Bài toán tựa cân bằng dạng blum – oettli tổng quát và ứng dụng tomtat t anh

THAI NGUYEN UNIVERSITY UNIVERSITY OF EDUCATION —————————————————— NGUYEN QUYNH HOA GENERAL QUASI-EQUILIBRIUM PROBLEMS OF THE BLUM - OETTLI TYPE AND THEIR APPLICATIONS Speciality: Mathematical Analysis Code: 9460102 DISSERTATION SUMMARY THAI NGUYEN - 2018 The dissertation was finished at: THAI NGUYEN UNIVERSITY - UNIVERSITY OF EDUCATION Supervisor: Prof Dr.Sc Nguyen Xuan Tan Referee 1: Referee 2: Referee 3: The dissertation will be defended in the university committee: THAI NGUYEN UNIVERSITY - UNIVERSITY OF EDUCATION At ,2018 The dissertation can be read at: - National library of Vietnam; - Thai Nguyen University - Learning Resource Center; - Library of University of Education Introduction When researching phenomena in nature and society, as well as in the sciences, we often encounter the questions: Exist or not exist? How to exist? In mathematics, the first question is related to the existence or non-existence of the solution of the equation, the problem is formulated: Find x ∈ D such that F (x) = 0, (1) in which, D ⊆ X be nonempty subset and F : D → Y with Y is a vector space This problem is called operator equation As for the second question, in mathematics, we can relate to the problem: Find x ∈ D such that f (x) ≤ f (x), for all x ∈ D, (2) with D ⊂ X and f : D → R This problem is called optimization problem Problem (1) and problem (2) play an improtant role in the application of mathematics to solving problems in real life Mathematicians have built theories to solve problem (1) and problem (2) The theory used solve the problem (1) is called theory of operator equation The theory to solve the problem (1) is called optimal theory These problems have central role of these two theories The theory of operator equation and optimal theory are interrelated and interactive In many cases, problem (1) can be taken on problem (2) and vice versa For example, when X is a Hilbert space, f is a convex function and has the derivative f , problem (2) is equivalent to problem: Find x ∈ D such that x = PD (x − f (x)), with PD (x) is orthogoral projection of x on the set D Or F (x) = 0, with F (x) = PD (x − f (x)) − x Thus, problem (1) is equivalent to problem (2) To solve problem (2), we classify them into problem classes based on the characteristics of function f and set D When f is a linear function and D convex polyhedron in Euclid space Rn , problem (2) is called linear programming problem In 1947, G B Danzig was an American mathematical scientist who found simplex algorithm to solve this problem When set D is a convex and closed in Rn and f is a convex function then the problem (2) is called convex programming problem Up to the 1960 - 1970, T Rockaffelar was an American mathematical scien- tist who gave lower differential defined of convex function to construct convex analysis to solve convex programming problems Next, when f is a locally Lipschitz function and D is a closed set, (2) is called Lipschitz programming problem After the 1970s, F H Clarke constructed lower differential of locally Lipschitz function to solve Lipschitz programming problems When f is a continuous function, D is a closed set, problem (2) is called continuous programming problem In the last years of the 20th century and the early years of the 21st century, D T Luc and V Jeyakumar gave approximate Jacobian theory to solve continuous programming problems Up to the 1990s of last century, Stampachia gave out variational inequality problem: Let D be a nonempty subset of space Rn , T : D → Rn Find x ∈ D such that T (x), x − x ≥ 0, với x ∈ D (3) Then, this problem was expanded into general variational inequality problem: Find x ∈ D such that T (x), x − x + φ(x) − φ(x) ≥ 0, for all x ∈ D, (4) with D is a nonempty subset of space Banach X, X ∗ is a dual space of X, T : D → X ∗ is a single valued mapping, φ : D → R is a real function In 1994, Blum and Oettli gave out equilibrium problem (EP): given mapping f : D × D → R, f (x, x) = 0, with x ∈ D Find x ∈ D such that f (t, x) ≥ 0, for all t ∈ D (5) To prove the existence of solutions of problem (5), the authors used the intersection of KKM mappings Theorem which is equivalent form of Browder fixed point Theorem As far as we know equilibrium problems generalize of variational inequalities and optimization problems, including also optimization-related problems such as fixed point, complementarity problems, Nash equilibrium, minimax problems, etc In recent years, the study of algorithms in order to find solutions tho these problems has been greatly expanded by both domestic and international mathematicians Next, minimax problems, variational inequalities and equilibrium problems were extended when relevant functions are vector functions and they are