Math Meth Oper Res DOI 10.1007/s00186-014-0462-0 ORIGINAL ARTICLE On topological existence theorems and applications to optimization-related problems Phan Quoc Khanh · Lai Jiu Lin · Vo Si Trong Long Received: 16 June 2013 / Accepted: 17 January 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies Keywords Continuous selections · Fixed points · Variational relations · Quasivariational inclusions · Nash equilibria · Traffic networks Mathematics Subject Classification 47H10 · 90C47 · 90C48 · 90C99 P Q Khanh Department of Mathematics, International University, Vietnam National University, Hochiminh City, Vietnam e-mail: pqkhanh@hcmiu.edu.vn P Q Khanh Federation University Australia, Ballarat, Victoria, Australia L J Lin Department of Mathematics, National Changhua University of Education, Changhua, Taiwan e-mail: maljlin@cc.ncue.edu.tw V S T Long (B) Department of Mathematics, Cao Thang College of Technology, Hochiminh City, Vietnam e-mail: vstronglong@gmail.com 123 P Q Khanh et al Introduction Existence of solutions takes a central place in the theory for any class of problems and plays also a vital role in applications Studies of the existence of solutions of a problem are based on existence results for important points in nonlinear analysis like fixed points, maximal points, intersection points, etc During a long period in the past, it was believed that such existence results needed both topological and linear/convex structures But, originated from Wu (1959) and Horvath (1991), two directions of dealing with pure topological existence theorems have been developed The first approach is based on replacing convexity assumptions by connectedness conditions, and the second one on replacing a convex hull by an image of a simplex through a continuous map Very recently, in Khanh and Quan (2013), a combination of the two ways was discussed This paper follows the idea of the second approach Recently, this idea was intensively developed in combination with the KKM theory (KKM means Knaster–Kuratowski–Mazurkiewicz) to obtain pure topological existence theorems and applications in the study of the existence solutions to optimization-related problems, (see, e.g., Ding 2005, 2007; Hai et al 2009; Khanh et al 2011; Khanh and Quan 2010; Khanh et al 2009; Park 2008; Park and Kim 1996) Inspired by these results and a definition in Chang and Zhang (1991), in this paper we propose a definition of a general type of KKM mappings in terms of a GFC-space (defined in Khanh and Quan (2010); Khanh et al (2009)) and use it to establish equivalent topological sufficient conditions for the existence of many important points in nonlinear analysis and apply these conditions to various optimization-related problems Our results improve or generalize a number of recent ones in the literature The outline of the paper is as follows Section contains definitions and preliminary facts for our later use In Sect 3, we establish purely topological sufficient conditions for the existence of important points in nonlinear analysis and prove the equivalence of these conditions Then, applications to investigating the solution existence for various optimization-related problems are presented Section is devoted to existence theorems on product GFC-spaces and applications to problems concerning systems of subproblems Preliminaries Recall first some definitions for our later use For a set X , by X and X we denote the family of all nonempty subsets, and the family of the nonempty finite subsets, respectively, of X N, Q, and R denote the set of the natural numbers, the set of rational numbers, and that of the real numbers, respectively, and R = R ∪ {−∞, +∞} If X is a topological space and A ⊂ X , then intA signify the interior of A Let X and Y be nonempty sets For F : Y → X we define F − : X → 2Y and F ∗ : X → 2Y , respectively, by F − (x) = {y ∈ Y : x ∈ F(y)} and F ∗ (x) = Y \ F − (x) F − and F ∗ are