Júlia SALAMON: Parametric vector equilibrium problems by Scientia Kiadó

JÚLIA SALAMON PARAMETRIC VECTOR EQUILIBRIUM PROBLEMS

SCIENTIA Publishing House Cluj-Napoca ·2011

JÚLIA SALAMON PARAMETRIC VECTOR EQUILIBRIUM PROBLEMS

SAPIENTIA BOOKS

SAPIENTIA ALAPÍTVÁNY SAPIENTIA ERDÉLYI MAGYAR TUDOMÁNYEGYETEM

PARTIUMI KERESZTÉNY EGYETEM

JÚLIA SALAMON

PARAMETRIC VECTOR EQUILIBRIUM PROBLEMS

S C I E N T I A Publishing House Cluj-Napoca·2011

SAPIENTIA BOOKS 72. Natural Sciences

This book was published with the support of the Sapientia Foundation. Published by Scientia Publishing House 400112 Cluj-Napoca, str. Matei Corvin 4. Tel./Fax: +40-264-593694 E-Mail: scientia@kpi.sapientia.ro www.scientiakiado.ro Publisher in Chief: Zoltán Kása The present volume is the PhD dissertation of the author who takes professional responsibility for it. Members of the PhD commission: Dr. József Kolumbán (thesis supervisor) Dr. Petru Jebelean Dr. Adrian Petruşel Dr. Daniela Inoan Dr. Wolfgang Breckner

First English edition: 2011 Scientia, 2011 All rights reserved, including the rights for photocopying, public lecturing, radio and television broadcast and translation of the whole work and of chapters as well. Descrierea CIP a Bibliotecii Naţionale a României JÚLIA SALAMON Parametric vector equilibrium problems / Júlia Salamon. – Cluj-Napoca: Scientia, 2011 Bibliogr. ISBN 978-973-1970-46-2 571