called vector minimax problems, vector variational inequalities and vector equilibrium problems In the late years of the 20th century and the early years of the 21st century, many authors as N X Tan, D T Luc, P N T, P H Sach, P Q Khanh, L J Lin, T T T Duong, B T Hung, N T Q Anh, formulated and solved the existence of solutions to these problems when relevant mappings are multivalued mappings Problem (1), (2) and generalized problems related to vector function and multivalued mapping can be put on the problem: Given multi-valued mapping F : D → 2Y Find x ∈ D such that ∈ F (x), (6) in which, X, Y be Hausdorff locally convex topological vector spaces, D is subset of X Problem (6) is called general equilibrium problem or multivalued equation In fact in many cases constrain fiela D changes and is depended by a mapping, P : D → 2D Then, we need to consider the problem: Find x ∈ D such that 1) x ∈ P (x); (7) 2) ∈ F (x) Problem (7) is called general quasi-equilibrium problem Sufficient condition for the existence of solution for this problem was studied in the case P is a continuous multivalued mapping with nonempty compact convex values and F is a u.s.c multivalued mapping with nonempty compact convex values In recent years, many mathematicians have studied the existence of the solutions to general quasi-equilibrium problem by minimizing continuity of mappings P, F Then, let X, Y, Z be Hausdorff locally convex topological vector spaces, D ⊆ X, K ⊆ Z, multivalued mappings P : D × K → 2D , Q : D × K → 2K , F : D × K → 2Y We are interested in the problem: Find (x, y) ∈ D × K such that 1) x ∈ P (x, y), y ∈ Q(x, y); (8) 2) ∈ F (x, y) The multivalued mappings P, Q are called constraint mappings, F is called utility multivalued mappings As can be seen, if we put D = D × K, P = P × Q, then problem (8) returns to form the problem (7) General quasi-equilibrium problem (8) encompassed a large class of problems of applied mathematics including quasi-optimization problems, quasi-variational inclusion, quasi-equilibrium problems, quasi-variational relation problems, etc Sufficient conditions for the existence of solutions to problem (8) were studied by many authors as L J Lin and S Park, M P Chen, L J Lin and S Park, S Park, Jian Wen Peng and Dao Li Zhu, Special, N X Tan and D T Luc, N X Tan and L J Lin, N X Tan and T T T Duong, N X Tan and B T Hung, N X Tan and N T Q Anh studied when P is a continuous multivalued mapping, Q is a u.s.c multivalued mapping and F is a u.s.c multivalued mapping or a l.s.c multivalued mapping All these mappings P, Q and F need to have nonempty convex and closed values To broaden the direction of this research, we consider problem (8) with the case the multivalued mapping F (x, y) = G(x, y) + H(x, y), which means that we are interested in the problem: Find (x, y) ∈ D × K such that 1) x ∈ P (x, y), y ∈ Q(x, y); 2) ∈ G(x, y) + H(x, y), with different continuous assumptions on the multivalued mappings G and H and we call "General quasi-equilibrium problems of the Blum - Oettli type " Many authors studied problems of the Blum - Oettli type, which means that multivalued problems with a utility function is a sum of two mappings as N X Tan and P N Tinh, T Y Fu, G Kassay and M Miholca, G Kassay, M Miholca and N T Vinh, etc The main purposes of the dissertation are: (1) To study sufficient conditions for the existence of solutions to general quasi-equilibrium problems of the Blum - Oettli type with utility function and constraint mappings are multivalued function and multivalued mappings in cases: - The utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping; - The utility multivalued mapping is a product of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping (2)Applied the obtained results in (1) to study the existence of solution of relevant problems: generalized quasi-equilibrium problem of type I, generalized quasi-equilibrium problem of type II and mixed generalized quasi-equilibrium problem For the reasons discussed above, we chose the research topic for the dissertation “General quasi-equilibrium problems of the Blum - Oettli type and their applications” In addition to an introduction, a section of conclusions and a list of references, the dissertation has three chapters Chapter collects some basic concepts needed for subsequent chapters The main results of this dissertation are presented in Chapter and Chapter Chapter studies the existence of solutions to general quasi-equilibrium problem Theorem 2.1.1 and Theorem 2.1.2 prove the existence to solution for general quasi-equilibrium problems with the utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping Theorem 2.2.1 and Theorem 2.2.2 prove the existence to solution for general quasi-equilibrium problems with the utility multivalued mapping is a product