Luận án: Định lý tồn tại trong giải tích phi tuyến & ứng dụng tối ưu - Võ Sĩ Trọng Long

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY UNIVERSITY OF SCIENCE VO SI TRONG LONG SEVERAL EXISTENCE THEOREMS IN NONLINEAR ANALYSIS AND APPLICATIONS TO OPTIMIZATION-RELATED PROBLEMS PhD THESIS IN MATHEMATICS Ho Chi Minh City - 2015 ' $ VIETNAM NATIONAL UNIVERSITY-HO CHI MINH CITY UNIVERSITY OF SCIENCE VO SI TRONG LONG SEVERAL EXISTENCE THEOREMS IN NONLINEAR ANALYSIS AND APPLICATIONS TO OPTIMIZATION-RELATED PROBLEMS Major: Optimization Theory Code: 62 46 20 01 Reviewer 1: Assoc. Lam Quoc Anh Reviewer 2: Dr. Nguyen Ba Thi Reviewer 3: Assoc. Nguyen Ngoc Hai Anonymous Reviewer 1: Assoc.

Lam Quoc Anh Anonymous Reviewer 2: Dr. Ha Binh Minh SCIENTIFIC SUPERVISOR: Prof. PHAN QUOC KHANH Ho Chi Minh City - 2015 & % To the memory of my mother. To my beloved children, Delta and Lambda.

Declaration I hereby declare that this dissertation, done under the supervision of Professor Phan Quoc Khanh, is entirely the result of my own work, and this study and its findings have never been published by any other researchers. I also obtained the consent of Nguyen Hong Quan, co-author of the joint paper [43] referred to in Chapter 2, to let me include in this thesis some of the results of the said joint paper, which were not included in his thesis defended two years ago. Ho Chi Minh City, november, 2015 The author Vo Si Trong Long i Acknowledgements First and foremost, I would like to express my sincere gratitude to my supervisors, Assoc. Tran Thi Hue Nuong and Prof.

Phan Quoc Khanh, who introduced me to the topic and provided me with continuous guidance throughout the course of this research. Second, I am thankful to the University of Science, Vietnam National University Ho Chi Minh City, for providing me with favorable working conditions and facilities during all the time of my PhD program. Many thanks also go to all the members of the Optimization Group of Southern Vietnam, especially Dr. Nguyen Hong Quan.

His comments and advice have been crucial for my research. Finally, I could have not done this work without my family’s support. I would like to thank my father, my wife, and all the family members for their love and encouragement. Ho Chi Minh City, november, 2015 The author Vo Si Trong Long ii Contents Declaration i Acknowledgements ii List of Symbols and Notations v Preface vii 1.

Background and Preliminaries 1 1. Basic definitions and properties. Abstract convexity structures and generalized KKM mappings. Stampacchia-type vector quasiequilibria.

Nash equilibria for non-cooperative games. Constrained minimization and maximization. Several existence theorems in nonlinear analysis related to generalized KKM mappings and applications 14 2. Existence theorems and applications to optimization-related problems.

Optimization-related problems. Existence theorems on product GFC-spaces and applications. Existence theorems on product GFC-spaces. Fixed points and existence of solutions of optimization-related prob- lems 35 3.

Locally GFC-spaces. Fixed points and coincidence points. Existence of solutions of optimization-related problems. The weak finite intersection property and characterizations of the solution existence in optimization 55 4.

The weak finite intersection property and its characterizations. Characterizations of the solution existence in optimization problems. Invariant-point theorems in metric spaces and applications to optimization- related problems 73 5. Invariant-point theorems in metric spaces.

Existence of solutions of variational relation and inclusion problems. 99 General Conclusions 100 List of the author’s papers related to the thesis 101 List of the author’s conference reports related to the thesis 102 References 103 iv List of Symbols and Notations 2X the family of all nonempty subsets of a set X hY i the family of nonempty finite subsets of a set Y clA the closure of a set A intA the interior of a set A coA the convex hull of a set A N the set of the natural numbers Q the set of the rational numbers R the set of the real numbers R = R ∪ {−∞, +∞} the set of the extended real numbers Rn a n-dimensional Banach space (X, d) a metric space |·| the absolute value ∅ the empty set ∆n the standard n-simplex ∆k a k-face of ∆n ∆M the face of ∆n corresponding to a finite set M (X, Y, Φ) or (X, Y, {ϕN }) a finite continuous topological space (GFC-space) (X, Y, U, Φ) a locally-GFC-space KKM (X, Y, Z) the class of the mappings enjoying the weak KKM property [·, ·] the close interval (·, ·) the open interval (·, ·] and [·, ·) the half-open intervals C[a,b] the space of the continuous functions on [a, b] v {xα } the net or sequence with elements xα xα → x xα converges to x N (x) the class of all the neighborhoods of an element x f :X→Y a single-valued mapping from X to Y F : X → 2Y a set-valued mapping from X to Y F − : Y → 2X the inverse mapping of F : X → 2Y F ∗ : Y → 2X the dual mapping of F : X → 2Y limsupx0 →x f (x0 ) infV ∈N (x) supx0 ∈V f (x0 ) for f : X → R liminfx0 →x f (x0 ) supV ∈N (x) infx0 ∈V f (x0 ) for f : X → R grF the graph of F ∀ for all ∃ there exists 2 the end of a proof vi Preface Optimization studies contains various topics. However, the existence of solutions of optimization problems and optimization-related problems such as variational inequal- ities, equilibrium problems, minimax problems, etc, always take a central place in the optimization theory. Main tools used to study the existence of solutions of a problem are existence theorems for important points in nonlinear analysis such as fixed points, coincidence points, maximal points, intersection points, etc.