CONTENTS

Preface

1 Preliminaries

1.1 Cones

1.2 Limit points of the nets

1.3 Vector topological pseudomonotonicity

1.4 Mosco convergence

1.5 Hadamard well-posedness

1.6 Existence results

1.6.1 Equilibrium problems

1.6.2 Weak vector equilibrium problems

1.6.3 Vector equilibrium problems

1.6.4 Strong vector equilibrium problems

1.6.5 Generalized vector equilibrium problems with trifunctions

1.7 Parametric equilibrium problems

2 Sensitivity analysis of vector equilibrium problems

2.1 Semicontinuity of the solution mapping

2.1.1 Upper semicontinuity of the solution mapping

2.1.2 Lower semicontinuity and continuity of solution mapping

2.2 Hรถlder continuity of the solution mapping

3 Closedness of the solution mapping

3.1 Parametric weak vector equilibrium problems

3.2 Parametric vector equilibrium problems

3.3 Parametric strong vector equilibrium problems

3.4 Parametric generalized vector equilibrium

3.4.1 Strong solutions

3.4.2 Weak solutions

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

1. Bevezető fogalmak

1.1. Kúpok

1.2. A hálók határértékpontjai

1.3. Vektor topológikus pszeudomonotonitás

1.4. Mosco konvergencia

1.5. Hadamard jól értelmezettség (well-posedness)

1.6. Létezési tételek

1.6.1. Egyensúlyi feladatok

1.6.2. Gyenge vektor egyensúlyi feladatok

1.6.3. Vektor egyensúlyi feladatok

1.6.4. Erős vektor egyensúlyi feladatok

1.6.5. Vektor egyensúlyi feladatok háromváltozós függvényekkel

1.7. Paraméteres egyensúlyi feladatok

2. Vektor egyensúlyi feladatok érzékenység vizsgálata

2.1. Megoldás függvény féligfolytonossága

2.1.1. Megoldás függvény felülről féligfolytonossága

2.1.2. Megoldás függvény alulról féligfolytonossága

2.2. Megoldás függvény Hölder folytonossága

3. Megoldás függvény graﬁkonjának zártsága

3.1. Paraméteres gyenge vektor egyensúlyi feladatok

3.2. Paraméteres vektor egyensúlyi feladatok

3.3. Paraméteres erős vektor egyensúlyi feladatok

3.4. Paraméteres vektor egyensúlyi feladatok háromváltozós függvényekkel

3.4.1. Erős megoldások

3.4.2. Gyenge megoldások

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 1. Noţiuni introductive

11 17

1.1. Conuri

1.2. Punctele de limită ale şirurilor generalizate

1.3. Pseudomonotonie vectorială topologică

1.4. Convergenţa Mosco

1.5. Proprietatea Hadamard bine pusă (well-posed)

1.6. Rezultate de existenţă

1.6.1. Probleme de echilibru

1.6.2. Probleme slabe de echilibru vectorial

1.6.3. Probleme de echilibru vectorial

1.6.4. Probleme tari de echilibru vectorial

1.6.5. Probleme de echilibru vectorial cu trifuncţii

1.7. Probleme parametrice de echilibru

2. Analiză de senzitivitate a problemelor de echilibru vectorial

2.1. Semicontinuitatea funcţiei soluţie

2.1.1. Superior semicontinuitatea funcţiei soluţie

2.1.2. Inferior semicontinuitatea funcţiei soluţie

2.2. Continuitatea Hölder a funcţiei soluţie

3. Închiderea graﬁcului funcţiei soluţie

3.1. Probleme parametrice slabe de echilibru vectorial

3.2. Probleme parametrice de echilibru vectorial

3.3. Probleme parametrice tari de echilibru vectorial

3.4. Probleme parametrice de echilibru vectorial cu trifuncţii

3.4.1. Soluţii tari

3.4.2. Soluţii slabe

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 general framework for the study of a large variety of problems such as optimization problems, fixed points problems, variational inequalities, Nash equilibria, saddle point problems and complementarity problems as special cases. The problems mentioned above often occur in mechanics, physics, finance, economics, network analysis, transportation and elasticity. The behavior of the solutions resulting by the change of the problems data is always of major concern. Sensitivity analysis examine the way how the solutions of such problems change when the data of the problems are modified. The reason why sensitivity analysis is important, is that the estimating problem data often introduces measurement errors and sensitivity results help in identifying sensitive parameters that have to be obtained with relatively high accuracy. Over the last decade, there has been increasing interest in studying the sensitivity analysis of variational inequalities, which has been studied by Tobin [179], Kyparisis [110], Dafermos [54], Qiu and Magnanti [154], Yen [188] and Liu [135], using quite different techniques. The techniques suggested so far vary with the problem studied. Many authors used the projection method to study the continuity or Lipschitz continuity of the solution set of variational inequalities in Euclidean spaces or Hilbert spaces (see Dafermos [54], Mukherjee-Verma [145] and Yen [187],[186]). Robinson [157] used the so-called normal mappings and an implicit function approach to deal with the solution sensitivity of variational inequalities satisfying some smoothness assumptions. Noor [146], Domokos [57], Ding and Luo [55] and Kassay and Kolumbán [94] considered the continuity or Lipschitz continuity of the solution set of variational or quasivariational inequality problems in infinite dimensional settings. The parametric equilibrium problems, what we already mentioned, we can formulate in the following way: Let X be a Hausdorff topological space and P, the set of parameters, another Hausdorff topological space. For a given p ∈ P we consider the following equilibrium 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.