of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping In this chapter, we also give some expanded results that connect Ky Fan Theorem and Fan - Browder Theorem (Corollary 2.1.7, Corollary 2.1.8, ) Chapter applies the obtained results in Chapter to get the existence to solutions for generalized quasi-equilibrium problem of type I (Theorem 3.1.1, Corollary 3.1.1), generalized quasi-equilibrium problem of type II (Theorem 3.2.1, Corollary 3.2.2, Corollary 3.2.3) and mixed generalized quasi-equilibrium problem (Theorem 3.3.1, Theorem 3.3.2) The main content of the thesis is written based on the articles in list of research papers published related to the dissertation Chương Preliminaries In mathematics as well as in natural and social life, to solve a problem, people often model in the form of a mathematical problem The problem must be set in a certain space, the solution of the problem must also be defined in a certain space Space must have structures to ensure that the problem is true and can be calculated strictly by the algorithm Therefore, before studying the mathematical problems mentioned in the thesis, it is necessary to reiterate the basic knowledge of normal spaces and concepts related to the problems that need to be studied This chapter collects some basic concepts needed for subsequent chapters The rest of this chapter is divided into two sections: Section 1.1 recalls some concepts of Hausdorff locally convex topological vector spaces Section 1.2 introduces the basic features of the cone, multi-valued mappings and some fixed point theory of continuous multivalued mapping Chương General quasi-equilibrium problem As shown in the Introduction, we can see that most problems in optimal theory are transformed in to general equilibrium problem: Find x¯ ∈ D such that ∈ F (¯ x), with D is subset of X space and F : D → 2Y , Y is a space The problem can be written: Find x¯ ∈ D such that 1) x¯ ∈ P (¯ x); 2) ∈ F (¯ x), with P : D → 2D The existence of solutions to this problem is studied in the case the multivalued mapping P is continuous with nonempty convex compact values, F is u.s.c In this chapter, let X, Y, Z be Hausdorff locally convex topological vector spaces, D ⊆ X, K ⊆ Z be nonempty subsets Given multi-valued mappings P : D × K → 2D , Q : D × K → 2K , F : D × K → 2Y We are interested in the problem: Find (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) ∈ F (x, y) This problem is called a general quasi-equilibrium problem The existence of solutions to this problem has been studied by the authors, especially, T T T Duong and N X Tan studied for the case in which the multivalued mapping P is continuous, the multivalued mapping Q is u.s.c and the multivalued mapping F is u.s.c, all these mappings P, Q and F need to have nonempty convex and closed values and the case the multivalued mapping P has open lower section, the multivalued mapping Q is l.s.c and the multivalued mapping F is u.s.c In this chapter, we study the special case of general quasi-equilibrium problem: i) The utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping; ii) The utility multivalued mapping is a product of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping 2.1 General quasi-equilibrium problems of the Blum - Oettli type In this section, we will give out some sufficient conditions for the existence of solutions of general quasi-equilibrium problems of the Blum - Oettli type This is general quasi-equilibrium problem when the utility is the sum of two mappings: F = G + H in cases: 1) G : D × K → 2X is scalar weakly lower semi-continous mapping, H : D × K → 2X is a scalar weakly upper semi-continous mapping and Y = X 2) G : D × K → 2X×Z is a scalar weakly lower semi-continous mapping, H : D × K → 2X×Z is a scalar weakly upper semi-continous mapping and Y = X × Z Firstly, we consider the case: F = G + H with G : D × K → 2X is a scalar weakly lower semi-continous mapping, H : D × K → 2X is a scalar weakly upper semi-continous mapping and Y = X Lemma 2.1.1 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) φ : K × D × D → R is a continuous function; φ(y, x, ) : D → R is a quasi-convex function and φ(y, x, x) = for all (y, x) ∈ K × D Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) φ(y, x, t) ≥ 0, for all t ∈ P (x, y) Theorem 2.1.1 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values, Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iii) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; iv) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values; v) For all (x, y) ∈ P (x, y) × Q(x, y), ∅ = G(x, y) + (H(x, y) ∩ TP (x,y) (x)) ⊂ TP (x,y) (x) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) ∈ G(x, y) + H(x, y) Corollary 2.1.1 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; v) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values; vi) For