VIETNAM NATIONAL UNIVERSITY - HCMC UNIVERSITY OF SCIENCE Nguyen Hong Quan EXISTENCE THEOREMS IN NONLINEAR ANALYSIS AND APPLICATIONS TO OPTIMIZATION-RELATED MODELS PhD THESIS IN MATHEMATICS Ho Chi Minh City - 2013 VIETNAM NATIONAL UNIVERSITY - HCMC UNIVERSITY OF SCIENCE Nguyen Hong Quan EXISTENCE THEOREMS IN NONLINEAR ANALYSIS AND APPLICATIONS TO OPTIMIZATION-RELATED MODELS Major: Mathematical Optimization Codes: 62 46 20 01 Referee 1: Assoc.Prof Dr Nguyen Dinh Referee 2: Dr Huynh Quang Vu Referee 3: Assoc.Prof Dr Lam Quoc Anh Independent Referee 1: Prof D.Sc Nguyen Dong Yen Independent Referee 2: Assoc.Prof Dr Mai Duc Thanh SCIENTIFIC SUPERVISOR Professor Phan Quoc Khanh Ho Chi Minh City - 2013 Confirmation I confirm that all the results of this thesis come from my work under the supervision of Professor Phan Quoc Khanh They have never been included in other papers in the literature Moreover, the paper (Q5) is a joint work with Dr Nguyen Xuan Hai and we have discussed in detail, and based on the contributions of each coauthor we have agreed that its whole content is used in this thesis, while the whole joint paper (Q1) was included in the thesis of Nguyen Xuan Hai in 2007 October 2013 Nguyen Hong Quan Contents Preface IV Existence theorems in nonlinear analysis and applications 1.1 Notions and definitions 1.2 Existence theorems in GFC-spaces 1.3 Applications 1.3.1 Variational inclusions 1.3.2 Minimax theorems 1 23 23 31 Nonlinear existence theorems for mappings on product GFC-spaces and applications 2.1 Existence theorems on product GFC-spaces 2.2 Applications 37 37 42 Topological characterizations of existence in nonlinear analysis and optimization-related problems 3.1 Topological existence theorems 3.2 The existence of solutions to optimization-related problems 3.2.1 Variational relation problems 3.2.2 Invariant-point theorems 3.2.3 Equilibrium problems of the Stampacchia and Minty types 3.2.4 Minimax theorems 3.2.5 Nash equilibria 52 52 70 70 74 77 80 88 Generic stability and essential components of generalized KKM points and applications 92 4.1 Notions and definitions 92 4.2 Generic essentialness of T -KKM mappings 94 4.3 Essential components of sets of T -KKM points 98 4.4 Applications to maximal elements and variational inclusions 99 4.4.1 Essential components of T0 -maximal elements 100 4.4.2 Essential components of solutions to variational inclusions 100 Contents III 4.4.3 Essential components of particular cases of variational inclusions 101 Conclusion 104 List of the author’s papers related to the thesis 105 List of the author’s conference reports related to the thesis 106 References 107 Glossary of Symbols 111 Preface The existence theorems, including theorems about various points like fixed points, coincidence points, intersection points, maximal elements, and other results as KKM theorems, minimax theorems, etc., constitute one of the most important parts of mathematics They are crucial tools in the solution existence study of wide-ranging problems of optimization and applied mathematics The existence theorems have a long history of development passing more than a century with the following major milestones: the Brouwer fixed point theorem (1912, [14]), Classical KKM principle (1929, [57]), Kakutani fixed point theorem (1941, [50]), KKM-Fan principle (1961, [33]) From 80s of the 20th century, to meet demands of practical situations, many classes of problems in optimization have appeared One of the first and most important issues of such a class is to know if solutions exist or not This requires more new effective mathematical tools Hence, existence theorems have been intensively developed to response that requirement Especially, in recent years the theory of existence theorems has obtained many significant achievements Like other mathematical theories, the existence theorems have been built from simple basic results by generalization and abstraction methods In early forms of these fundamental results, the convexity played a central role in formulating results Therefore, most of the later results mainly focus on improving assumptions on convexity or replacing them by purely topological assumptions According to our observations, for the last three decades the existence theorems have been developed in three ways First, some researchers renovated classical notions of convexity based on linear structures For instance, the KKM-Fan mapping (Fan [33]) was extended into the S-KKM mapping (e.g., Chang and Zhang [15], Chang and Yen [16], Chang, Huang, Jeng and Kuo [17]) In terms of these notions, new existence results were achieved Second, many authors replaced the classical convexity by abstract convexity notions, not using linear structures, and they extended earlier existing notions and results to these structures In the framework of this research direction, types of spaces with generalized convexity structures were proposed and studied Started with Lassonde ([60]) where a convex space was created, the following