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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 Prof Dr Lam Quoc Anh Reviewer 2: Dr Nguyen Ba Thi Reviewer 3: Assoc Prof Dr Nguyen Ngoc Hai Anonymous Reviewer 1: Assoc Prof Dr Lam Quoc Anh Anonymous Reviewer 2: Dr Ha Binh Minh SCIENTIFIC SUPERVISOR: Prof DSc 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 Prof 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 Background and Preliminaries 1.1 Basic definitions and properties 1.2 Abstract convexity structures and generalized KKM mappings 1.3 Problem settings 1.3.1 Variational relations 1.3.2 Quasivariational inclusions 10 1.3.3 Stampacchia-type vector quasiequilibria 10 1.3.4 Nash equilibria for non-cooperative games 11 1.3.5 Traffic networks 11 1.3.6 Constrained minimization and maximization 12 1.3.7 Saddle points 13 Several existence theorems in nonlinear analysis related to generalized KKM mappings and applications 14 2.1 Existence theorems and applications to optimization-related problems 15 2.1.1 Existence theorems 15 2.1.2 Optimization-related problems 22 iii 2.2 Existence theorems on product GFC-spaces and applications 28 2.2.1 Existence theorems on product GFC-spaces 28 2.2.2 Applications 31 2.3 Conclusions 34 Fixed points and existence of solutions of optimization-related problems 35 3.1 Locally GFC-spaces 36 3.2 Fixed points and coincidence points 39 3.3 Existence of solutions of optimization-related problems 46 3.4 Conclusions 54 The weak finite intersection property and characterizations of the solution existence in optimization 55 4.1 The weak finite intersection property and its characterizations 56 4.2 Characterizations of the solution existence in optimization problems 64 4.3 Conclusions 72 Invariant-point theorems in metric spaces and applications to optimizationrelated problems 73 5.1 Problem settings 74 5.2 Invariant-point theorems in metric spaces 76 5.3 Existence of solutions of variational relation and inclusion problems 80 5.4 Applications 85 5.5 Conclusions 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 Y the family of all nonempty subsets of a set X 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 limsupx →x f (x ) infV ∈N (x) supx ∈V f (x ) for f : X → R liminfx →x f (x ) supV ∈N (x) infx ∈V f (x ) for f : X → R grF the graph of F ∀ for all ∃ there exists ✷ 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 inequalities, 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 fixedpoint theorem and the Sperner lemma There have been many extensions and applications 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 fundamental 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 generalization to locally convex spaces of the well-known Kakutani fixed-point theorem, and vii H(¯ x, x¯, k) ⊂ −C(¯ x) \ {0} Therefore, there exists x¯ ∈ S1 (¯ x) such that, ∀y ∈ S2 (x), ¯ ⊂ −C(¯ ∃k¯ ∈ K(¯ x, y), H(¯ x, y, k) x) \ {0} ✷ The following three examples illustrate advantages of Corollary 5.4.6 over known results from the literature Example 5.4.5 Let X = [0,2], Z = Z = R, C(x) ≡ (−∞; 0], {0} if x = 0, (0, x] if x ∈ (0, 1], S1 (x) = [x, 2) if x ∈ (1, 2], {2} if x = 2, {0, 2} if x = 0, S2 (x) = [0, x] if x = 0, K(x, y) = [−(x + y), x + y], H(x, y, k) = [kx − ky, +∞) It is easy to see that the assumptions of Corollary 5.4.6 are satisfied By direct checking one sees that the solution set is the whole X = [0, 2] However, the results of [31, 63, 64] are not applicable, since many assumptions are not satisfied, e.g., S1 (x) is not closed for each x different from 0, 2, coS2 (x) ⊂ S1 (x) and S1 (x) = X for each x ∈ X, etc Example 5.4.6 Let X = [0, +∞), Y= Z = R, C(x) ≡ (−∞; 0], {0, 2} if x = 0, S1 (x) = S2 (x) = [0, x] if x = 0, K(x, y) = {x + y}, H(x, y, k) = [x − y, +∞) The assumptions of Corollary 5.4.6 are easily seen to be fulfilled Direct calculations yield the unique solution But, Theorem 3.1 of [27] and Theorem 3.1 of [79] are out of use, since S1 (0)) is not convex Example 5.4.7 Let X = [0, 1] ∪ [2, 3], Y = Z = R, C(x) ≡ (−∞; 0], S1 (x) ≡ S2 (x) ≡ X, K(x, y) = {x + y}, H(x, y, k) = {x − y} 96 Corollary 5.4.6 is easily employed to see the existence of solutions in this case By direct checking one sees that there is a unique solution But, Theorem of [5], Theorem 3.1 of [9], Theorem 3.4 of [14], Theorem 