all (x, y) ∈ P (x, y) × Q(x, y), ∅ = (G(x, y) − x) + (H(x, y) ∩ TP (x,y) (x)) ⊂ TP (x,y) (x) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) x ∈ G(x, y) + H(x, y) Corollary 2.1.2 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; v) H : D × K → 2X is a scalar weakly u.s.c mapping with closed convex values; vi) For all (x, y) ∈ P (x, y) × Q(x, y), x ∈ / G(x, y) + H(x, y) and ∅ = (G(x, y) − x) + (H(x, y) ∩ TP (x,y) (x)) ⊂ TP (x,y) (x) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) G(x, y) + H(x, y) = ∅ Corollary 2.1.3 We assume that the following conditions hold i) D is nonempty convex compact subset; ii) G0 : D → 2X is a scalar weakly l.s.c mapping with nonempty values; iii) H0 : D → 2X is a u.s.c multivalued mapping with nonempty closed convex values; iv) (G0 (x) − x) + (H0 (x) ∩ TD (x)) ⊂ TD (x), for all x ∈ D 10 iv) G : D × K → 2D is a scalar weakly l.s.c mapping; v) Với (x, y) ∈ P (x, y) × Q(x, y), x ∈ / G(x, y) and G(x, y) − x ⊂ TP (x,y) (x) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) G(x, y) = ∅ Corollary 2.1.7 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iii) F : D × K → 2D is a scalar weakly l.s.c mapping and với (x, y) ∈ D × K, y ∈ Q(x, y) đó, F (x, y) = ∅; Then there exists (x, y) ∈ D × K such that 1) y ∈ Q(x, y); 2) x ∈ F (x, y) Corollary 2.1.8 We assume that the following conditions hold i) D is nonempty convex compact subset; ii) F : D → 2D is a scalar weakly l.s.c mapping with nonempty values Then there exists x ∈ D such that x ∈ F (x) Remark 2.1.1 Corollary 2.1.8 is the expansion fixed point Theorem of X Wu In this result, mapping F has removed the closed and convex condition, only satisfied that the scalar weakly l.s.c mapping This corollary is also the expansion of fixed point Theorem of Fan - Browder Next, we studied general quasi-equilibrium when F = G + H with G : D × K → 2X×Z is a scalar weakly lower semi-continous mapping, H : D × K → 2X×Z is a scalar weakly upper semi-continous mapping and Y = X × Z Lemma 2.1.2 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D and Q : D × K → 2K are l.s.c multivalued mappings with nonempty values; iii) B = {(x, y) ∈ D × K|x ∈ P (x, y), y ∈ Q(x, y)} is a closed set; iv) φ : K × K × D × D → R is a u.s.c function such that for any fixed (y, x) ∈ K × D, φ(y, , x, ) : K × D → R is a convex function; 11 v) φ(y, y, x, x) = 0, for all (y, x) ∈ K × D Then there exists (x, y) ∈ D × K such that (x, y) ∈ P (x, y) × Q(x, y); φ(y, z, x, t) ≥ 0, for all t ∈ P (x, y), z ∈ Q(x, y) We have a theorem Theorem 2.1.2 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D , Q : D × K → 2K are l.s.c multivalued mappings with nonempty values and B = {(x, y) ∈ D × K|x ∈ P (x, y), y ∈ Q(x, y)} is a closed set; iii) G : D × K → 2X×Z is a scalar weakly l.s.c mapping with nonempty values; iv) H : D × K → 2X×Z is a scalar weakly u.s.c mapping with nonempty closed convex values; v) for all (x, y) ∈ B, ∅ = G(x, y) + (H(x, y) ∩ TP (x,y)×Q(x,y) (x, y)) ⊂ TP (x,y)×Q(x,y) (x, y) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) ∈ G(x, y) + H(x, y) Similarly the corollaries of Theorem 2.1.1, we have corollaries of Theorem 2.1.2 2.2 Problem with utility function is a product of two mappings In this section, we consider the existence of solutions of general quasi-equilibrium when the utility is a product of two mappings: F = G × H in cases: G : D × K → 2X is a scalar weakly lower semi-continous mapping, H : D × K → 2X ia s scalar weakly upper semi-continous mapping and Y = X × X G : D × K → 2X is a scalar weakly lower semi-continous mapping, H : D × K → 2Z is a scalar weakly upper semi-continous mapping and Y = X × Z Firstly, we consider the case F = G × H with G : D × K → 2X is a scalar weakly lower semicontinous mapping and H : D × K → 2X is a scalar weakly upper semi-continous mapping Y = X × X Lemma 2.2.1 We assume that the following conditions hold 12 i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a l.s.c multivalued mapping with nonempty values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) φ : K × K × D × D → R is a u.s.c function such that for any fixed (y, x) ∈ K × D, φ(y, , x, ) : K × D → R is a convex function; v) φ(y, y, x, x) = 0, for all (y, x) ∈ K × D Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) φ(y, z, x, t) ≥ 0, for all t ∈ P (x, y), z ∈ Q(x, y) Theorem 2.2.1 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; v) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values; vi) For all (x, y) ∈ P (x, y) × Q(x, y), G(x, y) ⊂ TP (x,y) (x), (H(x, y) ∩ TP (x,y) (y)) = ∅ Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) ∈ G(x, y) × H(x, y) Corollary 2.2.1 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; v) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values; vi) For all (x, y) ∈ P (x, y) × Q(x, y), (G(x, y) − x) ⊂ TP (x,y) (x), (H(x, y) − x) ∩ TP (x,y) (y)) = ∅ 13 Then there exists (x, y) ∈ D × K such that (x, y) ∈ (P (x, y) × Q(x, y)) ∩ (G(x, y) × H(x, y)) Corollary 2.2.2 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) G : D × K → 2X is a scalar weakly l.s.c mapping; v) For all (x, y) ∈ P (x, y) × Q(x, y), x ∈ / G(x, y) and G(x, y) − x ⊂ TP (x,y) (x) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) G(x, y) = ∅ Corollary 2.2.3 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) H : D × K → 2X is a scalar weakly u.s.c mapping with convex compact values; v) For all (x, y) ∈ P (x, y) × Q(x, y), x ∈ / H(x, y) and (H(x, y) − x) ∩ TP (x,y) (x) = ∅ Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ P (x, y) × Q(x, y); 2) H(x, y) = ∅ Corollary 2.2.4 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iii) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values and H(x, y) − x ⊆ TD (x), for all x ∈ D, y ∈ Q(x, y) Then there exists (x, y) ∈ D × K such that (x, y) ∈ H(x, y) × Q(x, y) Corollary 2.2.5 We assume that the following conditions hold i) D is a nonempty convex compact subset; 14 ii) P : D → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) G : D → 2X is a scalar weakly l.s.c mapping with nonempty values such that G(x) ⊂ TP (x) (x), for all x ∈ P (x) Then there exists x ∈ D such that x ∈ G(x) ∩ P (x) Corollary 2.2.6 We assume that the following conditions hold i) D is nonempty convex compact subset; ii) P : D → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) H : D → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values and H(x) ⊂ TP (x) (x), for all x ∈ P (x) Then there exists x ∈ D such that x ∈ H(x) ∩ P (x) Similarly, we consider general quasi-equilibrium problem with the utility function of the form F = G × H in which G : D × K → 2X is a scalar weakly lower semi-continous mapping, H : D × K → 2Z is a scalar weakly upper semi-continous mapping and Y = X × Z In this case, P and Q are l.s.c multivalued mappings We have a theorem Theorem 2.2.2 We assume that the following conditions hold i) D, K are nonempty convex compact subsets; ii) P : D × K → 2D is a l.s.c multivalued mapping with nonempty values; iii) Q : D × K → 2K is a l.s.c multivalued mapping with nonempty values and B = {(x, y) ∈ D × K|x ∈ P (x, y), y ∈ Q(x, y)} is a closed set; iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values; v) H : D × K → 2Z is a scalar weakly u.s.c mapping with nonempty closed convex values; vi) For all (x, y) ∈ B, G(x, y) ⊂ TP (x,y) (x), (H(x, y) ∩ TQ(x,y) (y)) = ∅ Then, there exists (x, y) ∈ D × K such that: 1) (x, y) ∈ P (x, y) × Q(x, y); 2) ∈ G(x, y) × H(x, y) The corollaries of Theorem 2.2.2 are quite similar to the corollaries of Theorem 2.2.1 15 Chương Relevant problems In this chapter, we present some problems related to general quasi-equilibrium problem Simultaneously, we apply the obtained results in Chapter to get sufficient conditions for the existence of solutions of relevant problems 3.1 3.1.1 Generalized quasi-equilibrium problem of type I Put the problem Let X, Y, Z be Hausdorff locally convex topological vector spaces, D ⊂ X, K ⊂ Z be nonempty subsets Given multi-valued mappings S : D × D → 2D , T : D × K → 2K and F1 : K × D × D × D → 2Y We are interested in the problem: Find (x, y) ∈ D × K such that 1) (x, y) ∈ S(x, y) × T (x, y); 2) ∈ F1 (y, x, x, t), for all t ∈ S(x, y) This problem is called a generalized quasi-equilibrium problem of type I, denoted by (GEP)I , in which the multivalued mappings S, T are called constraint mappings, F1 is called a utility multivalued mappings (GEP)I and general quasi-equilibrium problem are equivalent Indeed, we defined the multivalued mapping F : D × K → 2X by F (x, y) = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), ∀t ∈ S(x, y)} If exists (x, y) satisfies 1), 2) and x ∈ F (x, y) then ∈ x − F (x, y) Setting F (x, y) = x − F (x, y) Therefore, the solutions of (GEP)I are the solutions of general quasi-equilibrium problem and vice versa 16 3.1.2 The existence of solution theorem In the study results of N X Tan and T T T Duong, they used S Park fixed point Theorem or Lemma KKM to give the existence to solutions for generalized quasi-equilibrium problem of type I In this section, we apply the obtained new results in Chapter to get the existence to solutions for generalized quasi-equilibrium problem of type I Let X, Y, Z be Hausdorff locally convex topological vector spaces, D ⊂ X, K ⊂ Z be nonempty subsets Given multi-valued mappings S, T and F1 are defined in Section 3.1.1 We apply the Corollary 2.1.1 to have a theorem: Theorem 3.1.1 The following conditions are sufficient for (GEP)I to have a solution: i) D, K are nonempty convex compact subets; ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) T : D × K → 2K is a continuous multivalued mapping with nonempty closed convex values; iv) The set A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is closed; v) for any fixed (y, x) ∈ K × D, the set B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is a nonempty We have a lemma Lemma 3.1.1 We assume that the following conditions hold i) D, K are nonempty convex compact sets; ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed values; iii) T : D × K → 2K is a u.s.c multivalued mapping with nonempty closed values; iv) A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is a closed set; v) for any fixed (y, x) ∈ K × D, B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is a nonempty convex set Then, H : D × K → 2D : H(x, y) = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is a u.s.c multivalued mapping with nonempty closed convex values on D × K 17 Theorem 3.1.2 The following conditions are sufficient for (GEP)I to have a solution: i) D, K are nonempty convex compact sets; ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed values; iii) T : D × K → 2K is a u.s.c multivalued mapping with nonempty closed values; iv) A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is a closed set; v) for any fixed (y, x) ∈ K × D, B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is a nonempty closed set 3.2 3.2.1 Generalized quasi-equilibrium problem of type II Put the problem Let X, Y, Z be Hausdorff locally convex topological vector spaces, given multi-valued mappings P1 : D → 2D , P2 : D → 2D , Q : K × D → 2K and F : K × D × D → 2Y We are interested in the problem: Find x ∈ D such that 1) x ∈ P1 (x); 2) ∈ F (y, x, t), for all t ∈ P2 (x) and y ∈ Q(x, t) This problem is called a generalized quasi-equilibrium problem of type II, denoted by (GEP)II The multivalued mappings P1 , P2 , Q are called constraint mappings and F is called a utility multivalued mapping Extension of constraint conditions of (GEP)II with mappings S, P0 : D × K → 2D , T : D × K → 2K , Q0 : K × D × D → 2K and F2 : K × K × D × D → 2Y Then we have a problem: Find (x, y) ∈ D × K such that 1) (x, y) ∈ S(x, y) × T (x, y); 2) ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y) and v ∈ Q0 (y, x, t) The problem is called general (GEP)II General (GEP)II and general quasi-equilibrium are equivalent problems Indeed, we define multivalued mappings: H : D × K → 2X : H(x, y) = {z ∈ S(x, y)|0 ∈ F2 (y, v, x, t), ∀t ∈ P0 (x, y), v ∈ Q0 (y, x, t)} If, exists (x, y) such that 1), 2) are satisfied and x ∈ H(x, y) then ∈ x − H(x, y) Setting F (x, y) = x − H(x, y) Then, we see the solutions of general (GEP)II are the solutions of general quasi-equilibrium and vice versa 18 3.2.2 The existence of solution theorem Generalized quasi-equilibrium problem of type II was studied by T T T Duong, N X Tan, N T Q Anh, In this Section, we apply the obtained new results in Chapter to get the existence to solution for (GEP)II We have a theorem Theorem 3.2.1 The following conditions are sufficient for (GEP)II to have a solution: i) D is nonempty convex compact set; ii) P1 : D → 2D is a multivalued mapping with a nonempty closed fixed set B = {x ∈ X|x ∈ P1 (x}); iii) P2 : D → 2D is a multivalued mapping having open lower sections and P2 (x) is nonempty and contained in P1 (x) for each x ∈ D; iv) for any fixed t ∈ D, the set A = {x ∈ D|0 ∈ / F (y, x, t), for some y ∈ Q(x, t)} is open in D; v) ∈ F (y, x, x) for any (x, y) ∈ D × K Corollary 3.2.1 The following conditions are sufficient for (GEP)II to have a solution: i) D is nonempty convex compact subset; ii) P1 : D → 2D is a multivalued mapping with a nonempty closed fixed set B = {x ∈ X|x ∈ P1 (x}); iii) Ánh xạ P2 : D → 2D is a multivalued mapping having nonempty valued and lower sections, P2 (x)) ⊆ P1 (x) for each x ∈ D; iv) for any fixed t ∈ D, the multivalued mapping Q(., t) : D → 2D is l.s.c and the multivalued mapping F (., , t) : D × K → 2X is closed; v) ∈ F (y, x, x), for any (x, y) ∈ D × K Next, we study the existence of solution for general (GEP)II We have a lemma Lemma 3.2.1 We assume that the following conditions hold i) D is nonempty convex compact subset; ii) S : D × K → 2D is a multivalued mapping with a nonempty closed fixed set B = {(x, y) ∈ D × K|x ∈ S(x, y), y ∈ T (x, y)}; iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊂ S(x, y) for each (x, y) ∈ D × K; 19 iv) for any fixed t ∈ D, the set A = {(x, y) ∈ D × K|0 ∈ / F2 (y, v, x, t), for some v ∈ Q0 (x, t, y)} is open in D Then, the multivalued mapping G : D × K → 2D defined by G(x, y) = {t ∈ P0 (x, y)|0 ∈ / F2 (y, v, x, t), for some v ∈ Q0 (x, t, y)}, where (x, y) ∈ D × K, is l.s.c on D × K We assume that for any fixed (x, y) ∈ D × K, Q0 (x, , t) : D → 2K is a l.s.c multivalued mapping and F2 (., , , t) : K × K × D → 2Y is a closed mapping We have a lemma