PREFACE

The sensitivity or Hölder continuity of the solution mapping of parametric equilibrium 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 set of parameters. The semicontinuity of the solution multifunction of the parametric operator equilibrium problems was analyzed by Kim, Kum and Lee [100]. Most of decision situations require simultaneous consideration of more than one objective, which are often in conflict. The variables that optimize one objective may be far from optimal for the others. In practice, multiobjective optimization and - later - vector variational inequalities have been the appropriate methodologies for solving this conflict. Natural extensions to this case also include vector complementarity problems, vector best approximations, and cone saddle point theorems, for example the Nash equilibrium problems. Nash equilibrium is a solution concept of a game involving two or more players, in which each 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 or her strategy. If we assume that the players communicate, exchange information and cooperate, then it is possible to obtain better results for all the participants. Optimal strategies can be received by changing from one allocation to another that can make at least one player better off without making any other player worse off. An allocation is Pareto optimal when no further improvements can be made. The extension of the scalar parametric equilibrium problems to vector parametric equilibrium problems can be achieved in different ways. The problems under consideration are the following: Let X be a Hausdorff topological space and let P the set of parameters, be another Hausdorff topological space. Let Z be a real topological vector space and C be a cone in Z with C 6= Z and Int C 6= ∅, where Int C denotes the interior of C . We consider the following three parametric vector equilibrium problems: Find xp ∈ Dp such that

(P V EP )1 (P V EP )2 (P V EP )3

fp (xp , y) ∈ (− Int C) , ∀y ∈ Dp , fp (xp , y) ∈ / −C\ {0} , ∀y ∈ Dp , 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 problem [80], while the third one is known as parametric strong vector equilibrium problem [8]. If Z = R and C = R+ , then (P V EP )i for i = 1, 3 collapse to the parametric equilibrium problem (EP )p . Let us denote by Si (p) the set of the solutions for (P V EP )i i = 1, 3 for a ﬁxed p. If C is a cone, then the relationship between the problems mentioned above can be formulated as follows [90]:

PREFACE

– S2 (p) ⊆ S1 (p), for every p ∈ P. – If C is pointed, i.e. C ∩ − (C\ {0}) = ∅, then S3 (p) ⊆ S2 (p), for every p ∈ P. – If C is w-pointed, i.e. C ∩ − Int C = ∅, then S3 (p) ⊆ S1 (p), for every p ∈ P. – If C is connected, i.e. C ∪ − Int C = Z , then S1 (p) ⊆ S3 (p), for every p ∈ P. If C is a closed convex cone in Z with C 6= Z and Int C 6= ∅, then we have the 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 parametric 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 considering trifunctions from K × D × K into a real topological vector space Z with an ordering cone. By an ordering cone C ⊂ Z we mean that C is a closed convex cone in Z with Int C 6= ∅ and C 6= Z . Let Kp be a nonempty subset of X . Let fp : X × Y × X → Z be a trifunction and T : X → 2Y be a set-valued mapping. The following two problems are considered: ﬁnd a pair (xp , yp ) ∈ Kp × Y such that yp ∈ T (xp ) and c

fp (xp , yp , u) ∈ (− Int C) for all u ∈ Kp ; ﬁnd xp ∈ Kp such that for every u ∈ Kp there exists yu ∈ T (xp ) satisfying c

fp (xp , yp , u) ∈ (− Int 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 problem (GV EP )p . Denote by Ss (p) and Sw (p) the set of the strong solutions and weak solutions 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 problems.

PREFACE

In 2002, Domokos and Kolumbán [58] introduced and studied a class of operator variational inequalities. These operator variational inequalities include not only scalar and vector variational inequalities as special cases, but also have sufﬁcient evidence for their importance to study. Inspired by their work, Kum and Kim [107, 108] developed the scheme of operator variational inequalities from the single-valued case into the multi-valued one. The operator equilibrium problems was studied by Kazmi and Raouf [96], Kum and Kim [109]. The problem under consideration is the following: Let X and Y be Hausdorff topological vector spaces, L(X, Y ) be the space of all continuous linear operators from X to Y and let D ⊂ L(X, Y ) be a nonempty set. Let P the set of parameters be another Hausdorff topological space. Let C : D → 2Y be a set-valued mapping such that for each f ∈ D, C(f ) is a convex open cone with nonempty interior and C (f ) 6= Y . For a given p ∈ P the parametric operator equilibrium problem (OEP )p is to 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. We single 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 profiles and F1 , F2 : D1 × D2 → R are the payoff functions of the two players. A strategy profile (x1 , x2 ) ∈ D is a Nash equilibrium if no unilateral deviation in strategy by 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 (P V EP )3 . Vector optimization: Let ϕ : D → Z be a given vector valued function.