2.1 of [36], and Theorem 2.1 of [38] cannot be applied, since X is not convex Corollary 5.4.7 Assume for problem (QEP3 ) that (i) for all x ∈ X, x ∈ S2 (x) and ∃k ∈ K(x, x) such that H(x, x, k) ∩ C(x) = ∅; (ii) for all x ∈ X, the set {y ∈ S2 (x) : ∃k ∈ K(x, y), H(x, y, k) ∩ C(x) = ∅} is closed; (iii) if y ∈ S2 (x) and ∃k ∈ K(x, y), H(x, y, k) ∩ C(x) = ∅ and if z ∈ S2 (y) and ∃k ∈ K(y, z), H(y, z, k)∩C(x) = ∅, then z ∈ S2 (x) and ∃k ∈ K(x, z), H(x, z, k)∩ C(x) = ∅; (iv) if xn+1 ∈ S2 (xn ) and ∃k ∈ K(xn , xn+1 ), H(xn , xn+1 , k) ∩ C(xn ) = ∅ for all n, then limn→∞ d(xn , xn+1 ) = 0; (v) for each x ∈ X, there exists y ∈ S1 (x) ∩ S2 (x) and ∃k ∈ K(x, y) with H(x, y, k) ∩ C(x) = ∅ Then, problem (QEP3 ) is solvable Proof Apply Theorem 5.3.2 with ri = r3 , αj = α1 , and G(x, k) = C(x) Clearly, all the assumptions of Theorem 5.3.2 are satisfied Hence, there exists x¯ ∈ S(¯ x) such that, for all y ∈ S2 (¯ x) \ {¯ x} and k ∈ K(¯ x, y), H(¯ x, y, k) ∩ C(¯ x) = ∅ ✷ Example 5.4.8 Let X = Z = [1, 3], Z = R, C(x) ≡ [1, +∞), {2, 3} if x = 3, S1 (x) = [x, 3) if x = 3, {2, 3} if x = 3, S2 (x) = [x, 3] if x = 3, K(x, y) = {x}, H(x, y, k) = {x − y} Then, the assumptions of Corollary 5.4.7 are easy to be checked We can also verify directly that x¯ = is a solution If we denote Sˆ2 (x) = S2 (x) \ {x} to have 97 a usual quasiequilibrium problem and apply recent existing results, we will see that Theorem 3.3 of [30], Theorem 2.1, Corollary 3.2 of [31], and Theorem 3.1 of [63] are not applicable, since S1 (.) is not closed, coSˆ2 (x) ⊂ S1 (x) for each x = Theorem 5.2 of [61] does not work because S1 (x) = X, ∀x ∈ X \ {1} (if x = then is not a solution) Furthermore, since S1 (x) = X for each x, Theorem 4.1 of [64] is also out of use Moreover, If we consider the problem with S2 (x) in the place of S2 (x) \ {x}, then solutions not exist Corollary 5.4.8 Assume for problem (QEP4 ) that (i) for all x ∈ X, x ∈ S2 (x) and ∀k ∈ K(x, x), H(x, x, k) ∩ −C(x) = {0}; (ii) for all x ∈ X, the set {y ∈ S2 (x) : ∀k ∈ K(x, y), H(x, y, k) ∩ −C(x) = ∅} is closed; (iii) if y ∈ S2 (x) and ∀k ∈ K(x, y), H(x, y, k) ∩ −C(x) = ∅ and if z ∈ S2 (y) and ∀k ∈ K(y, z), H(y, z, t) ∩ −C(y) = ∅, then z ∈ S2 (x) and ∀k ∈ K(x, z), H(x, z, k) ∩ −C(x) = ∅; (iv) if xn+1 ∈ S2 (xn ) and ∀k ∈ K(xn , xn+1 ), H(xn , xn+1 , k) ∩ −C(xn ) = ∅, for all n, then limn→∞ d(xn , xn+1 ) = 0; (v) for each x ∈ X, there exists y ∈ S1 (x) ∩ S2 (x) such that, H(x, y, k) ∩ −C(x) = ∅ for all k ∈ K(x, y) Then, problem (QEP4 ) has solutions Proof We simply apply Theorem 5.3.2 with ri = r1 , αj = α2 , and G(x, k) = −C(x) ¯ ∩ Then, there exists x¯ ∈ S1 (¯ x) such that, ∀y ∈ S2 (¯ x) \ {¯ x}, ∃k¯ ∈ K(¯ x, y), H(¯ x, y, k) −C(¯ x) = ∅ By (i), H(¯ x, x¯, k) ∩ −C(¯ x) = {0} for all k ∈ K(¯ x, x¯) Hence, for all k ∈ K(¯ x, x¯), H(¯ x, x¯, k) ∩ [−C(¯ x) \ {0}] = ∅ So, there exists x¯ ∈ S1 (¯ x) such that, ∀y ∈ S2 (¯ x), ∃k¯ ∈ K(¯ x, y), H(¯ x, x¯, k) ∩ [−C(¯ x) \ {0}] = ∅ ✷ Finally, we apply Corollary 5.4.8 to a more practical problem of constrained minimization Let X be a complete metric space, and f : X → R and g : X × X → R be two real functions We consider the following constrained minimization problem (MP) find x¯ ∈ X such that f (y) ≥ f (¯ x) for all y satisfying g(¯ x, y) ≤ 98 Corollary 5.4.9 Assume for problem (MP) that (i) f is l.s.c.; (ii) for each x ∈ X, g(x, ) is l.s.c and g(x, x) ≤ 0; (iii) if g(x, y) ≤ and g(y, z) ≤ 0, then g(x, z) ≤ 0; (iv) if g(xn , xn+1 ) and f (xn+1 ) − f (xn ) ≤ 0, ∀n, then limn→+∞ d(xn , xn+1 ) = Then, problem (MP) has solutions Proof Employ Corollary 5.4.8 with S1 (x) ≡ X, S2 (x) = {y ∈ X : g(x, y) ≤ 0}, H(x, y, k) = f (y) − f (x), and C(x) ≡ [0, +∞) for all x, y ∈ X It is easy to see that (i) and (v) of Corollary 5.4.8 are fulfilled Since f and g(x, ) are l.s.c, the set {y ∈ S2 (x) : f (y) − f (x) ≤ 0} is closed for each x ∈ X So, (ii) of Corollary 5.4.8 is satisfied If g(x, y) ≤ and f (y) − f (x) ≤ 0, and if g(y, z) ≤ and f (z) − f (y) ≤ 0, then g(x, z) ≤ and f (z) − f (x) ≤ Thus, (iii) of Corollary 5.4.8 is fulfilled Finally, (iv) implies (iv) of Corollary 5.4.8 By Corollary 5.4.8, there is x¯ ∈ S1 (¯ x) such that, ∀y ∈ S2 (¯ x), F (¯ x, y, t¯)∩[−C(¯ x)\{0}] = ∅, i.e., there exists x¯ ∈ X such that f (y) ≥ f (¯ x) ✷ for all y ∈ X satisfying g(¯ x, y) ≤ 5.5 Conclusions This chapter considers some