezési tételek 103

4.2. Megoldás függvény grafikonjának zártsága 105

5. Alkalmazások 110

5.1. Paraméteres vektor optimalizációs feladatok 110

5.2. Paraméteres vektor variációs feladatok 112

5.3. Paraméteres vektor komplementaritási feladatok 117

Szakirodalom 124

Tárgymutató 139

Kivonat 141

A szerzőről 143

CUPRINS

Prefaţă 11

1. Noţiuni introductive 17

1.1. Conuri 17

1.2. Punctele de limită ale şirurilor generalizate 18

1.3. Pseudomonotonie vectorială topologică 22

1.4. Convergenţa Mosco 31

1.5. Proprietatea Hadamard bine pusă (well-posed) 32

1.6. Rezultate de existenţă 33

1.6.1. Probleme de echilibru 33

1.6.2. Probleme slabe de echilibru vectorial 33

1.6.3. Probleme de echilibru vectorial 34

1.6.4. Probleme tari de echilibru vectorial 36

1.6.5. Probleme de echilibru vectorial cu trifuncţii 36

1.7. Probleme parametrice de echilibru 38

2. Analiză de senzitivitate a problemelor de echilibru vectorial 42

2.1. Semicontinuitatea funcţiei soluţie 42

2.1.1. Superior semicontinuitatea funcţiei soluţie 44

2.1.2. Inferior semicontinuitatea funcţiei soluţie 47

2.2. Continuitatea Hölder a funcţiei soluţie 55

3. Închiderea graficului funcţiei soluţie 60

3.1. Probleme parametrice slabe de echilibru vectorial 62

3.2. Probleme parametrice de echilibru vectorial 75

3.3. Probleme parametrice tari de echilibru vectorial 79

3.4. Probleme parametrice de echilibru vectorial cu trifuncţii 88

3.4.1. Soluţii tari 88

3.4.2. Soluţii slabe 96

10 CUPRINS

4. Probleme parametrice de echilibru cu operatori 103

4.1. Rezultate de existenţă 103

4.2. Închiderea graficului funcţiei soluţie 105

5. Aplicaţii 110

5.1. Probleme parametrice de optimizare vectorială 110

5.2. Inegalităţi variaţionale vectoriale parametrice 112

5.3. Probleme parametrice de complementaritate vectorială 117

Bibliografie 124

Index 139

Rezumat 142

Despre autor 143

PREFACE

The equilibrium theory provides a unified, natural, innovative and generalframework for the study of a large variety of problems such as optimization prob-lems, fixed points problems, variational inequalities, Nash equilibria, saddlepoint problems and complementarity problems as special cases. The problemsmentioned above often occur in mechanics, physics, finance, economics, net-work analysis, transportation and elasticity.

The behavior of the solutions resulting by the change of the problems datais always of major concern.

Sensitivity analysis examine the way how the solutions of such problemschange when the data of the problems are modified. The reason why sensitivityanalysis is important, is that the estimating problem data often introduces mea-surement errors and sensitivity results help in identifying sensitive parametersthat have to be obtained with relatively high accuracy.

Over the last decade, there has been increasing interest in studying the sen-sitivity analysis of variational inequalities, which has been studied by Tobin[179], Kyparisis [110], Dafermos [54], Qiu and Magnanti [154], Yen [188] andLiu [135], using quite different techniques. The techniques suggested so far varywith the problem studied.

Many authors used the projection method to study the continuity or Lip-schitz continuity of the solution set of variational inequalities in Euclideanspaces or Hilbert spaces (see Dafermos [54], Mukherjee-Verma [145] and Yen[187],[186]). Robinson [157] used the so-called normal mappings and an im-plicit function approach to deal with the solution sensitivity of variational in-equalities satisfying some smoothness assumptions. Noor [146], Domokos [57],Ding and Luo [55] and Kassay and Kolumbán [94] considered the continuityor Lipschitz continuity of the solution set of variational or quasivariational in-equality problems in infinite dimensional settings.

The parametric equilibrium problems, what we already mentioned, we canformulate in the following way:

Let X be a Hausdorff topological space and P, the set of parameters, an-other Hausdorff topological space. For a given p ∈ P we consider the followingequilibrium problem:

(EP )p Find an element xp ∈ Dp such that

fp (xp, y) ≥ 0, ∀y ∈ Dp,

where Dp is a nonempty subset of X and fp : X ×X → R is a given function.Let S : P → 2X be the solution mapping, where S (p) denotes the set of

solutions for a fixed p.