Lemma 3.2.2 We assume that the following conditions hold i) D is a nonempty convex compact subset; ii) S : D × K → 2D is a multivalued mapping with nonempty closed values fixed set B = {(x, y) ∈ D × K|x ∈ S(x, y), y ∈ T (x, y)}; iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊆ S(x, y) for each (x, y) ∈ D × K; iv) for any fixed t ∈ D, Q0 (., t, ) : D × K → 2K is a l.s.c multivalued mapping and F2 (., , , t) : K × K × D → 2Y is a closed multivalued mapping Then G : D × K → 2D : G(x, y) = {t ∈ P0 (x, y)|0 ∈ / F2 (y, v, x, t), với v ∈ Q0 (x, t, y) đó}, (x, y) ∈ D × K is a l.s.c multivalued mapping on D × K Theorem 3.2.2 The following conditions are sufficient for general (GEP)II to have a solution: i) D is a nonempty convex compact subset; ii) S : D × K → 2D is a multivalued mapping with nonempty closed values fixed set B = {(x, y) ∈ D × K|x ∈ S(x, y), y ∈ T (x, y)}; iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊂ S(x, y) for each (x, y) ∈ D × K; iv) for any fixed t ∈ D, the set A = {(x, y) ∈ D × K|0 ∈ / F2 (y, v, x, t), for some v ∈ Q0 (x, t, y)} is open in D; 20 v) ∈ F2 (y, v, x, x) for any y ∈ K and v ∈ Q0 (x, x, y) Theorem 3.2.3 The following conditions are sufficient for general (GEP)II to have a solution: i) D is a nonempty convex compact subset; ii) S : D × K → 2D is a multivalued mapping with nonempty closed values fixed set B = {(x, y) ∈ D × K|x ∈ S(x, y), y ∈ T (x, y)}; iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊆ S(x, y) for each (x, y) ∈ D × K; iv) for any fixed t ∈ D, Q0 (., t, ) : D × K → 2K is a l.s.c multivalued mapping and F2 (., , , t) : K × K × D → 2Y is a closed multivalued mapping; v) ∈ F2 (y, v, x, x) for any y ∈ K and v ∈ Q0 (x, x, y) 3.3 3.3.1 Mixed generalized quasi-equilibrium problem Put the problem Let X, Y, Y1 , Y2 , Z be Hausdorff locally convex topological vector spaces, D ⊂ X, K ⊂ Z be nonempty subsets Given multi-valued mappings S : D × D → 2D , T : D × K → 2K , P : D → 2D , Q : K × D → 2K and F1 : K × D × D × D → 2Y1 , F : K × D × D → 2Y2 We have a problem: Find (x, y) ∈ D × K such that 1) x ∈ S(x, y); 2) y ∈ T (x, y); 3) ∈ F1 (y, x, x, t), for all t ∈ S(x, y); 4) ∈ F (y, x, t), for all t ∈ P (x) and y ∈ Q(x, t) The problem is called mixed generalized quasi-equilibrium problem, denoted by (M GQEP ), in which the multivalued mappings S, T, P, Q are call constraint mappings, F1 , F are called utility multivalued mappings Besides, it is the combination of generalized quasi-equilibrium problem of type I and general generalized quasi-equilibrium problem of type II to have a problem: Find (x, y) ∈ D × K such that 1) x ∈ S(x, y); 2) y ∈ T (x, y); 3) ∈ F1 (y, x, v, x), for all v ∈ T (x, y); 4) ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y) and v ∈ Q0 (y, x, t), 21 where, the multivalued mappings: S : D × D → 2D , T : D × K → 2K , P0 : D × K → 2D , Q0 : K × D × D → 2K and F1 : K × D × D × D → 2Y1 , F2 : K × K × D × D → 2Y2 The problem is called general mixed generalized quasi-equilibrium problem, in which the multivalued mappings S, T, P0 , Q0 are call constraint mappings, F1 , F2 are called utility multivalued mappings 3.3.2 The existence of solution theorem The existence of solutions for mixed generalized quasi-equilibrium problem was studied by T T T Duong and N X Tan to rely on the some lemma of P H Sach and L A Tuan In this section, we apply the results obtained in Chapter to get the existence of solutions for general mixed generalized quasi-equilibrium problem We have a theorem Theorem 3.3.1 The following conditions are sufficient for general (M GQEP ) to have a solution: i) D, K are nonempty convex compact sets; ii) S : D × K → 2D is a l.s.c multivalued mapping with nonempty convex values; iii) T : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) P0 : D × K → 2D is a multivalued mapping with nonempty values and has open lower sections; coP0 (x, y) ⊆ S(x, y), for any (x, y) ∈ D × K; v) A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is a closed set; vi) for any fixed (y, x) ∈ K × D, the set B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is nonempty convex; vii) for any fixed t ∈ D, the set A1 = {(x, y) ∈ D × K|0 ∈ / F2 (y, v, x, t), for some v ∈ Q0 (y, x, t)} is open in D viii) for any fixed y, v ∈ K, ∈ F2 (y, v, x, x), for any x ∈ D Corollary 3.3.1 We assume that the following conditions hold i) D, K are nonempty convex compact sets; ii) S : D × K → 2D , T : D × K → 2K are continuous multivalued mapping with nonempty convex values; iii) ψ : K × K × D × D → R is a real function such that 22 a) for any fixed t ∈ D, the function ψ(., , , t) : K × K × D → R are u.s.c; b) ψ(y, v, x, x) ≥ 0, for all y, v ∈ K, x ∈ D Then, there exists (x, y) ∈ D × K such that (x, y) ∈ S(x, y) × T (x, y) and ψ(y, v, x, t) ≥ 0, for all (t, v) ∈ S(x, y) × T (x, y) Next, we