(W V OP )

Find x ∈ D such that ϕ (y) − ϕ (x) ∈ / − Int C for all y ∈ D.

If we set f (x, y) = ϕ (y) − ϕ (x) for all x, y ∈ D, then (W V OP ) coincides with (P V EP )1 . Note that if X = Rm , Z = Rn and C = Rn + for all x ∈ D ,

PREFACE

then solutions to (W V OP ) are known as weak Pareto (or efficient) solutions. Roughly speaking, (W V OP ) appears when one attempts to make decisions by optimizing a multiobjective. In the literature, some properties of the solution map defined on the set of parameters for a (P V EP ) have been investigated. Anh and Khanh [3], Huang, Li, and Thompson [85], Long, Huang and Teo [137] studied the stability for set-valued vector quasiequilibrium problems or parametric implicit vector equilibrium problems. Bianchi and Pini [30], Kimura and Yao [101], Sach, Tuan and Lee [161] studied the sensitivity for parametric vector equilibria. Anh and Khanh [2], Li, Li, Wang and Teo [119], Chen, Li and Teo [42], Gong [80], Xu and Li [182] considered the continuity or Hölder continuity of the solution mappings for parametric vector equilibrium problems. These results are detailed in chapter 2. The main purpose of this book is the extension of the closedness results of scalar parametric equilibrium problems obtained in [35] to parametric vector equilibrium 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 introduced 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 recalled. In Section 1.3 generalizations of the topological pseudomonotonicity, notion introduced by Brézis in [37], are discussed. Each type of parametric vector equilibrium problem presented above requires a new notion of vector topological pseudomonotonicity. Another basic notion referred to is a slight generalization of the Mosco convergency, it is presented in section 1.4. Section 1.5 focuses on the relations between closedness, upper semicontinuity and Hadamard well-posedness of the parametric vector equilibrium problems. In order to obtain closedness results, it is necessary to exist solutions at each parameter. The section 1.6 presents existence results for every previously mentioned vector equilibrium problems. Section 1.7 reviews the starting point of this book, the closedness results of the solution map for scalar parametric equilibrium problems presented in [35]. Chapter 2 is devoted to the presentation of some stability results for parametric vector equilibrium problems. These results target only the weak parametric vector equilibria. Generally speaking, the notion of stability or sensitivity of the solution mapping refers to upper or lower semicontinuity. Section 2.1 analyses the semicontinuity of the solution mapping by invoking Huang, Li and Thompson [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 mapping for weak parametric vector equilibrium problems, based on Li, Li, Wang and Teo [119], Anh and Khanh [2], Bednarczuk [24] results. Another regularity property, the closedness of the solution mapping for parametric vector equilibrium problems, is studied in chapter 3. Each section

PREFACE

focuses on a different parametric vector equilibrium problem. In cases of the solution mapping for weak and strong parametric vector equilibrium problems and strong solution mapping for generalized parametric vector equilibrium problems two closedness theorems are presented. From them, only the second theorems generalize the results in [35]. Here also the Hadamard well-posedness of the mentioned problems is analyzed. Two closedness results are given for the weak solution mapping of the parametric vector equilibrium problems with trifunctions. An extension of the closedness results is presented for the problems (P V EP )2 in section 3.2. Chapter 4 is devoted to the study of the parametric operator equilibrium problems, which it is a two-way generalization of (P V EP )1 . In section 4.1 some known existence results are presented. In section 4.2 the closedness of the solution mapping for parametric operator equilibrium problems are analyzed. In chapter 5 the applications of the closedness results obtained in chapter 3 are treated. Parametric vector optimization problems, parametric vector variational inequalities and parametric vector complementarity problems are studied. Section 5.1 presents the closedness results of weak efficient and so called absolute solution mapping for vector optimization problems. Even in the case of parametric vector variational inequalities we can formulate three type of problems similar to those described in the case of parametric vector equilibrium problems. In section 5.2 the closedness of solution mapping for these type of problems are discussed. In section 5.3 the parametric vector complementarity problems 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 different 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 equilibrium problems are studied; – Corollaries 3.7, 3.16, 3.17, 3.23, 3.24, 4.8 where Hadamard wellposedness of the parametric vector equilibrium problems are analyzed; – some applications of the closedness results for parametric vector optimization problems, parametric vector variational inequality and parametric vector complementarity problems are presented in chapter 5. The elaboration of this book is based on my Ph.D. thesis, that was possible due to the guidance of my scientific advisor Prof. Dr. Kolumbán József. I am very grateful to Dr. Bogdan Marcel, who provided me useful suggestions. I also wish to 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 for improving the manuscript. Finally, I want to thank my family for the support during my studies.