12 PREFACE

The sensitivity or Hölder continuity of the solution mapping of parametricequilibrium problem presented above was studied by Anh and Khanh [4, 5],Bianchi and Pini [27, 29], Mansour and Riahi [139]. Bogdan and Kolumbán [35]gave sufficient conditions for closedness of the solution map defined on the setof parameters. The semicontinuity of the solution multifunction of the paramet-ric operator equilibrium problems was analyzed by Kim, Kum and Lee [100].

Most of decision situations require simultaneous consideration of morethan one objective, which are often in conflict. The variables that optimize oneobjective may be far from optimal for the others. In practice, multiobjective op-timization and - later - vector variational inequalities have been the appropriatemethodologies for solving this conflict. Natural extensions to this case also in-clude vector complementarity problems, vector best approximations, and conesaddle point theorems, for example the Nash equilibrium problems. Nash equi-librium is a solution concept of a game involving two or more players, in whicheach player is assumed to know the equilibrium strategies of the other players,and none of the players has anything to gain by unilaterally changing his orher strategy. If we assume that the players communicate, exchange informationand cooperate, then it is possible to obtain better results for all the participants.Optimal strategies can be received by changing from one allocation to anotherthat can make at least one player better off without making any other playerworse off. An allocation is Pareto optimal when no further improvements canbe made.

The extension of the scalar parametric equilibrium problems to vector para-metric equilibrium problems can be achieved in different ways. The problemsunder consideration are the following: Let X be a Hausdorff topological spaceand let P the set of parameters, be another Hausdorff topological space. LetZ bea real topological vector space and C be a cone in Z with C 6= Z and IntC 6= ∅,where IntC denotes the interior of C.

We consider the following three parametric vector equilibrium problems:Find xp ∈ Dp such that

(PV EP )1 fp (xp, y) ∈ (− IntC)c, ∀y ∈ Dp,

(PV EP )2 fp (xp, y) /∈ −C\ {0} , ∀y ∈ Dp,(PV EP )3 fp (xp, y) ∈ C, ∀y ∈ Dp,

where Dp is a nonempty subset of X and fp : X ×X → Z is a given function.The first problem is also called weak parametric vector equilibrium prob-

lem [80], while the third one is known as parametric strong vector equilibriumproblem [8].

If Z = R and C = R+, then (PV EP )i for i = 1, 3 collapse to the parametricequilibrium problem (EP )p.

Let us denote by Si (p) the set of the solutions for (PV EP )i i = 1, 3 for afixed p. If C is a cone, then the relationship between the problems mentionedabove can be formulated as follows [90]:

PREFACE 13

– S2(p) ⊆ S1(p), for every p ∈ P.– If C is pointed, i.e. C ∩ − (C\ {0}) = ∅, then S3 (p) ⊆ S2(p), for everyp ∈ P.

– If C is w-pointed, i.e. C ∩ − IntC = ∅, then S3 (p) ⊆ S1(p), for everyp ∈ P.

– If C is connected, i.e. C ∪ − IntC = Z, then S1 (p) ⊆ S3(p), for everyp ∈ P.

If C is a closed convex cone in Z with C 6= Z and IntC 6= ∅, then we havethe following relations:

S2(p) ⊆ S1(p), for every p ∈ P ;

S3(p) ⊆ S1(p), for every p ∈ P.

In the following, we always suppose that C is a closed convex cone.Another type of problem which is also investigated is the generalized para-

metric vector equilibrium problem with trifunctions [49].Let X,Y be Hausdorff topological spaces and let K ⊂ X and D ⊂ Y be

nonempty subsets.These vector equilibrium problems (for short GV EP ) are obtained by con-

sidering trifunctions fromK×D×K into a real topological vector space Z withan ordering cone. By an ordering coneC ⊂ Z we mean thatC is a closed convexcone in Z with IntC 6= ∅ and C 6= Z.

Let Kp be a nonempty subset of X. Let fp : X×Y ×X → Z be a trifunctionand T : X → 2Y be a set-valued mapping. The following two problems areconsidered:

find a pair (xp, yp) ∈ Kp × Y such that yp ∈ T (xp) and

fp (xp, yp, u) ∈ (− IntC)c for all u ∈ Kp;

find xp ∈ Kp such that for every u ∈ Kp there exists yu ∈ T (xp) satisfying

fp (xp, yp, u) ∈ (− IntC)c.

