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 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV 1 Existence theorems in nonlinear analysis and applications . . . . . . . . 1 1.1 Notions and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Existence theorems in GFC-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Variational inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.2 Minimax theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Nonlinear existence theorems for mappings on product GFC-spaces and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Existence theorems on product GFC-spaces . . . . . . . . . . . . . . . . . . . 37 2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Topological characterizations of existence in nonlinear analysis and optimization-related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 Topological existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 The existence of solutions to optimization-related problems . . . . . 70 3.2.1 Variational relation problems . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.2 Invariant-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.3 Equilibrium problems of the Stampacchia and Minty types 77 3.2.4 Minimax theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.5 Nash equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 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 T 0 -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 re- sults as KKM theorems, minimax theorems, etc., constitute one of the most im- portant 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 re- sults. 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 classi- cal convexity by abstract convexity notions, not using linear structures, and they extended earlier existing notions and results to these structures. In the frame- work of this research direction, types of spaces with generalized convexity struc- tures 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]), G- convex 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 equiva- lence relations between existence results. For examples, in Ha [40], an extension of Kakutani fixed-point theorem was proved to be equivalent to a section theo- rem 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 sit- uations. 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 re- lation problems, or practical problems as traffic networks, Nash equilibria, ab- stract economies, etc. One of the important topics in nonlinear optimization, which have been at- tracting many mathematicians recently, is properties of solution sets and solu- tion 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 prop- erties, 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 prob- lems. 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 GFC- spaces and applications” is based on the paper (Q5); Chapter 3: “Topological characterizations of existence in analysis and opti- mization 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 Insti- tute 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 1 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 e 0 = (1,0, ,0), e 1 = (0,1, ,0), , e n = (0,0, ,1) of a basis of R n+1 . For N = {y 0 ,y 1 ,y n } ∈ Y  and M = {y i 0 ,y i 1 , ,y i k } ⊂ N, let ∆ |N| ≡ ∆ n , and ∆ M ≡ ∆ k be the face of ∆ |N| corresponding to M, i.e., ∆ M = co{e i 0 ,e i 1 , e i k }. If A,B ⊂ X, X being a topological space, then A (or clA), A B (or cl B A), intA, int B A and A c 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 2 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 open- valued (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 (K ∩ F(y)) =  y∈Y (K ∩ cclF(y)) =  y∈Y (K ∩ cl K F(y))   y∈Y (K ∩ F(y)) =  y∈Y (K ∩ cintF(y)) =  y∈Y (K ∩ int K F(y)), resp  . We propose the following definition of a GFC-space to unify a number of ear- lier 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 continu- ous 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 ϕ N : ∆ |N| → X of the family Φ (we also use (X,Y,{ϕ N }) to denote (X,Y,Φ)). 