spaces (in the chronological order) were in- Preface V troduced: S-contractible spaces (Horvath [45]), H-spaces (Horvath [46-47]), Gconvex spaces (Park and Kim [79-85]), and FC-spaces (Ding [24-32]) These spaces have been used in studying existence theorems and nonlinear problems In the third approach, a number of authors (e.g., Wu [93], Tuy [91-92], Geraghty and Lin [38], Kindler and Trost [54], Kindler [55-56], Konig [59], Tarafdar and Yuan [90]) proved existence results which did not require any convexity structures, where convexity conditions were replaced by connectedness conditions which are purely topological conditions One of the main purposes of this thesis is to develop further the theory of existence theorems, focusing on the last two approaches Based on analyzing the earlier notions and results, we introduce several new structures and use them to formulate new definitions and establish new or more general results By providing illustrative examples, we show the existence of our structures and their usefulness in many applications Moreover, our notions and results in this thesis improve or include as special cases a number of known notions and results It is worthwhile noticing that the equivalence of mathematical theorems is meaningful in applications because it allows us to approach to a problem from different angles Therefore, researchers pay much attention on proving equivalence relations between existence results For examples, in Ha [40], an extension of Kakutani fixed-point theorem was proved to be equivalent to a section theorem and a minimax result, an equivalence between the KKM-Fan theorem and a Browder-type fixed-point theorem was shown in Tarafdar [89] Later on many researchers discovered similar equivalence relations for other kinds of results like coincidence theorems, matching theorems, intersection theorems, maximal-point theorems, section theorems and some geometric results One of our attempts in this thesis is to show the equivalence between many of our existence theorems On the other hand, any mathematical result needs be applicable for certain situations Typical applications of existence theorems are in optimization problems Therefore, using our existence theorems to establish solution existence results for optimization-related models is also a purpose of this thesis The harvested results include many new solution existence theorems for various problems as minimax problems, equilibrium problems, generalized inclusions, variational relation problems, or practical problems as traffic networks, Nash equilibria, abstract economies, etc One of the important topics in nonlinear optimization, which have been attracting many mathematicians recently, is properties of solution sets and solution maps The properties of solution sets such as closedness, connectedness, convexity, etc, were studied in many papers (e.g., Fort [37], Jones and Gowda [49], Khanh and Luc [53], Papageorgiou and Shahzad [78], Rapcsak [86], Zhong, Huang and Wong [105]) The properties of solution maps, like semicontinuities, continuity, differentiability, etc., which are commonly called the stability properties, have also been intensively investigated during recent years (e.g., Anh and Khanh [1-7], Khanh and Luc [53], Xiang, Liu and Zhou [94], Yang and Yu [96], Preface VI Yu and Xiang [98], Yu, Yang and Xiang [99], Zhou, Xiang and Yang [101]) Based on relationships between sets of particular points of set-valued maps and solution sets of optimization problems, we propose solution map notions for these points, consider their stability and apply the obtained results to optimization problems Stability issues considered in this thesis are included in the generic stability study The thesis consists of four chapters and contains the results of 10 papers (from the list of 14 related papers of the author): Chapter 1: “Existence theorems in nonlinear analysis and applications” is based on the papers (Q2), (Q3), (Q4), (Q6), (Q7); Chapter 2: “Nonlinear existence theorems for mappings on product GFCspaces and applications” is based on the paper (Q5); Chapter 3: “Topological characterizations of existence in analysis and optimization related problems” is based on the papers (Q10), (Q11), (Q12); Chapter 4: “Generic stability and essential components of generalized KKM points and applications” is based on the paper (Q8) Acknowledgments I express my deep gratitude to Professor Phan Quoc Khanh, my supervisor, for a continuous guidance, encouragement and valuable suggestions I would like to thank very much the University of Science of Hochiminh City for providing me all conditions and facilities for my work I am also indebted to the Vietnam Institute for Advanced Study in Mathematics (VIASM) and its members During my stay there as a visiting young researcher, they facilitated me with both a financial support and a perfect research environment for the completion of a part of this thesis Ho Chi Minh City, October 2013 Nguyen Hong Quan Existence