The solution of the first problem is called strong solution of the problem(GV EP )p in the sense that yp does not depend on u ∈ Kp.

The solutions of the second problem are called weak solutions of the prob-lem (GV EP )p.

Denote by Ss (p) and Sw (p) the set of the strong solutions and weak solu-tions for a fixed p, respectively. It is clear that

Ss(p) ⊆ Sw(p), for every p ∈ P.

In a separate chapter, we studied the parametric operator equilibrium prob-lems.

14 PREFACE

In 2002, Domokos and Kolumbán [58] introduced and studied a class of op-erator variational inequalities. These operator variational inequalities includenot only scalar and vector variational inequalities as special cases, but also havesufficient evidence for their importance to study. Inspired by their work, Kumand Kim [107, 108] developed the scheme of operator variational inequalitiesfrom the single-valued case into the multi-valued one. The operator equilibriumproblems was studied by Kazmi and Raouf [96], Kum and Kim [109].

The problem under consideration is the following:LetX and Y be Hausdorff topological vector spaces, L(X,Y ) be the space of

all continuous linear operators fromX to Y and letD ⊂ L(X,Y ) be a nonemptyset. Let P the set of parameters be another Hausdorff topological space. LetC : D → 2Y be a set-valued mapping such that for each f ∈ D, C(f) is a convexopen cone with nonempty interior and C (f) 6= Y .

For a given p ∈ P the parametric operator equilibrium problem (OEP )p isto find fp ∈ D such that

Fp(fp, g) /∈ −C(fp),∀g ∈ D,

where Fp : D ×D → Y is a given function.For the sake of convenience, let us develop the reasons why the study of

vector equilibrium problems is important from a theoretical point of view. Wesingle out just some special cases.

Nash equilibria: Let (D, f) be a game with two players, where D1 and D2

are the strategy sets for the players, D = D1 ×D2 is the set of strategy profilesand F1, F2 : D1×D2 → R are the payoff functions of the two players. A strategyprofile (x1, x2) ∈ D is a Nash equilibrium if no unilateral deviation in strategyby any single player is profitable for that player, that is

F1 (y1, x2) ≥ F1 (x1, x2) for every y1 ∈ D1;

F2 (x1, y2) ≥ F2 (x1, x2) for every y2 ∈ D2.

If we set f (x, y) = (f1 (x, y) , f2 (x, y)) where

f1 (x, y) = F1 (y1, x2)− F1 (x1, x2)

f2 (x, y) = F2 (x1, y2)− F2 (x1, x2)

for all x = (x1, x2) , y = (y1, y2) ∈ D and C = R2+, then the Nash equilibrium

problem coincides with (PV EP )3.Vector optimization: Let ϕ : D → Z be a given vector valued function.

(WVOP ) Find x ∈ D such that ϕ (y)− ϕ (x) /∈ − IntC for all y ∈ D.

If we set f (x, y) = ϕ (y) − ϕ (x) for all x, y ∈ D, then (WVOP ) coincideswith (PV EP )1. Note that if X = Rm, Z = Rn and C = Rn+ for all x ∈ D,

PREFACE 15

then solutions to (WVOP ) are known as weak Pareto (or efficient) solutions.Roughly speaking, (WVOP ) appears when one attempts to make decisions byoptimizing a multiobjective.

In the literature, some properties of the solution map defined on the set ofparameters for a (PV EP ) have been investigated. Anh and Khanh [3], Huang,Li, and Thompson [85], Long, Huang and Teo [137] studied the stability forset-valued vector quasiequilibrium problems or parametric implicit vector equi-librium problems. Bianchi and Pini [30], Kimura and Yao [101], Sach, Tuanand Lee [161] studied the sensitivity for parametric vector equilibria. Anh andKhanh [2], Li, Li, Wang and Teo [119], Chen, Li and Teo [42], Gong [80], Xu andLi [182] considered the continuity or Hölder continuity of the solution map-pings for parametric vector equilibrium problems. These results are detailed inchapter 2.

The main purpose of this book is the extension of the closedness resultsof scalar parametric equilibrium problems obtained in [35] to parametric vectorequilibrium problems and to parametric operator equilibrium problems.