1.1 Notions and definitions 3 The above mentioned existing spaces are the examples for GFC-space. In par- ticular, a convex subset A of a topological vector space is a GFC-space, where X = Y = A and each N = {a 0 ,a 1 , ,a n } ∈ A, there is ϕ N : ∆ |N| → A which is defined by ϕ N (e) = ∑ n i=0 λ i a i for all e = ∑ n i=0 λ i e i ∈ ∆ |N| . GFC-spaces are properly more general than known existing spaces. 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 ϕ N : ∆ |N| → ϒ (N) such that, for each M ∈ N, ϕ N (∆ M ) ⊂ ϒ (M). A G-convex space (X,Y,ϒ ) is called trivial iff, for all N ∈ Y , ϒ (N) = X. Of course, any above-mentioned space can be made into a trivial G-convex space, but a trivial G-convex space has no use. Example 1.1.1 Let Y =]0,+∞[, X = {(x,x) | x ∈]0,+∞[} ⊂ R 2 and, for each N ∈ Y , ϕ N (e) = (α(N),α(N)) ∈ X for all e ∈ ∆ |N| , where α(N) = min N.maxN. Then, (X,Y,{ϕ N }) is a GFC-space. Suppose there exists ϒ : Y  ⇒ X such that (X,Y,ϒ ) is a nontrivial G-convex space, and (X,Y,{ϕ N }) can be made into (X,Y,ϒ ). Then, since (X,Y,ϒ ) is a G-convex space, we have, for all N ∈ Y  and y ∈ Y , ϕ N∪{y} (∆ |N| ) = (α(N ∪ {y}),α(N ∪ {y})) ∈ ϒ (N). It follows that, for all N ∈ Y, Λ :=  (α(N ∪ {y}),α(N ∪ {y})) | y ∈ Y  ⊂ ϒ (N). (1.1.1) On the other hand, for all N ∈ Y and x ∈]0,+∞[, taking y = x maxN if x ≤ α(N) and y = x minN if x > α(N), we have (x, x) = (α(N ∪{y}),α(N ∪{y})) ∈ Λ. Thus, X ⊂ Λ . This and (1.1.1) imply that, for all N ∈ Y , ϒ (N) = X, i.e., (X,Y,ϒ ) is a trivial G-convex space. Next, we define several concepts in a GFC-space. Definition 1.1.2 Let (X,Y,Φ) be a GFC-space, D,C ⊂ Y and S : Y ⇒ X be given. (i) D is called an S-subset of Y (S-subset of Y wrt C) if for all N ∈ Y  and for all M ⊂ N ∩ D (for all M ⊂ N ∩C, resp), ϕ N (∆ M ) ⊂ S(D). (ii)If in addition to (i), S −1 (ϕ N (∆ M )) ⊂ D, then D is called an S GFC -subset of Y . The GFC-hull wrt S of C is defined by GFC S (C) =  {D ⊂ Y | D is S GFC - subset of Y containing C}. Clearly, if D is an S GFC -subset of Y , D must be an S-subset of Y . Roughly speaking, if D is an S-subset of Y then (S(D),D,Φ) is a ”pre” GFC-space, and if D is an S GFC -subset of Y then (S(D),D,Φ) is a ”full” GFC-space. When X = Y , i.e, (X,Y,Φ) = (X,Φ) is an FC-space, and S = I is the identity map, then being an S GFC -subset or an S-subset of X coincides with being an FC-subspace of X ([30]). Moreover, the notion of a GFC-hull wrt S of a set extends the notions of 1.1 Notions and definitions 4 an FC-hull of a set in a FC-space ([25]) and a G-convex hull of a set in a G- convex space ([64]). The extension of concepts of subspaces and convex hulls as above also shows the rationality of the GFC-spaces. The following Lemma 1.1.2 extends Lemma 2.1 of [95] from FC-spaces to GFC-spaces. Lemma 1.1.2 Let (X,Y,Φ) be a GFC-space, C a nonempty subset of Y and S : Y ⇒ X. Then, (a) GFC S (C) is an S GFC -subset of Y ; (b) GFC S (C) is the smallest S GFC -subset of Y containing C; (c) GFC S (C) =  N∈C GFC S (N). Proof. (a) and (b) are obvious. (c) For every N ∈ C, it is clear that GFC S (N) ⊂ GFC S (C). Hence, C ⊂  N∈C GFC S (N) ⊂ GFC S (C). Therefore, it suffices to show that the union in these inclusions, which is now denoted by B, is an S GFC -subset of Y. Assume that N 0 ∈ Y  and M 0 ⊂ N 0 ∩B. By the definition of B, there is {N 1 , ,N l } ∈ C, such that M 0 ⊂  l i=1 GFC S (N i ) ⊂ B. Since  l i=1 N i ∈ C, one has GFC S (  l i=1 N i ) ⊂ B. As, ∀ j = 1, ,l, GFC S (N j ) ⊂ GFC S (  l i=1 N i ), one has further M 0 ⊂ GFC S (  l i=1 N i ). Since each GFC - hull wrt S is S GFC -subset, by Definition 1.1.2, S −1 (ϕ N 0 (∆ M 0 )) ⊂ GFC S   l i=1 N i  ⊂ B. Hence, again by this definition, B is an S GFC -subset of Y .  