apply Corollary 2.1.8 to prove the existence of solutions to general mixed generalized quasi-equilibrium problem Theorem 3.3.2 We assume that the following conditions hold i) D, K are nonempty convex compact sets; ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) T : D × K → 2K is a u.s.c continuous multivalued mapping with nonempty closed convex values; iv) The set A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is closed; v) for any fixed (y, x) ∈ K × D, the set B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)} is nonempty and convex; vi) P0 : D × K → 2D is a multivalued mapping having open lower sections and for each (x, y) ∈ P0 (x, y) × T (x, y), on has ∈ F1 (y, x, x, t), for all t ∈ S(x, y); vii) Q0 : D × D × K → 2K , and F2 : K × K × D × D → 2Y are multivalued mappings such that for any fixed t ∈ D, the set C = {(x, y) ∈ D × K|0 ∈ / F2 (y, v, x, t), với v ∈ Q0 (x, t, y) đó} is open in D; viii) ∈ F2 (y, v, x, x), for all (x, y) ∈ S(x, y) × T (x, y), v ∈ Q0 (x, x, y) Then there exists (x, y) ∈ D × K such that 1) (x, y) ∈ S(x, y) × T (x, y); 2) ∈ F1 (y, x, x, t), for all t ∈ S(x, y); 3) ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y), v ∈ Q0 (x, t, y) 23 Nhận xét 3.3.1 We can understand Theorem 3.3.2 as solving general (GEP )II on set of solution of (GEP )I In particular, for the scalar case, we consider the functions φ1 : K × D × D × D → R, φ2 : K × K × D × D → R We have a corollary Corollary 3.3.2 We assume that the following conditions hold i) D, K are nonempty convex compact sets; ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values; iii) T : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values; iv) A = {(y, x, z, t) ∈ K × D × D × D|φ1 (y, x, z, t) ≤ 0} is a closed set; v) for any fixed (y, x) ∈ K × D, the set B = {z ∈ S(x, y)|φ1 (y, x, z, t) ≤ 0, for all t ∈ T (x, t)} is nonempty and convex; vi) P0 : D × K → 2D is a multivalued mapping having open lower sections and for each (x, y) ∈ P0 (x, y) × T (x, y), φ1 (y, x, x, t) ≥ 0, for all t ∈ S(x, y); vii) Q0 : D × D × K → 2K and φ2 : K × K × D × D → 2Y are multivalued mappings such that for any fixed t ∈ D, the set C = {(x, y) ∈ D × K|φ2 (y, v, x, t) < 0, for some v ∈ Q0 (x, t, y)} is open in D; viii) φ2 (y, v, x, x) ≥ 0, for all (x, y) ∈ S(x, y) × T (x, y), v ∈ Q0 (x, x, y) Then, there exists (x, y) ∈ D × K such that 1) (x, y) ∈ S(x, y) × T (x, y); 2) φ1 (y, x, x, t) ≤ 0, for all t ∈ S(x, y); 3) φ2 (y, v, x, t) ≥ 0, for all t ∈ P0 (x, y) and v ∈ Q0 (x, t, y) It follows from the following remark that Corollary 3.3.3 generalizes Ky Fan’s result and Minty’s result Remark 3.3.1 1) In the case when φ1 is a l.s.c function and for any fixed (y, x, t) ∈ K ×D×D, the function φ1 (y, x, , t) : D → R is a quasi-convex function, then conditions v) and vi) are satisfied 2) In the case when Q0 : D × D × K → 2K is a l.s.c multivalued mapping and φ2 : K × K × D × D → 2Y is a function such that for any fixed t ∈ D, the function φ2 (., , , t) : K ×D ×D → R is u.s.c, then condition vii) is satisfied 24 Conclusions and recommendations A Conclusions The main content of the dissertation is to study general quasi-equilibrium problems of the Blum - Oettli type New findings are as following: 1) Create the sufficient conditions for the existence of solutions to general quasi-equilibrium problems with the utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping 2) Create the sufficient conditions for the existence of solutions to general quasi-equilibrium problems with the utility multivalued mapping is a product of scalar weakly lower semicontinous mapping and scalar weakly upper semi-continous mapping 3) Applied the obtained results in (1) and (2) to study the existence of solutions of relevant problems: generalized quasi-equilibrium problem of type I, generalized quasi-equilibrium problem of type II and mixed generalized quasi-equilibrium problem B New development orientations of the dissertation 1) Study the applications of general quasi-equilibrium problem in control theory, economic problems, etc 2) Study fixed point theorems of separately continous mappings 3) Study the general quasi-equilibrium problem related to the intersection of multivalued mappings ... of the Blum - Oettli type " Many authors studied problems of the Blum - Oettli type, which means that multivalued problems with a utility function is a sum of two mappings as N X Tan and P N Tinh,... Exist or not exist? How to exist? In mathematics, the first question is related to the existence or non-existence of the solution of the equation, the problem is formulated: Find x ∈ D such that... problems of the Blum - Oettli type In this section, we will give out some sufficient conditions for the existence of solutions of general quasi-equilibrium problems of the Blum - Oettli type This is