The book includes five chapters.The notions and auxiliary results necessary for studying the properties of

the solution mapping for parametric vector equilibrium problems are intro-duced in Chapter 1. Section 1.1 contains some properties of the ordering cones.In section 1.2 the definitions of the lower and upper limits of the nets are re-called. In Section 1.3 generalizations of the topological pseudomonotonicity,notion introduced by Brézis in [37], are discussed. Each type of parametricvector equilibrium problem presented above requires a new notion of vectortopological pseudomonotonicity. Another basic notion referred to is a slightgeneralization of the Mosco convergency, it is presented in section 1.4. Sec-tion 1.5 focuses on the relations between closedness, upper semicontinuity andHadamard well-posedness of the parametric vector equilibrium problems. Inorder to obtain closedness results, it is necessary to exist solutions at eachparameter. The section 1.6 presents existence results for every previously men-tioned vector equilibrium problems. Section 1.7 reviews the starting point ofthis book, the closedness results of the solution map for scalar parametric equi-librium problems presented in [35].

Chapter 2 is devoted to the presentation of some stability results for para-metric vector equilibrium problems. These results target only the weak paramet-ric vector equilibria. Generally speaking, the notion of stability or sensitivity ofthe solution mapping refers to upper or lower semicontinuity. Section 2.1 anal-yses the semicontinuity of the solution mapping by invoking Huang, Li andThompson [85], Chen, Li and Teo [42], Anh and Khanh [3], Kimura and Yao[101], Gong [80]. Section 2.2 studies the Hölder continuity of the solution map-ping for weak parametric vector equilibrium problems, based on Li, Li, Wangand Teo [119], Anh and Khanh [2], Bednarczuk [24] results.

Another regularity property, the closedness of the solution mapping forparametric vector equilibrium problems, is studied in chapter 3. Each section

16 PREFACE

focuses on a different parametric vector equilibrium problem. In cases of thesolution mapping for weak and strong parametric vector equilibrium problemsand strong solution mapping for generalized parametric vector equilibriumproblems two closedness theorems are presented. From them, only the secondtheorems generalize the results in [35]. Here also the Hadamard well-posednessof the mentioned problems is analyzed. Two closedness results are given forthe weak solution mapping of the parametric vector equilibrium problemswith trifunctions. An extension of the closedness results is presented for theproblems (PV EP )2 in section 3.2.

Chapter 4 is devoted to the study of the parametric operator equilibriumproblems, which it is a two-way generalization of (PV EP )1. In section 4.1some known existence results are presented. In section 4.2 the closedness of thesolution mapping for parametric operator equilibrium problems are analyzed.

In chapter 5 the applications of the closedness results obtained in chap-ter 3 are treated. Parametric vector optimization problems, parametric vectorvariational inequalities and parametric vector complementarity problems arestudied. Section 5.1 presents the closedness results of weak efficient and socalled absolute solution mapping for vector optimization problems. Even in thecase of parametric vector variational inequalities we can formulate three type ofproblems similar to those described in the case of parametric vector equilibriumproblems. In section 5.2 the closedness of solution mapping for these type ofproblems are discussed. In section 5.3 the parametric vector complementarityproblems are presented.

The author’s original contributions are the following:– Definitions 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 4.5 where different

types of vector topological pseudomonotonicity notions are introduced;– Examples 1.2, 1.3, 1.4 and Proposition 4.5 where relations between dif-

ferent types of vector topological pesudomonotonicities are presented;– Theorems 3.2, 3.3, 3.9, 3.11, 3.12, 3.18, 3.19, 3.27, 3.28, 4.6 where

closedness results for different type of parametric vector equilibriumproblems are studied;

– Corollaries 3.7, 3.16, 3.17, 3.23, 3.24, 4.8 where Hadamard well-posedness of the parametric vector equilibrium problems are analyzed;

– some applications of the closedness results for parametric vector opti-mization problems, parametric vector variational inequality and para-metric vector complementarity problems are presented in chapter 5.

The elaboration of this book is based on my Ph.D. thesis, that was possibledue to the guidance of my scientific advisor Prof. Dr. Kolumbán József. I am verygrateful to Dr. Bogdan Marcel, who provided me useful suggestions. I also wishto thank the official referees of my Ph.D. thesis, Prof. Dr. Jebelean Petru, Prof.Dr. Petruşel Adrian, Conf. Dr. Inoan Daniela, for their valuable suggestions forimproving the manuscript. Finally, I want to thank my family for the supportduring my studies.