One of the most celebrated existence theorems in nonlinear analysis is the classical KKM theorem (KKM being an abbreviation of Knaster-Kuratowski-Mazurkiewicz; the theorem is known also as the Three Polish lemma) Knaster-Kuratowski-Mazurkiewicz [56] in finite-dimensional spaces, which is equivalent to the well-known Brouwer fixed- point theorem and the Sperner lemma. There have been many extensions and appli- cations of this KKM theorem in optimization theory. First, it was extended to general topological vector spaces by Fan [25]. Later, we will mention the subsequent main direction of developing this result.

Various generalized linear/convex structures have been proposed and corresponding types of KKM mappings have been defined together with these spaces. Lassonde [58] introduced convex spaces and obtained many funda- mental results. A well known generalized convex structure was developed by Horvath [34] by replacing a convex hull with a continuous map from a simplex to a contractible set. Park-Kim [71] proposed the notation of a G-convex space.

Ding [18] introduced the concept of a FC-space and then Khanh-Quan [48] generalized and unified the previous spaces into a notion called a GFC-space. The Fan-Glicksberg fixed-point theorem in Fan [24] and Glicksberg [29], a general- ization to locally convex spaces of the well-known Kakutani fixed-point theorem, and vii many classical fixed-point results, have been also developed by various researchers. A major attention has been paid to generalizing it to abstract convex spaces. The first result of this direction appeared in Himmelberg [35].

Later, Tarafdar [73] dealt with the case of a locally H-convex space. So far, it has been considered for other general spaces such as locally G-convex uniform spaces and locally FC-spaces, etc. Moreover, it has been also extended in another way by replacing the convex-valuedness of a map with more general conditions (see for example Jeng-Hsu-Huang [37]; Khanh-Long [41] and Kuo-Jeng-Huang [57]). Another kind of existence theorems, which has also been attracting increasing at- tention, is the invariant-point theorem in Dancs-Hegedus-Medvegyev [17], which is equivalent to the Ekeland variational principle Ekeland [22].

Several efforts have been made to establish sufficient conditions for the existence of solutions of optimization- related problems by using this result (see Khanh-Lin-Long [39]; Lin-Chuang [60] and Lin-Chuang-Wang [62]). On the other hand, many contributions have been made in recent decades to intro- duce general setting problems related to optimization and then deriving as consequences corresponding results for particular cases. For these problems, the above existence the- orems have been employed as crucial mathematical tools to achieve existence results. Among such general models, variational relation problems and variational inclusion problems have been attracting increasing interests of researchers.

Variational relation problems were investigated first in Luc [63] for existence and in Khanh-Luc [44] for sta- bility and have been developed in Balaj-Lin [8, 9]; Khanh-Long [40] and Khanh-Long- Quan [43]. It is worth stressing that this model contains many important problems in optimization such as constrained minimization, variational inequalities, equilibrium problems, minimax problems, Nash equilibria, etc. The variational inclusion problem was proposed in Hai-Khanh-Quan [31] and has also been used, e., in Khanh-Long- Quan [43] and Khanh-Quan [49]. This problem also includes special cases, as many as the variational relation model.

The major focus of this thesis is to develop several existence theorems in nonlin- ear analysis in order to apply them to considering the existence of solutions of various viii optimization-related problems as mentioned above. It includes five chapters. We begin, in Chapter 1, with several basic concepts, definitions and preliminary facts for our later use. In chapter 2, we consider some continuous selection results, and use them to estab- lish purely topological-based sufficient conditions for the existence of important points in nonlinear analysis and prove the equivalence of these conditions.

Then, we apply them to investigating the solution existence of optimization-related problems stated in Chapter 1. The results in Chapter 2 were published in Khanh-Long-Quan [43] and Khanh-Lin-Long [39]. Chapter 3 is devoted to an extension of the Fan-Glicksberg fixed- point theorem and applications to problems concerning optimization. The contents of this chapter are based on the paper Khanh-Long [41].

In Chapter 4, we propose a gen- eral notion, called FI-map (finite-intersection map) and use it to establish necessary and sufficient conditions for the solution existence of various optimization-related prob- lems such as variational relation problems, minimax problems, saddle-point problems, etc, in pure topological settings. This chapter includes the results of the submitted paper Khanh-Long [42]. Finally, Chapter 5 develops several invariant-point theorems in metric spaces also as tools for establishing conditions for the existence of solutions to many of the above-mentioned problems. However, applications of results on invariant points are based on techniques quite different from those on other encountered points.

Chapter 5 also contains discussions about the equivalence of these conditions. Many examples and comparison remarks are provided to illustrate advantages of the obtained results. Chapter 5 is written by using results of the paper Khanh-Long [40] which has already been published. ix Chapter 1 Background and Preliminaries 1.1 Basic definitions and properties In this thesis, we use standard notations.

For a set X, by 2X and hXi we denote the family of all the subsets and the family of all the nonempty finite subsets, respectively (resp), of X. If X is a topological space and A ⊂ X, then intA and clA signify the interior and closure, resp, of A. Let X and Y be nonempty sets. For F : Y → 2X , we define F − : X → 2Y and F ∗ : X → 2Y , resp, by F − (x) = {y ∈ Y : x ∈ F (y)} and F ∗ (x) = Y \ F − (x).

F − and F ∗ are called the inverse and dual, resp, map of F. grF := {(y, x) ∈ Y × X : x ∈ F (y)} denotes the graph of F. N, Q, and R stand for the set of the natural numbers, the rational numbers, and the real numbers, resp, and R = R ∪ {−∞, +∞}. For n ∈ N, ∆n signifies the standard n-simplex, whose vertices are the points e0 = (1, 0, .1 Let X be a topological space.