theorems in nonlinear analysis and applications In this chapter, we propose a definition of GFC-spaces to encompass G-convex spaces, FC-spaces and many earlier existing spaces with generalized convexity structures Existence theorems are then established with underlying structures being GFC-spaces under relaxed assumptions These results contain, as properly particular cases, a number of counterparts which were recently developed in the literature As applications, using these results we prove the existence of solutions to a general variational inclusion problem, which contains most of the existing results of this type, and develop in detail general types of minimax theorems Examples are given to explain advantages of our results 1.1 Notions and definitions We recall notions used in the whole thesis Let Y be a nonempty set, Y stands for the set of all finite subsets of Y For n ∈ N, the set of all natural numbers, ∆n stands for the n-simplex with the vertices being the unit vectors e0 = (1, 0, , 0), e1 = (0, 1, , 0), , en = (0, 0, , 1) of a basis of Rn+1 For N = {y0 , y1 , yn } ∈ Y and M = {yi0 , yi1 , , yik } ⊂ N, let ∆|N| ≡ ∆n , and ∆M ≡ ∆k be the face of ∆|N| corresponding to M, i.e., ∆M = co{ei0 , ei1 , eik } If A, B ⊂ X, X being a topological space, then A (or clA), AB (or clB A), intA, intB A and Ac signify the closure, closure in B, interior, interior in B and complement X \ A, respectively (shortly, resp), of A Let X, Y be nonempty sets and F : X ⇒ Y be a set-valued map For x ∈ X and y ∈ Y , the sets F(x), F −1 (y) = {x ∈ X | y ∈ F(x)} and F ∗ (y) = X \ F −1 (y) are called an image, a fiber (or inverse image) and a cofiber, resp The map F −1 (F ∗ ) is called the inverse map (dual map, resp) of F The graph of F is GphF := {(x, y) ∈ X × Y | y ∈ F(x)} Now let X and Y be topological spaces, F : X ⇒ Y , and f : X → R F is called closed (open, resp) if its graph is closed (open, resp) F is said to be upper semicontinuous (usc, for short) (resp, lower semicontinuous (lsc)) if for each open (resp, closed) subset U of Y , the set {x ∈ X | F(x) ⊂ U} is open (resp, closed) F is said to be continuous if it is both usc and lsc f is said to be usc (resp, lsc), if, for all α ∈ R, the set {x ∈ X | f (x) ≥ α} (resp, {x ∈ X | f (x) ≤ α}) is closed 1.1 Notions and definitions The following concepts are taken from [22, 25] A subset A of a topological space X is called compactly open (compactly closed, resp) if, for each nonempty compact subset K of X, A ∩ K is open (closed, resp) in K The compact interior and compact closure of A are defined by, resp, cintA = {B ⊂ X : B ⊂ A and B is compactly open in X}, cclA = {B ⊂ X : B ⊃ A and B is compactly closed in X} F : Y ⇒ X is called transfer open-valued (transfer closed-valued, resp) if ∀y ∈ Y , ∀x ∈ F(y) (∀x ∈ / F(y), resp), ∃y ∈ Y such that x ∈ int(F(y ) (x ∈ / cl(F(y ), resp) F is termed transfer compactly open-valued (transfer compactly closed-valued) if ∀y ∈ Y , ∀K ⊂ X: nonempty and compact, ∀x ∈ F(y) ∩ K (∀x ∈ / F(y) ∩ K), ∃y ∈ Y such that x ∈ cintF(y ) (x ∈ / cclF(y ), resp) Of course intA ⊂ cintA (clA ⊃ cclA), and transfer open-valuedness (transfer closed-valuedness) implies transfer compact open-valuedness (transfer compact closed-valuedness, resp) Moreover, a set-valued mapping has open values (closed values) then it is transfer openvalued (transfer closed-valued) We will use these notions in order to compare our results directly with many known existing ones Lemma 1.1.1 (e.g., [22]) Let Y be a set, X be a topological space and F : Y ⇒ X The following statements are equivalent (i) F is transfer compactly closed-valued (transfer compactly open-valued, resp); (ii) for each compact subset K ⊂ X, y∈Y y∈Y (K ∩ intK F(y)), resp (K ∩ cintF(y)) = (K ∩ F(y)) = y∈Y y∈Y (K ∩ clK F(y)) (K ∩ cclF(y)) = (K ∩ F(y)) = y∈Y y∈Y We propose the following definition of a GFC-space to unify a number of earlier existing notions of spaces with generalized convexity structures, but without linear structures This notion is proposed based on observing that although the abstract convexity structures associated with the earlier existing spaces such as convex spaces ([60]), H-spaces ([46-47]), G-convex spaces ([79-85]), FC-spaces ([24-32]) are different, all of them use the image of a simplex through a continuous map Definition 1.1.1 Let X be a topological space, Y be a nonempty set and Φ be a family of continuous mappings ϕ : ∆n → X, n ∈ N Then a triple (X,Y, Φ) is said to be a generalized finitely continuous topological space (GFC-space in short) if for each finite subset N ∈ Y , there is ϕN : ∆|N| → X of the family Φ (we also use (X,Y, {ϕN }) to denote (X,Y, Φ)) 4.3 Essential components of sets of T -KKM points 98 4.3 Essential components of sets of T -KKM points We have seen from the preceding section that, for u ∈ M and KKM-map G : M ⇒ Z, if u ∈ Q, i.e., u is essential, then every point of G(u) is essential (and Q is a dense residual set of M ) Now we will show that, if u is nonessential, then there exists