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INDEX

C

C-property, 49Closedness, 32, 38, 43, 60, 104, 105Cone, 17Convexity

C-convex, 35, 36, 44, 52C-quasiconcave like, 36C-quasiconvex, 50C-quasiconvex-like, 34C-weakly quasiconcave, 50concave, 49convex, 49, 104H-convex, 104natural quasi H-convex, 104vector 0-diagonally convex, 33, 36

H

Hölder continuity, 55h.β-Hölder-strongly

pseudomonotone, 56Hölder of order m, 58l.α-Hölder, 56lower Hölder of order m, 58upper Hölder of order m, 58

Hadamard well-posedness, 32, 41, 75,87, 96, 109

Hemicontinuity, 104

L

Limit points, 18limit inferior, 19limit superior, 19

M

Monotonity

C-strictly monotone, 52monotone, 35quasimonotone, 56

Mosco convergence, 31

P

PseudomonotonicityB-C(f)-pseudomonotone, 105C(f)-pseudomonotone, 103

S

SemicontinuityC-lower semicontinuity, 35C-upper semicontinuity, 25lower semicontinuity, 36, 42, 47termed Hausdorff upper

semicontinuous, 43upper semicontinuity, 22, 32, 42, 44

T

Topological pseudomonotonicity, 22A1-vector topological

pseudomonotonicity, 23A2-vector topological

pseudomonotonicity, 27a2-vector topologically

pseudomonotone, 114A3-vector topological

pseudomonotonicity, 27a3-vector topologically

pseudomonotone, 115B1-vector topological

pseudomonotonicity, 23B3-vector topological

pseudomonotonicity, 27

140 INDEX

b3-vector topologicallypseudomonotone, 116

class (GPM), 30class (GPM2), 30class (SPM), 29class (SPM2), 29topological pseudomonotonicity, 22vector topological

h-pseudomonotonicity, 112vector topological

h-pseudomonotonicity of type II,113

vector topologicalpseudomonotonicity, 23, 105

KIVONAT

Az egyensúlyi feladatok több feladat típust foglalnak magukban, mint avariációs egyenlőtlenségeket, komplementaritási feladatokat, optimalizálási fel-adatokat, fixpontok elméletét, Nash egyensúlyi feladatokat. Ezen feladatoknaknagyon sok hasznos alkalmazásai vannak különböző területeken, és matemati-kailag modelleznek jelenségeket közgazdaságtanból, pénzügyből, logisztikából,optimalizálásból, mechanikából illetve mérnöki tudományokból.

A könyv elsődleges célja a paraméteres egyensúlyi feladatok zártságra vo-natkozó eredményeinek általánosítása paraméteres vektor egyensúlyi illetve pa-raméteres operátor egyensúlyi feladatokra.

A könyv öt fejezetet, egy könyvészetet és egy tárgymutatót foglal magába.Az első fejezetben azon bevezető fogalmak és eredmények kerülnek bemu-

tatásra, amelyek a paraméteres vektor egyensúlyi feladatok tanulmányozásánaka kiindulási pontját képezik. Itt találhatók meg a kúpok és vektoriális terekenértelmezett határérték pontok értelmezései és tulajdonságai. Az 1.3 alfejezet-ben több vektor topológikus pszeudomonotonitási fogalom van bevezetve. AMosco konvergencia és a Hadamard jólértelmezettségi (well-posedness) fogal-mak az 1.4 és 1.5 alfejezetekben találhatók meg. Az 1.6 alfejezetben a létezésitételek vannak bemutatva, míg az 1.7 alfejezet Bogdan és Kolumbán által meg-adott zártsági eredményeket tartalmazza valós egyensúlyi feladatokra.

A második fejezetben a vektor egyensúlyi feladatok érzékenységi eredmé-nyei vannak ismertetve. A megoldás függvény féligfolytonosságára vonatkozóeredmények a 2.1 alfejezetben találhatók meg, míg a 2.2 alfejezet a Hölder foly-tonosságra vonatkozó eredményeket tárgyalja.

A szerző legjelentősebb eredményeit a harmadik fejezet foglalja magában.Ezen fejezet, a vektor egyensúlyi feladatok paramétertől függő halmazértékűmegoldás függvényének gyenge zártsági tulajdonságának van szentelve. Egyesesetekben a feladatok Hadamard jólértelmezettsége (well-posedness) is elem-zésre kerül.

A negyedik fejezet a paraméteres operátor egyensúlyi feladatok vizsgála-tának van szentelve. A 4.1 alfejezetben a már ismert létezési tételek vannakbemutatva, míg a 4.2 alfejezetben a megoldás függvény zártsági tulajdonságavan vizsgálva.