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INDEX

C C-property, 49 Closedness, 32, 38, 43, 60, 104, 105 Cone, 17 Convexity C-convex, 35, 36, 44, 52 C-quasiconcave like, 36 C-quasiconvex, 50 C-quasiconvex-like, 34 C-weakly quasiconcave, 50 concave, 49 convex, 49, 104 H-convex, 104 natural quasi H-convex, 104 vector 0-diagonally convex, 33, 36

H Hölder continuity, 55 h.β -Hölder-strongly pseudomonotone, 56 Hölder of order m, 58 l.α-Hölder, 56 lower Hölder of order m, 58 upper Hölder of order m, 58 Hadamard well-posedness, 32, 41, 75, 87, 96, 109 Hemicontinuity, 104

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

M Monotonity

C-strictly monotone, 52 monotone, 35 quasimonotone, 56 Mosco convergence, 31

P Pseudomonotonicity B-C(f)-pseudomonotone, 105 C(f)-pseudomonotone, 103

S Semicontinuity C-lower semicontinuity, 35 C-upper semicontinuity, 25 lower semicontinuity, 36, 42, 47 termed Hausdorff upper semicontinuous, 43 upper semicontinuity, 22, 32, 42, 44

T Topological pseudomonotonicity, 22 A1 -vector topological pseudomonotonicity, 23 A2 -vector topological pseudomonotonicity, 27 a2 -vector topologically pseudomonotone, 114 A3 -vector topological pseudomonotonicity, 27 a3 -vector topologically pseudomonotone, 115 B1 -vector topological pseudomonotonicity, 23 B3 -vector topological pseudomonotonicity, 27

140

b3 -vector topologically pseudomonotone, 116 class (GPM), 30 class (GPM2 ), 30 class (SPM), 29 class (SPM2 ), 29 topological pseudomonotonicity, 22 vector topological h-pseudomonotonicity, 112 vector topological h-pseudomonotonicity of type II, 113 vector topological pseudomonotonicity, 23, 105