The notion of a GFC-space helps us also to extend the notion of generalized KKM maps which plays an essential 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 (ϕ N (∆ M )) ⊂  y∈M F(y). The definition of T -KKM mappings was introduced for X being a convex sub- set of a topological vector space in [16] and extended to FC-spaces in [24]. Defini- tion 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,{ϕ N }) be an FC-space, Y be a nonempty set and s : Y → X be a mapping. We define a GFC-space (X,Y,{ϕ N }) by setting ϕ N = ϕ s(N) for each N ∈ Y . Then, a generalized s-KKM mapping wrt T introduced in [25] becomes a T -KKM map- ping 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 map- pings wrt to T in [61] is as well a particular case of Definition 1.1.3. [...]... 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 problems Firstly, the problem settings have been more and more general and leading to unifying studies for diverse... 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 used tool is existence theorems, which have been generalized by many authors This subsection aims at applying above existence. .. Q1 in the places of (X,Y, Φ), Z, S, T, H, P and Q, respectively By this theorem a point (x, y, z) ∈ S(LN ) × LN × T (S(LN )) exists such that x ∈ S|LN (y), y ∈ Q1 (z) and z ∈ T |S(LN ) (x) This point is also a required point of Theorem 1.2.19 1.3 Applications In this subsection, as applications, we use our existence theorems to establish sufficient conditions for the solution existence of a general inclusion... a 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,... ∈ N and choosing y = y0 we have y ∈ GFCS (Ω (z)) = ˆ ˆ ˆ ˆ −1 (y) Moreover, z ∈ int Ω −1 (y) ∩ K (GFCS ◦ Ω )(z), i.e., z ∈ (GFCS ◦ Ω ) ˆ ˆ K −1 (y) ∩ K Thus, (GFC ◦ Ω )−1 is transfer compactly open⊂ intK (GFCS ◦ Ω ) ˆ S valued 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... 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(¯) = 0 z / 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... containing N such that S(LN ) is compact and for each z ∈ T (S(LN )) \ K, there exists y ∈ LN such that z ∈ cint(GFCS ◦ Ω )−1 (y) Then, there is z ∈ Z such that Ω (ˆ) = 0 ˆ z / Theorem 1.2.6 (Coincidence points) Let (X,Y, {ϕN }) be a GFC-space, Z a topological space, T ∈ KKM(X,Y, Z) and S : Y X a multivalued mapping such that Y is an S-subset of itself For multivalued mappings H and Ω from Z into Y... where ϕs(N) : ∆|N| → X j=0 is the mapping of the family Φ, corresponding to {s(y0 ), s(y1 ), , s(yn )} in the definition of an FC-space [24] We define a GFC-space (X,Y, Φ) as follows: for N ∈ Y , as the 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... theorems in GFC-spaces Then, T (S(Y ) ∩ y∈Y 21 F(y) = 0 / Proof The proof is similar to that of Theorems 1.2.13 and 1.2.14 and hence omitted Remark 1.2.6 A particular case of Theorem 1.2.14 is Theorem 3.4 of [25] and a consequence of Theorem 1.2.15 is Theorem 3.2 of [27] Using KKM type theorems we establish coincidence theorems and some their geometric versions These results are either equivalent to or... −1 (z) = Y and all the assumptions of Theorem 1.2.18 are satisfied For a coincidence point (x, y, z) existing by this theorem we see that z ∈ T (S(y)) and y ∈ Q(z) = Y \ M −1 (z), i.e z ∈ M(y), con/ tradicting (ii2 ) Theorem 1.2.19 (Coincidence point theorem) Let (X,Y, Φ), Z, S and T be as in Theorem 1.2.17 except the compactness of T (S(Y )) Let (ii), (iii) of Theorem 1.2.17 and the following conditions . 111 Preface The existence theorems, including theorems about various points like fixed points, coincidence points, intersection points, maximal elements, and other re- sults as KKM theorems, minimax theorems, . 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 GFC- spaces and. 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