a connected essential set included in G(u) In this section we fix some T0 ∈ T (X,Y, Z) and the corresponding subspace M0 of M , i.e., M0 := u = (F, T0 )| F : Y ⇒ Z is T0 −KKM and compact−valued M0 , with the Hausdorff metric induced by that of M , is clearly a complete metric space Instead of (F, T0 ) ∈ M0 we will write simply F ∈ M0 For such an F, let G0 (F) = T0 (X) ∩ y∈Y F(y) Then we have a KKM-point map G0 : M0 ⇒ Z By Lemma 4.2.2, G0 is obviously usco Theorem 4.3.1 There exist essential components of G0 (F) for each F ∈ M0 Proof As G0 is usco, G0 (F) is an essential set of itself Let P be the family of all essential sets of G0 (F), ordered partially by the inclusion relation Then every decreasing chain of sets in P possesses a lower bound, which is the intersection of all these sets Hence Zorn’s lemma yields a minimal element µ(F) of P, which is essential Suppose µ(F) be not connected, i.e there were two nonempty closed subsets µ1 (F), µ2 (F) of µ(F) and two open subsets U1 ,U2 of Z, such that µ(F) = µ1 (F) ∪ µ2 (F), µ1 (F) ⊂ U1 , µ2 (F) ⊂ U2 , clU1 ∩ clU2 = / As µ(F) is the minimal essential set, both µ1 (F) and µ2 (F) are not essential Then, we can assume that, for all positive δ , there are F1 , F2 ∈ B(F, δ ) such that, for k = 1, 2, G0 (Fk ) ∩Uk = / (4.3.4) As µ(F) ⊂ U1 ∪U2 and is essential, there exists a positive δ , which is taken less than a := inf{ z − z : z ∈ U1 , z ∈ U2 } such that, for any F ∗ ∈ B(F, δ ), G0 (F ∗ ) ∩ (U1 ∪U1 ) = / (4.3.5) Fk ∈ B(F, δ /4) exist satisfying (4.3.4) for k = 1, We define F : Y ⇒ Z by, for y ∈ Y, F (y) = F1 (y) \U2 ∪ F2 (y) \U1 Clearly, F is compact-valued We will show that F ∈ M0 by checking that F is T0 -KKM Suppose F be not T0 -KKM, i.e there are N ∈ Y , M = {yi0 , yi1 , , yil } ⊂ N, x ∈ ϕN (∆M ) and z ∈ T0 (x), such that z ∈ / F (yi j ) for all j ∈ {0, 1, , l} Since U1 ∩U2 = 0, / z∈ / U1 or z ∈ / U2 As z ∈ / F2 (yi j ) \U1 , if z ∈ / U1 l we have z ∈ / j=0 F2 (yi j ), which contradicts the fact that F2 is T0 -KKM Similarly, if z ∈ / U2 , we also arrive at a contradiction Thus F ∈ M0 Next we show 4.4 Applications to maximal elements and variational inclusions 99 that h(F, F ) ≤ δ For z ∈ F (y), one has d z, F1 (y) = if z ∈ F1 (y) \ U2 If z ∈ F2 (y) \U1 , then d z, F1 (y) ≤ e F2 (y), F1 (y) ≤ h F1 (y), F2 (y) Hence e F (y), F1 (y) = supz∈F (y) d z, F1 (y) ≤ h F1 (y), F2 (y) (4.3.6) To get the corresponding inequality for e F1 (y), F (y) , we consider any t ∈ / U2 , i.e., t ∈ F1 (y) \ U2 , one has d t, F (y) = ≤ h F1 (y), F2 (y) F1 (y) If t ∈ If t ∈ U2 , t ∈ / U1 since U1 ∩U2 = / For all t ∈ F2 (y) ∩U1 , one has (by the definition of a) d t, F2 (y) ≤ h F1 (y), F2 (y) ≤ h(F1 , F2 ) ≤ h(F , F1 ) + h(F , F2 ) < δ /2 < a ≤ t − t Then, d t, F2 (y) = d t, F2 (y) \U1 Therefore, d t, F (y) ≤ d t, F2 (y) \U1 = d t, F2 (y) ≤ e F1 (y), F2 (y) ≤ h F1 (y), F2 (y) Thus, e F1 (y), F (y) = supt∈F (y) d t, F (y) ≤ h F1 (y), F2 (y) (4.3.7) From (4.3.6) and (4.3.7), we have h F (y), F1 (y) ≤ h F1 (y), F2 (y) for all y ∈ Y Hence h(F , F1 ) ≤ h(F1 , F2 ) ≤ δ /2 Similarly, one has h(F , F2 ) ≤ δ /2 It follow that h(F, F ) ≤ h(F, F1 ) + h(F , F1 ) ≤ δ By (4.3.5), there exists z¯ ∈ G0 (F ) ∩ (U1 ∪ U2 ) If z¯ ∈ G0 (F ) ∩ U1 , then z¯ ∈ / F2 (y) \U1 for all y ∈ Y Since G0 (F1 ) ∩U1 = 0, / z¯ ∈ / G0 (F1 ) = T0 (X) ∩ y∈Y F1 (y), i.e., y¯ ∈ Y exists such that z¯ ∈ / T0 (X) ∩ F1 (y) ¯ ⊃ T0 (X) ∩ F1 (y) ¯ \ U2 Hence, z¯ ∈ / T0 (X) ∩ F1 (y) ¯ \ U2 ∪ F2 (y) ¯ \ U1 ⊃ G0 (F ), which is a contradiction Similarly, if z¯ ∈ G0 (F ) ∩U2 we also get a contradiction Consequently, µ(F) is connected, whence is an essential component of G0 (F) Remark 4.3.1 If X = Y = Z are a nonempty compact, convex subset of a normed space and T is the identity map, then Theorem 4.3.1 collapses to Theorem 4.1, the main result of [98], and Theorem 4.1 of [100] 4.4 Applications to maximal elements and variational inclusions In this section, applying Theorem 4.3.1 we deduce the existence of essential components of sets of maximal elements and solution sets of optimization-related problems To discuss the essential components for these particular cases, we need 4.4 Applications to maximal elements and variational inclusions 100 the following lemma, which can be found in [96] Lemma 4.4.1 ([96]) Let X,Y, Z be metric spaces and G : Z ⇒ Y Let K : Z → X be continuous Let H : X ⇒ Y be a usco map with the values H(x) possessing essential components If G(z) ⊃ H(K(z)) for each z ∈ Z, then G(z) has at least one essential component for each z ∈ Z 4.4.1 Essential components of T0 -maximal elements Let (X,Y, {ϕN }) and Z be as above, T0 ∈ T (X,Y, Z) We call Mˆ0 := the set of mappings F : Z ⇒ Y satisfying (i) F −1 (y) be open for each y ∈ Y ; (ii) for N ∈ Y and M ⊂ N, T0 (ϕN (∆M )) ∩ y∈M F −1 (y) = / Then, for F ∈ Mˆ0 , by Theorem 1.2.3 (Chapter 1), there exists a maximal-element x¯ of F We call this x¯ a T0 -maximal element of F and denote the set of all such elements by Gˆ (F) Then, we have a maximal-element map Gˆ : Mˆ0 ⇒ Z For F1 and F2 in Mˆ0 , define h(F1 , F2 ) := supy∈Y h Z \ F1−1 (y), Z \ F2−1 (y) Then, Mˆ becomes a complete metric space It is not hard to see that, for F ∈ Mˆ0 , the set-valued map y ⇒ Z \ F −1 (y) belongs to M0 and J : Mˆ0 → M0 , defined by F → Z \ F −1 , conserves distances, i.e., h Z \ F1−1 , Z \ F2−1 = h(F1 , F2 ) Corollary 4.4.1 There exists at least one essential component of Gˆ (F) for each F ∈ Mˆ0 Proof For each F ∈ Mˆ0 we have Z \ F −1 (y) Gˆ (F) = z ∈ Z : F(z) = 0/ = z ∈ Z : z ∈ ∪y∈Y F −1 (y) = y∈Y Z \ F −1 (y) = G0 (Z \ F −1 ) = G0 J(F) ⊃ T0 (X) ∩ y∈Y By Theorem 4.3.1, G0 J(F) has essential components for each J(F) ∈ M0 Applying Lemma 4.4.1 completes the proof 4.4.2 Essential components of solutions to variational inclusions Let r, α, Y ≡ X, Z, Ω , D, S1 , S2 , T , f and g be as in data of the problem (IPrα ) (see Section 1.3, Chapter 1), where X be a compact metric space, Z be a compact, convex set in a normed space Let H0 ∈ T (X, X, Z) We fix r, α and not include them in some notations for the sake of simplicity Let M˜0 := v = (S1 , S2 , S, f , g)|the assumptions (i) (ii) (iii) below are satisfied , where 4.4 Applications to maximal elements and variational inclusions 101 (i) E = z ∈ Z : z ∈ S1 (z) be nonempty and closed and, for z ∈ Z \ E, N ∈ X and M ⊂ N ∩ S2 (z), H0 ϕN (∆M ) ⊂ S1 (z); (ii) f be (g, T, rα)-quasiconvex wrt H0 ; (iii) for each x ∈ X, S2−1 (x) be open and the set frα (x) := z ∈ Z : α t, T (z, x) , r f (t, z, x), g(t, z) be closed For each v = (S1 , S2 , S, f , g) ∈ M˜0 , denote the set of the solutions of the corresponding (IPrα ) by G˜ (v) Then, G˜ (v) = 0/ (by Theorem 1.3.1), and we have a solution map G˜ : M˜0 ⇒ Z Denote further, for each v ∈ M˜0 , v ∈ M˜0 and x ∈ X, Rv (x) = Z \ S2−1 (x) ∪ E ∩ frα (x) , h(v, v ) = supx∈X h Rv (x), Rv (x) Then, M˜0 becomes a complete metric space It is not hard to verify the following lemma Lemma 4.4.2 For each v ∈ M˜0 , the following assertions hold (i) G˜ (v) = x∈X Rv (x) (ii) The map Rv : X ⇒ Z is T0 -KKM and hence belongs to M0 ˜ = Rv , is isometric (iii) The map J˜ : M˜0 → M0 , defined by J(v) Now we obtain the existence of essential components of solutions to problem (IPrα ) as follows Corollary 4.4.2 There exist essential components of G˜ (v) for each v ∈ M˜0 Proof For v ∈ M˜0 , by Lemma 4.4.2, one has G˜ (v) = ˜ Rv (x) = G0 (Rv ) = G0 (J(v)) Rv (x) ⊃ T0 (X) ∩ x∈X x∈X Applying now Theorem 4.3.1 and Lemma 4.4.1 leads to the required conclusion 4.4.3 Essential components of particular cases of variational inclusions In this section we derive the existence of essential components of Nash equilibria and Pareto efficient points, as particular cases of Corollary 4.4.2 for (IPrα ) 4.4.3.1 Essential components of Nash equilibria For an n-person non-cooperative game G (which is defined at the final part of Chapter 3), we define ΦΓ : Z × Z → R by ΦG (z, x) = ∑ni=1 ( fi (z) − fi (ziˆ, xi )) Then, z¯ is a Nash equilibrium point of G if and only if z¯ is a solution of (IPrα ) with S1 (z) := S1G (z) = Z, S2 (z) := S2G (z) = Z, T (z, x) := TG (z, x) = {x}, f (t, z, x) := ΦG (z, x), g(t, z) := gG (t, z) = [0, +∞) Let Zi be a nonempty compact, convex subset of a normed space and let M¯ be the set of all games G satisfying the following conditions 4.4 Applications to maximal elements and variational inclusions 102 (i) ΦG is (gG , SG , r1 α1 )-quasiconvex wrt H0 = I (the identity map on Z); (ii) for each x ∈ Z, the set {z ∈ Z|ΦG (z, x) ≥ 0} is closed By these conditions (i), (ii) and Theorem 1.3.1, for each G ∈ M¯ , the set of the Nash equilibrium points of G , denoted by G¯0 (G ), is nonempty The solution mapping G¯0 : M¯ ⇒ Z is well-defined Moreover, G¯0 (G ) = x∈X RvG (x), where vG = (S1G , S2G , SG , ΦG , gG ) ∈ M˜0 and RvG (x) = z ∈ Z | ΦG (z, x) ≥ For G1 , G2 ∈ M¯ , we define h(G1 , G2 ) ≡ h(vG1 , vG2 ) := supx∈Z h RvG1 (x), RvG2 (x) Then, M¯ becomes a metric space As an implication of Corollary 4.4.2 we obtain Corollary 4.4.3 There exist essential components of G¯0 (Γ ) for each Γ ∈ M¯ 4.4.3.2 Essential components of solutions to quasi-optimization problems In this subsection let Z be a nonempty compact, convex subset of a normed space, Ω be a set, D be a locally convex space, C ⊂ D be a closed, pointed convex cone, and Sˆ : Z ⇒ Z, S : Z ⇒ Ω , f : Ω × Z × Z ⇒ D We use Min{A/C} to denote the set of the Pareto efficient points of set A ⊂ D (with respect to the ordering ˆ cone C) Set E = z ∈ Z|z ∈ S(z) and ( f +C)(t, z) = f (t, z, z) +C We consider the following quasi-optimization problem ˆ z) such that, for all t ∈ S(¯z), (QOP) Find z¯ ∈ S(¯ ˆ z), z¯)/C} = f (t, z¯, z¯) ∩ Min{ f (t, S(¯ / In the sequel let D∗ be the dual of D, B ⊂ D∗ be bounded and C∗ := coneB (the cone generated by B) Lemma 4.4.3 Assume that f (t, z, z) be compact for all (t, z) ∈ S(Z) × Z and z¯ be a solution of the following variational inclusion ˆ z) such that, for all z ∈ S(¯ ˆ z) and all t ∈ S(¯z), (IP) Find z¯ ∈ S(¯ f (t, z¯, z) ⊂ f (t, z¯, z¯) +C Then z¯ is a solution of (QOP) Proof ˆ z) and all t ∈ S(¯z), Let z¯ be a solution of (IP) Then, for all z ∈ S(¯ f (t, z¯, z) ⊂ f (t, z¯, z¯) +C (4.4.8) By the compactness of f (t, z¯, z¯), Min f (t, z¯, z¯)/C = / Suppose that d¯ ∈ Min f (t, ˆ z), z¯)/C z¯, z¯)/C but d¯ ∈ / Min f (t, S(¯z), z¯)/C Then there exists d ∈ Min f (t, S(¯ such that d¯ − d ∈ C \ (−C) By virtue of (4.4.8), d ∈ f (t, z¯, z¯) + C, i.e d = dˆ + c for some dˆ ∈ f (t, z¯, z¯) and c ∈ C Hence d¯ − dˆ ∈ c + C \ (−C)=C \ (−C), contradicting the fact that 4.4 Applications to maximal elements and variational inclusions 103 d¯ ∈ Min f (t, z¯, z¯)/C ˆ S, f ) Let M˘ be the collection of all problems (QOP) corresponding to p = (S, (we will call such a problem (QOP) simply problem p later on) such that the following conditions hold (i) f (t, z, z) is compact for all (t, z) ∈ S(Z) × Z; ˆ is nonempty and convex, and (ii) E is nonempty and closed; for all z, x ∈ Z, S(z) −1 Sˆ (x) is open; (iii) f is ( f +C, S, r1 α1 )-quasiconvex wrt T0 = I (the identity map on Z); (iv) for each x ∈ Z, the set fr1 ,α1 (x) := z ∈ Z|∀t ∈ S(z), f (t, z, x) ⊂ f (t, z, z) +C is closed ˆ S, f ) ∈ M˘ By Denote by G˘0 (p) the set of the solutions of problem p = (S, (i)-(iv), Theorem 1.3.1 and Lemma 4.4.3, G˘0 (p) = 0/ for all p ∈ M˘ p ⇒ G˘0 (p) defines a multifunction G˘0 : M˘ ⇒ Z Moreover, G˘0 (p) ⊃ x∈Z Rv p (x), where ˆ S, ˆ S, f , f +C) ∈ M˜ and Rv p (x) = Z \ Sˆ−1 (x) ∪ E ∩ fr ,α (x) v p = (S, 1 For each p1 , p2 ∈ M˘0 , define h(p1 , p2 ) := h(v p1 , v p2 ) = supx∈Z h Rv p1 (x), Rv p2 (x) Then, M˘ becomes a metric space Furthermore, the map J˘ : M˘ → M˜0 , defined ˘ = v p , is isometric and satisfies the relation G˘0 (p) ⊃ G˜ (v p ) = G˜ (J(p)) ˘ by J(p) As an implication of Lemma 4.4.1 and Corollary 4.4.2 we obtain Corollary 4.4.4 There exist essential components of G˘0 (p) for each p ∈ M˘ Concluding remarks Chapter is devoted not directly to existence issues, but to studies of the existence of essential components and generic stability First, we investigated generalized KKM points and generic stability of T-KKM mappings Then, we applied the obtained results to studying the existence of essential components for various sets: maximal-element sets of set-valued maps, solution sets of general variational inclusions, sets of Nash equilibrium points, and solution sets of quasi-optimization problems It is worth noting that, unlike many existing papers on essential components and generic stability, where one kind of points/solutions was studied, here a unified way for considering the existence of essential components for various points/solutions is presented For possible direct developments of the results in this chapter, we expect that, using the techniques of Section 4.4, we can develop new results on the existence of essential components of solutions to other optimization-related problems such as quasiequilibrium problems, quasivariational inequalities, complementarity problems, and problems on coincidence and fixed points, minimax and saddle points, or Ky Fan points, etc Conclusions The thesis is an effort to develop, in a systematical and comprehensive way, a unified study of the existence of most of important points in applied mathematical analysis and of solutions to various optimization-related problems, in pure topological settings (without linear structures) It includes new results on the existence and stability of points, for set-valued maps, like fixed points, intersection points, coincidence points, maximal elements, etc, and applications to problems, from general models, including those on product spaces, to particular practical cases They are general variational relation problems, general variational inclusions, various quasi-equilibrium models, different quasi-optimization problems, quasivariational inequalities, non-cooperative multi-player games, minimax problems, traffic networks, and some systems of such problems Our results are evaluated by some authors who cited them or by reviewers of our papers as significant contributions to the two approaches to existence study without linear and convex structures The first one is based on the replacement of convex hulls by a family of continuous mappings from a simplex to our underlying space The other uses connectedness instead of the classical and more restrictive notion of convexity The remarkable features of our study are the following First, we always try to prove the equivalence between the important theorems about different objects Second, the results are stated as general as possible, and we usually go from general settings and results to particular and special cases to encompass various problems, including practical situations Third, besides sufficient conditions included in the first two chapters, full (two-way) characterizations are explored for the both encountered approaches in detailed in the long Chapter Such necessary and sufficient conditions were developed earlier only for minimax problems by only Kindler, while we develop them for all of our objects and prove also