A harmadik fejezetben bemutatott zártsági eredmények alkalmazásait azötödik fejezet tartalmazza. A vizsgált feladatok a következők: paraméteres vek-tor optimalizálási feladatok, paraméteres vektor variációs egyenlőtlenségek, pa-raméteres vektor komplementaritási feladatok.

A könyvet egyaránt használhatja a téma iránt érdeklődő, az eredményeitfelhasználni vágyó kutató, doktorandus és magiszteri hallgató.

REZUMAT

Problemele de echilibru sunt probleme generale, care la rândul lor, cu-prind multe tipuri de probleme cum ar fi: inegalităţi variaţionale, problemede complementaritate, probleme de optimizare, teoria punctelor fixe, problemede echilibru Nash. Acestea au o mulţime de aplicaţii importante în diferitedomenii şi modelează matematic fenomene din economie, finanţe, transport,optimizare, mecanică şi inginerie.

Scopul principal al acestei cărţi este extinderea rezultatelor de închidere aproblemelor parametrice de echilibru scalar la probleme parametrice de echili-bru vectorial şi la probleme parametrice de echilibru cu operatori.

Lucrarea este organizată în cinci capitole, o listă bibliografică şi o listă deindice.

În primul capitol sunt introduse noţiunile şi rezultatele care alcătuiescpunctul de pornire al studiului problemelor parametrice de echilibru vectorial.Aici sunt date definiţiile şi proprietăţile conurilor, punctele de limită definite pespaţii vectoriale. În paragraful 1.3 sunt introduse mai multe tipuri de pseudo-monotonităţi vectoriale topologice. În paragrafele 1.4 şi 1.5 sunt redate noţiunilede convergenţă Mosco şi proprietatea de Hadamard bine pusă (well-posed) aproblemelor de echilibru. În paragraful 1.6 sunt prezentate rezultatele de exis-tenţă, iar paragraful 1.7 conţine rezultatele lui Bogdan şi Kolumbán, obţinutepentru probleme de echilibru parametrice reale.

În capitolul al doilea sunt prezentate rezultatele de senzitivitate deja cunos-cute pentru probleme de echilibru parametrice vectoriale. În paragraful 2.1 suntredate rezultate referitoare la semicontinuitatea funcţiei soluţie, iar paragraful2.2 conţine rezultatele de Hölder continuitatea a funcţiei soluţie.

Capitolul al treilea conţine cele mai importante rezultate proprii obţinutede autor. Acest capitol este dedicat închiderii slabe a aplicaţiei de soluţie multi-vocă, definită pe mulţimea parametrilor pentru probleme de echilibru vectorial.În unele cazuri este analizată chiar şi proprietatea de Handmard bine pusă (wellposed) a problemelor.

Capitolul al patrulea este dedicat studiului problemelor parametrice deechilibru cu operatori. În secţiunea 4.1 sunt prezentate nişte rezultate cunos-cute de existenţă, iar în secţiunea 4.2 sunt descrise rezultatele de închidere afuncţiei soluţie.

În capitolul al cincilea sunt tratate aplicaţiile rezultatelor de închidere ob-ţinute în capitolul 3. Problemele studiate sunt cele de optimizare vectorialăparametrice, inegalităţi variaţionale vectoriale parametrice şi probleme para-metrice de complementaritate vectorială.

Cartea se adresează cercetătorilor, doctoranzilor şi masteranzilor cu dome-niul de cercetare în probleme de echilibru.

ABOUT THE AUTHOR

The Author is lecturer at the Sapientia University, Faculty of Business andHumanities, Chair of Mathematics and Informatics.

Place and date of birthMiercurea Ciuc (Csíkszereda), Romania, April 27, 1980.

Education- 1994-1998 Márton Áron High-school, Miercurea Ciuc (Csíkszereda),

High-school graduation in 1998;- 1998-2002 Faculty of Mathematics and Computer Science, Babeş-Bolyai

University, Cluj-Napoca (Kolozsvár), Licentiate in Mathematics and in-formatics, 2002;

- 2002-2003 Faculty of Mathematics and Computer Science, Babeş-BolyaiUniversity, Cluj-Napoca (Kolozsvár), Master in Real and complex anal-ysis, 2003;

- 2003-2009 Ph.D. student at the Faculty of Mathematics and ComputerScience, Babeş-Bolyai University, Cluj-Napoca (Kolozsvár);

- 2009 Ph.D. degree in Mathematics. Title of the Ph.D. thesis: Contribu-tions to the theory of parametric vector equilibrium

Research interest- operational research;- complex analysis;- computer graphics;

Teaching activities- Bachelor level courses: Informatics I (Matlab), Operational research, De-

cision analysis (all in Hungarian).