INDEX

KIVONAT

Az egyensúlyi feladatok több feladat típust foglalnak magukban, mint a variációs egyenlőtlenségeket, komplementaritási feladatokat, optimalizálási feladatokat, ﬁxpontok elméletét, Nash egyensúlyi feladatokat. Ezen feladatoknak nagyon sok hasznos alkalmazásai vannak különböző területeken, és matematikailag modelleznek jelenségeket közgazdaságtanból, pénzügyből, logisztikából, optimalizálásból, mechanikából illetve mérnöki tudományokból. A könyv elsődleges célja a paraméteres egyensúlyi feladatok zártságra vonatkozó eredményeinek általánosítása paraméteres vektor egyensúlyi illetve paraméteres operátor egyensúlyi feladatokra. A könyv öt fejezetet, egy könyvészetet és egy tárgymutatót foglal magába. Az első fejezetben azon bevezető fogalmak és eredmények kerülnek bemutatásra, amelyek a paraméteres vektor egyensúlyi feladatok tanulmányozásának a kiindulási pontját képezik. Itt találhatók meg a kúpok és vektoriális tereken értelmezett határérték pontok értelmezései és tulajdonságai. Az 1.3 alfejezetben több vektor topológikus pszeudomonotonitási fogalom van bevezetve. A Mosco konvergencia és a Hadamard jólértelmezettségi (well-posedness) fogalmak az 1.4 és 1.5 alfejezetekben találhatók meg. Az 1.6 alfejezetben a létezési tételek vannak bemutatva, míg az 1.7 alfejezet Bogdan és Kolumbán által megadott zártsági eredményeket tartalmazza valós egyensúlyi feladatokra. A második fejezetben a vektor egyensúlyi feladatok érzékenységi eredményei vannak ismertetve. A megoldás függvény féligfolytonosságára vonatkozó eredmények a 2.1 alfejezetben találhatók meg, míg a 2.2 alfejezet a Hölder folytonosságra vonatkozó eredményeket tárgyalja. A szerző legjelentősebb eredményeit a harmadik fejezet foglalja magában. Ezen fejezet, a vektor egyensúlyi feladatok paramétertől függő halmazértékű megoldás függvényének gyenge zártsági tulajdonságának van szentelve. Egyes esetekben a feladatok Hadamard jólértelmezettsége (well-posedness) is elemzésre kerül. A negyedik fejezet a paraméteres operátor egyensúlyi feladatok vizsgálatának van szentelve. A 4.1 alfejezetben a már ismert létezési tételek vannak bemutatva, míg a 4.2 alfejezetben a megoldás függvény zártsági tulajdonsága van vizsgálva. A harmadik fejezetben bemutatott zártsági eredmények alkalmazásait az ötödik fejezet tartalmazza. A vizsgált feladatok a következők: paraméteres vektor optimalizálási feladatok, paraméteres vektor variációs egyenlőtlenségek, paraméteres vektor komplementaritási feladatok. A könyvet egyaránt használhatja a téma iránt érdeklődő, az eredményeit felhasználni vágyó kutató, doktorandus és magiszteri hallgató.

REZUMAT

Problemele de echilibru sunt probleme generale, care la rândul lor, cuprind multe tipuri de probleme cum ar fi: inegalităţi variaţionale, probleme de complementaritate, probleme de optimizare, teoria punctelor fixe, probleme de echilibru Nash. Acestea au o mulţime de aplicaţii importante în diferite domenii şi modelează matematic fenomene din economie, finanţe, transport, optimizare, mecanică şi inginerie. Scopul principal al acestei cărţi este extinderea rezultatelor de închidere a problemelor parametrice de echilibru scalar la probleme parametrice de echilibru vectorial şi la probleme parametrice de echilibru cu operatori. Lucrarea este organizată în cinci capitole, o listă bibliografică şi o listă de indice. În primul capitol sunt introduse noţiunile şi rezultatele care alcătuiesc punctul de pornire al studiului problemelor parametrice de echilibru vectorial. Aici sunt date definiţiile şi proprietăţile conurilor, punctele de limită definite pe spaţii vectoriale. În paragraful 1.3 sunt introduse mai multe tipuri de pseudomonotonităţi vectoriale topologice. În paragrafele 1.4 şi 1.5 sunt redate noţiunile de convergenţă Mosco şi proprietatea de Hadamard bine pusă (well-posed) a problemelor de echilibru. În paragraful 1.6 sunt prezentate rezultatele de existenţă, iar paragraful 1.7 conţine rezultatele lui Bogdan şi Kolumbán, obţinute pentru probleme de echilibru parametrice reale. În capitolul al doilea sunt prezentate rezultatele de senzitivitate deja cunoscute pentru probleme de echilibru parametrice vectoriale. În paragraful 2.1 sunt redate rezultate referitoare la semicontinuitatea funcţiei soluţie, iar paragraful 2.2 conţine rezultatele de Hölder continuitatea a funcţiei soluţie. Capitolul al treilea conţine cele mai importante rezultate proprii obţinute de autor. Acest capitol este dedicat închiderii slabe a aplicaţiei de soluţie multivocă, definită pe mulţimea parametrilor pentru probleme de echilibru vectorial. În unele cazuri este analizată chiar şi proprietatea de Handmard bine pusă (well posed) a problemelor. Capitolul al patrulea este dedicat studiului problemelor parametrice de echilibru cu operatori. În secţiunea 4.1 sunt prezentate nişte rezultate cunoscute de existenţă, iar în secţiunea 4.2 sunt descrise rezultatele de închidere a funcţiei soluţie. În capitolul al cincilea sunt tratate aplicaţiile rezultatelor de închidere obţinute în capitolul 3. Problemele studiate sunt cele de optimizare vectorială parametrice, inegalităţi variaţionale vectoriale parametrice şi probleme parametrice de complementaritate vectorială. Cartea se adresează cercetătorilor, doctoranzilor şi masteranzilor cu domeniul de cercetare în probleme de echilibru.