the needed equivalence Forth, our results are new or improve/generalize recent ones in the literature Comparisons to justify this assertion or for other purposes are included in many examples and remarks Fifth, further developments of both theoretical and application results of the thesis may be seen relatively clearly In fact, we are working on some of them, and a new paper has already submitted for publication We have observed as well a number of papers in the literature, which cite and develop our results List of the author’s papers related to the thesis (Q1) Nguyen Xuan Hai, Phan Quoc Khanh, Nguyen Hong Quan, On the existence of solutions to quasivariational inclusion problems, Journal of Global Optimization, 45, 565 - 581 (2009) (Q2) Phan Quoc Khanh, Nguyen Hong Quan, Existence conditions for quasivariational inclusion problems in G-convex spaces, Acta Mathematica Vietnamica, 34 n.1, 1-10 (2009) (Q3) Phan Quoc Khanh, Nguyen Hong Quan, Intersection theorems, coincidence theorems and maximal-element theorems in GFC-spaces, Optimization, 59, 115-124 (2010) (Q4) Phan Quoc Khanh, Nguyen Hong Quan, Jen-Chih Yao, Generalized KKM type theorems in GFC-spaces and applications, Nonlinear Analysis: Theory, Methods and Applications, 71, 1227-1234 (2009) (Q5) Nguyen Xuan Hai, Phan Quoc Khanh, Nguyen Hong Quan, Some existence theorems in nonlinear analysis for mappings on product GFC-spaces and applications, Nonlinear Analysis: Theory, Methods and Applications, 71, 6170-6181 (2009) (Q6) Phan Quoc Khanh, Nguyen Hong Quan, General existence theorems, alternative theorems and applications to minimax problems, Nonlinear Analysis: Theory, Methods and Applications, 72, 2706-2715 (2010) (Q7) Phan Quoc Khanh, Nguyen Hong Quan, Existence results for general inclusions using generalized KKM theorems with applications to minimax problems, Journal of Optimization Theory and Applications, 146, 640-653 (2010) (Q8) Phan Quoc Khanh, Nguyen Hong Quan, Generic stability and essential components of generalized KKM points and applications, Journal of Optimization Theory and Applications, 148, 488-504 (2011) (Q9) Phan Quoc Khanh, Vo Si Trong Long, Nguyen Hong Quan, Continuous selections, collectively fixed points and weak Knaster-Kuratowski-Mazurkiewicz mappings in optimization, Journal of Optimization Theory and Applications, 151, 522-572 (2011) (Q10) Phan Quoc Khanh, Nguyen Hong Quan, A unified study of topological existence theorems and applications, Proceedings of American Mathematical Society, submitted for publication (Q11) Phan Quoc Khanh, Nguyen Hong Quan, A topological characterization of the existence of fixed points and applications, Mathematische Nachrichten, 287, 281-289 (2014) (Q12) Phan Quoc Khanh, Nguyen Hong Quan, Topological characterizations of the existence of solutions to optimization-related problems, Journal of Optimization Theory and Applications, accepted for publication (2013) (Q13) Phan Quoc Khanh, Nguyen Hong Quan, A fixed-component-point theorem and applications, Mathematische Nachrichten, submitted for publication (Q14) Nguyen Hong Quan, An equilibrium-existence result for abstract economies, Vietnam Journal of Mathematicical Applications, accepted for publication (2013) List of the author’s conference reports related to the thesis (QR1) Phan Quoc Khanh, Nguyen Hong Quan, Generalized KKM type theorems in GFC-spaces and applications, The 7th International Workshop on Mathematical Optimization Theory and Applications, Hanoi, Vietnam, July 31- August (2008) (QR2) Phan Quoc Khanh, Nguyen Hong Quan, Some existence theorems in nonlinear analysis, The 7th Vietnam Mathematical Congress, Quinhon, Vietnam, August 04-08 (2008) (QR3) Nguyen Xuan Hai, Phan Quoc Khanh, Nguyen Hong Quan, Existence theorems in GFCspaces with applications to minimax problem, The 7th Workshop on Optimization and Computing, Bavi, Vietnam, 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Huang, N.J., Wong, M.M, Connectedness and path-connectedness of solution sets to symmetric vector equilibrium problems Taiwanese J Math 13, 8211-836 (2009) Glossary of Symbols A |N| [·, ·] ]·, ·[ ]·, ·], [·, ·[ := and =: ≡ · or | · | xα → x N R := ]-∞, +∞[ R := [-∞, +∞] Rn 0/ ∆n ∆|N| := ∆n ∆k ∆M clA and A clB A and AB intA intB A Ac , X \ A cclA cintA coA the set of all finite subsets of the set A the cardinality of N a closed interval an open interval half-open intervals equal by definition identically equal norms xα converges to x the set of all natural numbers the real line the extended real line the n-dimensional Euclidean space the empty set the end of a proof the standard n-simplex the n-simplex corresponding to N with |N| = n+1 a k-face of ∆n the face of ∆n corresponding to M the closure of A the closure of A in B the interior of A the interior of A in B the complement of A the compact closure of A the compact interior of A the convex hull of A Glossary of Symbols (X,Y, Φ) or (X,Y, {ϕN }) GFCS (A) B(X,Y, Z) KKM(X,Y, Z) F := (ΦX , ℑY ) Γ GKKM(X;Y ) Γ (X;Y ) K(Z) CK(Z) f :X →Y F :X ⇒Y F −1 : Y ⇒ X F∗ : Y ⇒ X GFCS ◦ F F(x) F −1 (y) F ∗ (y) F ◦G GphF e(A, B) h(A, B) a finite continuous topological space (GFC-space) the GFC-hull with respect to the set-valued map S the class of all better admissible maps from X to Z the class of all mappings enjoying the generalized KKM property a KKM-structure a connectedness structure the class of all GKKM-maps of (X;Y ) the class of all Γ -maps between X and Y the set of the nonempty compact subsets of Z the set of nonempty compact convex subsets of Z a single-valued mapping from X to Y a set-valued mapping from X to Y the inverse mapping of F : X ⇒ Y the dual mapping of F : X ⇒ Y the GFC-hull mapping wrt S of F image the fiber the inverse image of y cofiber at y composition of mappings f and G the graph of F the excess of the set A from the set B the Hausdorff distance between sets A and B 112 [...]... s-KKM mapping wrt T introduced in [25] becomes a T -KKM mapping by Definition 1.1.3 A multivalued mapping F : Y ⇒ X, being an R-KKM mapping as defined in [20], is a special case of T -KKM mappings on GFC-space when X = Z and T is the identity map The definition of generalized KKM mappings wrt to T in [61] is as well a particular case of Definition 1.1.3 1.1 Notions and definitions 5 Definition 1.1.4 Let... subsection, as applications, we use our existence theorems to establish sufficient conditions for the solution existence of a general inclusion and develop in detail general types of minimax theorems 1.3.1 Variational inclusions For the last decade, two features can be recognized in the increasingly intensive study of optimization- related problems, especially as concerns with the existence of solutions to such... problem settings have been more and more general and leading to unifying studies for diverse classes of problems which may occur in practice We explain the second feature according to the purpose of this subsection, in terms of the solution existence issue The assumptions imposed for getting existence sufficient conditions are becoming more and more 1.3 Applications 24 relaxed The most powerful and frequently... role in the theory of existence Definition 1.1.3 Let (X,Y, Φ) be a GFC-space, Z be a topological space, F : Y ⇒ Z and T : X ⇒ Z be set-valued mappings F is said to be a generalized KKM mapping with respect to (shortly, w.r.t) T (T -KKM mapping in short) if, for each N ∈ Y and each M ⊂ N, one has T (ϕN (∆M )) ⊂ y∈M F(y) The definition of T -KKM mappings was introduced for X being a convex subset of a topological... topological vector space in [16] and extended to FC-spaces in [24] Definition 1.1.3 includes these definitions as particular cases It encompasses also many other kinds of generalized KKM mappings We mention here some of them Let (X, {ϕN }) be an FC-space, Y be a nonempty set and s : Y → X be a mapping We define a GFC-space (X,Y, {ϕN }) by setting ϕN = ϕs(N) for each N ∈ Y Then, a generalized s-KKM mapping wrt... yˆ0 ∈ N and choosing yˆ = yˆ0 we have yˆ ∈ GFCS (Ω (z)) = (GFCS ◦ Ω )(z), i.e., z ∈ (GFCS ◦ Ω )−1 (y) ˆ Moreover, z ∈ intK Ω −1 (y) ˆ ∩K −1 −1 ⊂ intK (GFCS ◦ Ω ) (y) ˆ ∩ K Thus, (GFCS ◦ Ω ) is transfer compactly openvalued 1.2 Existence theorems in GFC-spaces Using two elementary topological tools: the finite intersection property of compact sets and the existence, for a finite covering of a compact... G-convex space and S ≡ I then Theorem 1.2.13 implies Theorem 3 of [61] and Theorem 5.1 of [19] For S-KKM mappings with respect to T and the class S-KKM(X,Y, Z) defined in [17] we have the following consequence which is Theorem 5.1 of [17] 1.2 Existence theorems in GFC-spaces 20 Corollary 1.2.2 Let X, Y be convex spaces and Z be a Hausdorff topological space Let S : Y ⇒ X, F : Y ⇒ Z and T ∈ S-KKM(X,Y,... Therefore it is reasonable and valuable to study existence theorems and nonlinear problems in GFC-spaces without linear structure The following example shows that GFC-spaces are properly more general than G-convex spaces Recall that, a G-convex space is [79-85] a triple (X,Y,ϒ ), where X and Y are as Definition 1.1.1 and ϒ : Y ⇒ X is such that, for each N ∈ Y , there exists a continuous map ϕN : ∆|N| →... corresponding mapping from Φ we take ϕs(N) Then an s-KKM mapping wrt T acting on the FC-space (X, Φ) becomes a T-KKM mapping acting on the GFC-space (X,Y, Φ), according to Definition 1.1.3 Therefore, Theorem 3.2 of [25] is a special case of Theorem 1.2.12, where S ≡ s If in addition Y = X, S ≡ s ≡ I (the identity map) and T is a compact mapping, our Theorem 1.2.12 collapses to Theorem 3.3 of [25] and Theorem... 1.2 Existence theorems in GFC-spaces ⊂ K ∩ T (X) ∩ Z\ 11 (Oy ∩ K) y∈Y (F −1 (y) ∩ K) = K ∩ T (X) ∩ Z\ y∈Y Consequently, there is z¯ ∈ K such that z¯ ∈ / F −1 (y) for each y ∈ Y , which means that F(¯z) = 0 / The above theorems are very close to each other, but they are not completely equivalent The coincidence point and maximal-element theorems are deduced from intersection theorems To underline a