PUBLICATIONSIN THE SAPIENTIA BOOKS SERIES

Books published:

1. Tonk Márton–Veress Károly (Eds.)

Értelmezés és alkalmazás. Hermeneutikai és alkalmazottfilozófiai vizsgálódások. 2002.

2. Pethő Ágnes (Ed.)

Képátvitelek. Tanulmányok az intermedialitás tárgyköréből.2002.

3. Nagy László

Numerikus és közelítő módszerek az atomfizikában. 2002.

4. Egyed Emese (Ed.)

Theátrumi Könyvecske. Színházi zsebkönyvek és szerepüka régió színházi kultúrájában. 2002.

5. Vorzsák Magdolna–Kovács Liciniu Alexandru

Mikroökonómiai kislexikon. 2002.

6. Köllő Gábor (Ed.)

Műszaki szaktanulmányok. 2002.

7. Szenkovits Ferenc–Makó Zoltán–Csillik Iharka–Bálint Attila

Mechanikai rendszerek számítógépes modellezése. 2002.

8–10. Tánczos Vilmos–Tőkés Gyöngyvér (Eds.)

Tizenkét év. Összefoglaló tanulmányok az erdélyimagyar tudományos kutatások 1990–2001 közöttieredményeiről. I–III. 2002.

11. Sorbán Angella (Ed.)

Szociológiai tanulmányok erdélyi fiatalokról. 2002.

12. Gábor Csilla–Selyem Zsuzsa (Eds.)

Kegyesség, kultusz, távolítás.Irodalomtudományi tanulmányok. 2002.

13. Salat Levente (Ed.)

Kínlódni ebben az országban. . . ? Ankét a romániai magyarságmegmaradásának szellemi feltételeiről. 2002.

14. Németi János–Molnár Zsolt

A tell telepek elterjedése a Nagykárolyi-síkságon és az Érvölgyében. 2002.

15. Nagy László (Ed.)

Tanulmányok a természettudományok tárgyköréből. 2002.

16. Bocskay István–Matekovits György–Székely Melinda–

Kovács-Kuruc J. Szabolcs

Magyar–román–angol fogorvosi szakszótár. 2003.

17. Brassai Attila (Ed.)

Orvostudományi tanulmányok. 2003.


Köztes képek. A filmelbeszélés színterei. 2003.

19. Kiss István

Erodált talajok enzimológiája. 2003.

20. Nagy László (Ed.)

Korszerű kísérleti és elméleti fizikatanulmányok. 2003.

21. Ujvárosi Lujza (Ed.)

Erdély folyóinak természeti állapota. Kémiai és ökológiaivízminősítés a rekonstrukció megalapozására. 2003.

22. Kolumbán József et alii

Lectures on Nonlinear Analysis and Its Applications. 2003.


„Szabadon fordította. . . ” Fordítások a magyar színjátszáscéljaira a XVIII–XIX. században. 2003.

24. Bajusz István (Ed.)

Mindennapi élet a római Dáciában. 2003.

25. Selyem Zsuzsa–Balázs Imre József (Eds.)

Humor az avantgárdban és a posztmodernben. 2004.

26. Gábor Csilla (Ed.)

Devóciók, történelmek, identitások. 2004.

27. Roth-Szamosközi Mária (Ed.)

Válassz okosan. . . Készségfejlesztő program az agresszivitáscsökkentésére. 2004.

28. Sipos Gábor (Ed.)

A kolozsvári Akadémiai Könyvtár Régi MagyarKönyvtár-gyűjteményeinek katalógusa. 2004.

29. Neményi Ágnes (Ed.)

A rurális bevándorlók. Az elsőgenerációs kolozsvárivároslakók társadalma. 2004.


A Csíki-medence településtörténete a neolitikumtól a XVII.század végéig a régészeti adatok tükrében. 2004.

31. Kovács Zsolt (Ed.)

Erdély XVII–XVIII. századi építészetének forrásaiból. 2004.

32. Kiss István

A mikroorganizmusokkal beoltott talajok enzimológiája.2004.