equivalent versions of invariant-point theorems The equivalence between these results and the above-mentioned invariant-point theorems are also proved Next, they are employed to derive sufficient conditions for the existence of solutions for two general models of variational relations and inclusions, and various optimization-related problems In applications, the consequences of these results on a wide range of particular cases are discussed, from relatively general inclusion problems to classical results such as Ekeland’s variational principle, and practical situations such as traffic networks and non-cooperative games, to illustrate application possibilities of our general results Many examples are provided to explain the advantages of the obtained results and also to motivate in detail our problem settings 99 General Conclusions The main scope of this thesis is to develop several existence theorems in nonlinear analysis and applications to optimization-related problems It includes the following Several theorems on continuous selections, fixed points, sectional points, intersection points, collectively fixed points, collectively coincidence points, etc, and their applications in various optimization-related problems are presented A new extension of the Fan-Glicksberg fixed-point theorem is proved Sufficient conditions for the existence of a fixed point of a composition, and of a coincidence point are provided Full characterizations of the weak finite intersection property are obtained Necessary and sufficient conditions for the solution existence of variational relation problems are established without KKM-structures or connectedness structures (unlike [50, 51], where they were largely used) To stress applications possibilities of our results, characterizations of the solution existence for the repeatedly considered typical particular optimization problems are derived (of course they are different from the results in the other places of the thesis) Some equivalent versions of invariant-point theorems are demonstrated Sufficient conditions for the solution existence again of the two general models of variational relations and inclusions are obtained The results obtained in the thesis are new or improve/include many known ones in the literature 100 List of the author’s papers related to the thesis Khanh P Q., Lin L J., Long V S T (2014), On topological existence theorems and applications to optimization-related problems, Math Meth Oper Res., 79, pp 253-272 Khanh P Q., Long V S T (2014), Invariant-point theorems and existence of solutions to optimization-related problems, J Global Optim., 58, pp 545-564 Khanh P Q., Long V S T., Fixed points, continuous selections, and existence of solutions of optimization-related problems, submitted to Journal of the Australian Mathematical Society Khanh P Q., Long V S T., The weak finite intersection property and characterizations of the solution existence in optimization, submitted to Topological Methods in Nonlinear Analysis Khanh P Q., Long V S T, Quan N H (2011), Continuous selections, collectively fixed points and weakly Knaster-Kuratowski-Mazurkiewicz mappings in Optimization, J Optim Theory Appl., 151, pp 552-572 101 List of the author’s conference reports related to the thesis Khanh P Q., Long V S T., Fixed-point, coincidence-point and alternative theorems in locally GFC-uniform spaces and applications, The 8th Workshop on Optimization and Computing, Ba Vi, Vietnam, April 20-23 (2010) Khanh P Q., Long V S T., Quan N H., Continuous selections, collectively fixed points and weakly T -KKM theorems in GFC-spaces, CIMPA-UNESCO-MICINNVIETNAM research School “Variational inequalities and related problems” Ha Noi, Vietnam, May 10-21 (2010) Long V S T., General 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The variational inclusion problem was proposed in Hai-Khanh -Quan [31] and has also been used, e.g., in Khanh-LongQuan [43] and Khanh -Quan [49] This problem also includes special cases, as many... path flow x¯ is called a strong equilibrium flow if (i) is satisfied with ∃z ∈ L(¯ x) replaced by ∀z ∈ L(¯ x) In [45, 46], it is proved that a feasible path flow x¯ is a strong (or weak) equilibrium... 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
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