ABOUT THE AUTHOR

The Author is lecturer at the Sapientia University, Faculty of Business and Humanities, Chair of Mathematics and Informatics. Place and date of birth Miercurea 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 informatics, 2002; - 2002-2003 Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca (Kolozsvár), Master in Real and complex analysis, 2003; - 2003-2009 Ph.D. student at the Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca (Kolozsvár); - 2009 Ph.D. degree in Mathematics. Title of the Ph.D. thesis: Contributions 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, Decision analysis (all in Hungarian).

PUBLICATIONS IN THE SAPIENTIA BOOKS SERIES Books published: 1. Tonk Márton–Veress Károly (Eds.) Értelmezés és alkalmazás. Hermeneutikai és alkalmazott filozófiai 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 szerepük a 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 erdélyi magyar tudományos kutatások 1990–2001 közötti eredményeiről. 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 magyarság megmaradásának szellemi feltételeiről. 2002. 14. Németi János–Molnár Zsolt A tell telepek elterjedése a Nagykárolyi-síkságon és az Ér völgyében. 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. 18. Pethő Ágnes (Ed.) 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 ökológiai vízminősítés a rekonstrukció megalapozására. 2003. 22. Kolumbán József et alii Lectures on Nonlinear Analysis and Its Applications. 2003. 23. Egyed Emese (Ed.) „Szabadon fordította. . . ” Fordítások a magyar színjátszás céljaira 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 agresszivitás csökkentésére. 2004. 28. Sipos Gábor (Ed.) A kolozsvári Akadémiai Könyvtár Régi Magyar Könyvtár-gyűjteményeinek katalógusa. 2004. 29. Neményi Ágnes (Ed.) A rurális bevándorlók. Az elsőgenerációs kolozsvári városlakók társadalma. 2004. 30. Bajusz István (Ed.) 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 végtagi verőérszűkületek kezelésében. 2004. 34. Horváth István (Ed.) Erdély és Magyarország közötti migrációs folyamatok. 2005. 35. Gábor Csilla (Ed.) A történetmondás rétegei a kora újkorban. 2005. 36. Egyed Emese (Ed.) Ismeretség: Interkulturális kapcsolatok a színház révén: XVII–XIX. század. 2005. 37. Berszán István (Ed.) Alternatív mozgásterek: Működés és/vagy gyakorlás a kognitív folyamatokban. 2005. 38. Egyed Péter (Ed.) A közösségről – a hagyományos, valamint a kommunitarista felfogásban. 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. 42. Bajusz István (Ed.) 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, halál szokásvilágának 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 az Ér völgyében. 2006. 49. Tamási Zsolt-József Az erdélyi római katolikus egyházmegye az 1848–49-es forradalomban. 2007. 50. Pethő Ágnes (Ed.) Film. Kép. Nyelv. 2007. 51. Szilágyi Györgyi–Flóra Gábor–Ari Gyula Bihar megye gazdasági-társadalmi fejlődése. Eredmények és távlatok. 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 gazdasági átalakulásában. 2008. 55. Szabó Árpád A romániai gazdasági átalakulás esettanulmányokban. 2008. 56. Tőkés Gyöngyvér Szakma vagy hivatás? A kolozsvári magyar egyetemi oktatók státuszcsoportja. 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 zu deutsch-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 diplomas ﬁatalok körében. 2011. 71. Bálint Gyöngyvér Foglalkoztatási stratégiák Hargita megyében. 2011.

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