33. Brassai Zoltán et alii

A kovásznai szénsavas fürdők és mofetták a végtagiverőérszűkületek kezelésében. 2004.

34. Horváth István (Ed.)

Erdély és Magyarország közötti migrációs folyamatok. 2005.

35. Gábor Csilla (Ed.)

A történetmondás rétegei a kora újkorban. 2005.


Ismeretség: Interkulturális kapcsolatok a színház révén:XVII–XIX. század. 2005.

37. Berszán István (Ed.)

Alternatív mozgásterek: Működés és/vagy gyakorlás akognitív folyamatokban. 2005.

38. Egyed Péter (Ed.)

A közösségről – a hagyományos, valamint a kommunitaristafelfogásban. 2005.

39. Jancsó Miklós

Csiky Gergely színpadi világa. 2005.

40. Zoltán Benyó–Béla Paláncz–László Szilágyi

Insight Into Computer Science with Maple. 2005

41. Keszeg Vilmos (Ed.)

Specialisták. Életpályák és élettörténetek. 2006.


Téglás István jegyzetei. 2005.

43. Zoltán Makó

Quasi-triangular Fuzzy Numbers, Theory and Applications.2006.

44. Berszán István (Ed.)

Gyakorlat–etika–pragmatizmus. 2006.

45. Bege Antal

Régi és új számelméleti függvények. 2006.

46. Balázs Lajos

A vágy rítusai – rítusstratégiák. A születés, házasság, halálszokásvilágának lelki hátteréről. 2006.

47. Albert-Lőrincz Enikő

Átfesthető horizont. 2006.

48. Németi János–Molnár Zsolt

A tell telepek fejlődése és vége a Nagykárolyi-síkságon és azÉr völgyében. 2006.

49. Tamási Zsolt-József

Az erdélyi római katolikus egyházmegye az 1848–49-esforradalomban. 2007.


Film. Kép. Nyelv. 2007.

51. Szilágyi Györgyi–Flóra Gábor–Ari Gyula

Bihar megye gazdasági-társadalmi fejlődése. Eredmények éstávlatok. 2007.

52. Ozsváth Imola (Ed.)

Néptanítók. Életpályák és élettörténetek. 2008.

53. István Urák

Date despre arahnofauna din bazinul superior al Oltului.2008.

54. Szabó Árpád

A tulajdonváltás folyamata Románia gazdaságiátalakulásában. 2008.

55. Szabó Árpád

A romániai gazdasági átalakulás esettanulmányokban. 2008.

56. Tőkés Gyöngyvér

Szakma vagy hivatás? A kolozsvári magyar egyetemi oktatókstátuszcsoportja. 2008.

57. Zsigmond István

Metakognitív stratégiák – összetevőik és mérésük. 2008.

58. Andrea Virginás

Crime Genres and the Modern – Postmodern Turn: Canons,Gender, Media. 2008.

59. Pletl Rita (Ed.)

Az anyanyelvoktatkás metamorfózisa. 2008.

60. Makó Zoltán–Lázár Ede–Máté Szilárd

Előrejelző módszerek gazdasági és műszaki alkalmazásai.2009.

61. Sándor Miklós Szilágyi

Dynamic modeling of the human heart. 2009.

62. László Szilágyi

Novel Image Processing Methods Based on Fuzzy Logic.2009.

63. Éva György

Studiu anatomic al unor cormofi te din zona Ciucului. 2009.

64. Beáta Ábrahám

Evoluţia turbăriilor din judeţul Harghita. 2009.

65. Imre Attila

A Cognitive Approach to Metaphorical Expressions. 2010.

66. Ştefania Maria Custură

Marin Sorescu. Poezia teatrului şi teatralitatea poeticului.2010.

67. Gizella Boszák (Szerk.)

Fehlertypologie im DaF-Unterricht Band 1. Studien zudeutsch-rumänish-ungarischen Interferenzerscheinungen.2010.

68. Pletl Rita (Szerk.)

Az anyanyelvoktatás mozaikjai. 2010.

69. Veress Emőd

Államfő és kormány a hatalommegosztás rendszerében. 2011.

70. Bálint Blanka

A túlképzettség okainak vizsgálata az erdélyi diplomasfiatalok körében. 2011.

71. Bálint Gyöngyvér

Foglalkoztatási stratégiák Hargita megyében. 2011.

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júlia salamon: parametric vector equilibrium problems

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