called the inverse and dual, respectively, map of F For x ∈ X , F − (x) is called the inverse image, or the fiber, of F at x, F ∗ (x) is called the cofiber of F at x n , n ∈ N, denotes the standard n-simplex, i.e., the simplex with vertices being the points e0 = (1, 0, , 0), , en = (0, , 0, 1) of Rn+1 123 On topological existence theorems and applications Definition 2.1 (Khanh and Quan 2010) Let X be a topological space, Y a nonempty set, and a family of continuous mappings ϕ : n → X, n ∈ N A triple (X, Y, ) is said to be a generalized finitely continuous topological space (GFC-space in short) if, for each finite subset N = {y0 , y1 , , yn } ∈ Y , there is ϕ N : n → X of the family If X is compact, (X, Y, ) is called a compact GFC-space Later we also use (X, Y, {ϕ N }) to denote (X, Y, ) Observe that a GFC-space is equipped only with topological structures, without linear or convex structures The same notion was introduced in Park (2008) under the name “ A -space” (independently and earlier than the GFC-space) These spaces are generalizations of other topological structures as G-convex spaces Park and Kim (1996), FC-spaces Ding (2005, 2007), etc, in order to study topics of existence, mainly in optimization-related problems, without linear or convex structures Note that if Y = X , then (X, Y, ) (written as (X, )) collapses to a FC-space Allowing to take Y different from X may help to have a suitable family {ϕ N : N ∈ Y } in many situations, see, e.g., Example 3.1 below Definition 2.2 (Hai et al 2009; Khanh et al 2011; Khanh and Quan 2010; Khanh et al 2009) Let (X, Y, ) be a GFC-space, Z a topological space, F : Y → Z and T : X → Z F is said to be a KKM mapping with respect to T (T -KKM mapping in short) if, for each N = {y0 , , yn } ∈ Y and each {yi0 , , yik } ⊂ N , one has T (ϕ N ( k )) ⊂ ∪kj=0 F(yi j ), where ϕ N ∈ is corresponding to N and k is the face of n formed by {ei0 , , eik } Definition 2.3 Let (X, Y, ) be a GFC-space, Z a topological space, A a nonempty set, F : A → Z , and T : X → Z F is said to be a general KKM mapping with respect to T (g-T -KKM mapping in short) if, for each N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {i , , i k } ⊂ {0, , n}, one has T (ϕ N ( k )) ⊂ kj=0 F(ai j ), where ϕ N ∈ is corresponding to N and k is the face of n formed by {ei0 , , eik } Note that Definition 2.3 is a natural generalization of Definition 2.1 of Chang and Zhang (1991), where X = Y = Z is a topological vector space, A is a convex subset of another topological vector space, T is the identity map, and ϕ N (·) = co(·) (the usual convex hull) Consequently, it also generalizes Definition 2.1 of Ansari et al (2000) We also see that every T -KKM mapping is a g-T -KKM when A = Y , but the converse is not true as explained by the following example Example 2.1 Let X = Z = R and Y = Q For each N = {y0 , , yn } ∈ Y let ϕ N n n be defined by ϕ N (e) = i=0 λi yi for all e = i=0 λi ei ∈ n Clearly, (X, Y, {ϕ N }) Z is a GFC-space Let F : Y → be given by F(y) ≡ [0, +∞) and T be the identity map Let N = {−1} Then, T (ϕ N ( )) = {−1} ⊂ F(−1) = [0, +∞) Hence, F is not a T -KKM mapping Now, for each N A = {a0 , , an } ∈ A = Y , we take N = {y0 , , yn } = {|a0 |, , |an |} ∈ Y , where | · | denotes absolute value It is easy to see that 123 P Q Khanh et al T (ϕ N ( n )) = [minN , maxN ] ⊂ [0, +∞) = F(ai ), ∀i ∈ {0, , n} This means that F is a g-T -KKM Lemma 2.1 Let (X, Y, {ϕ N }) be a GFC-space, Z a topological space, A a nonempty set, H : Z → A , and T : Z → X Then, the following statements are equivalent (i) for each z ∈ Z and N A = {a0 , , an } ∈ A , there exists N = {y0 , y1 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (z), one has ϕ N ( k ) ⊂ T (z), where k is the simplex formed by {ei0 , , eik }; (ii) H ∗ is a g-T ∗ -KKM mapping Proof (i) ⇒ (ii) Suppose to the contrary that N A = {a0 , , an } ∈ A exists such that, for each N = {y0 , y1 , , yn } ∈ Y , there exists {i , , i k } ⊂ {0, , n}, k T ∗ (ϕ N ( k )) ⊂ H ∗ (ai j ) (1) j=0 We choose N given in condition (i) associated with N A By (1), there are x0 ∈ ϕ N ( k ) and z ∈ T ∗ (x0 ) such that k / z0 ∈ H ∗ (ai j ) = j=0 k Z \ H − (ai j ) = Z \ j=0 k H − (ai j ) j=0 It follows that {ai0 , , aik } ⊂ N A ∩ H (z ) Hence, from (i) one has ϕ N ( k ) ⊂ / Z \ T − (x0 ) = T ∗ (x0 ), a contradiction T (z ) Hence, z ∈ T − (x0 ), and so z ∈ (ii) ⇒ (i) Suppose there exist z ∈ Z and N A = {a0 , , an } ∈ A such that, for each N = {y0 , , yn } ∈ Y , there exists {ai0 , , aik } ⊂ N A ∩ H (z ) such that ϕN ( k) ⊂ T (z ) (2) Since H ∗ is a g-T ∗ -KKM mapping and N is arbitrary, one can take N associated with N A such that T ∗ (ϕ N ( k k )) ⊂ H ∗ (ai j ) (3) j=0 Since {ai0 , , aik } ⊂ N A ∩ H (z ), one has z ∈ k z0 ∈ / Z\ j=0 By (2), there is x0 ∈ ϕ N ( k) H − (ai j ) = k H − (ai j ), i.e., H ∗ (ai j ) (4) j=0 such that x0 ∈ / T (z ) This means that z ∈ T ∗ (x0 ) ⊂ T ∗ (ϕ N ( In view of (3), (5) contradicts (4) 123 k j=0 k )) (5) On topological existence theorems and applications To end this section, we state our variational relation problem For a set U and a point x under consideration, we adopt the notations α1 (x; U ) means ∀x ∈ U ; α2 (x; U ) means ∃x ∈ U Let X and Z be nonempty sets, S : X → X and F : X × X → Z have nonempty values, and R(x, w, z) be a relation linking x ∈ X , w ∈ X and z ∈ Z Our problem is, for α ∈ {α1 , α2 }, (VRα ) find x¯ ∈ X such that , ∀w ∈ S(x), ¯ α(z, F(x, ¯ w)), R(x, ¯ w, z) holds Note that the first variational relation problem was investigated in Khanh and Luc (2008); Luc (2008) Developments have been obtained in some papers, (e.g., Balaj and Lin 2010, 2011; Khanh and Long 2013; Khanh et al 2011; Lin 2012; Luc et al 2010) Topological existence theorems and applications to optimization-related problems 3.1 Topological existence theorems In this subsection, we establish the existence of important objects in applied analysis in pure topological settings of GFC-spaces Let us begin with the existence of continuous selections of set-valued maps For a set-valued map T : X → Z between two topological spaces X and Z , recall that a (single-valued) continuous map t : X → Z is called a continuous selection of T if t (x) ∈ T (x) for all x ∈ X Theorem 3.1 (continuous selections) Let Z be a compact topological space, (X, Y, ) be a GFC-space, and T : Z → X be a set-valued mapping with nonempty values Assume that there are a nonempty set A and H : Z → A such that the following conditions hold (i) H ∗ is a g-T ∗ -KKM mapping; (ii) Z = a∈A int H − (a) Then, T has a continuous selection of the form t = ϕ ◦ ψ for continuous maps ϕ : n → X and ψ : Z → n , for some n ∈ N Proof Since Z is compact, by (ii), there exists N A = {a¯ , , a¯ n } ∈ A such that n n int H − (a¯i ) Then, there is a continuous partition of unity {ψi }i=0 of Z Z = i=0 n −1 associated with the finite open cover {int H (a¯ i )}i=0 From (i) there exists N = {y0 , , yn } ∈ Y associated with N A = {a¯ , , a¯ n } Moreover, due to the GFC-space structure, there is ϕ N : n → X corresponding to N Now, we define the continuous maps ψ : Z → n and t : Z → X , respectively, by n ψ(z) = ψi (z)ei , t (z) = (ϕ N ◦ ψ)(z) i=0 123 P Q Khanh et al Suppose to the contrary that t is not a selection of T , i.e., there exists z ∈ Z and / T (z ), or equivalently, z ∈ Z \ T − (x0 ) = T ∗ (x0 ) t (z ) = ϕ N (ψ(z )) := x0 ∈ Furthermore, one has n ψ(z ) = ψi (z )ei = ψ j (z )e j ∈ J (z ) , j∈J (z ) i=0 where J (z ) := { j ∈ {0, , n} : ψ j (z ) = 0} Since H ∗ is a g-T ∗ -KKM mapping, one has z ∈ T ∗ (x0 ) = T ∗ (ϕ N (ψ(z ))) ⊂ T ∗ (ϕ N ( J (z ) )) H ∗ (a¯ i ) ⊂ i∈J (z ) Hence, there exists j0 ∈ J (z ), z ∈ H ∗ (a¯ j0 ) = Z \ H − (a¯ j0 ), i.e., / H − (a¯ j0 ) z0 ∈ (6) n , On the other hand, by the definition of J (z ) and the partition {ψi }i=0 z ∈ {z ∈ Z : ψ j0 (z) = 0} ⊂ int H − (a¯ j0 ) ⊂ H − (a¯ j0 ), contradicting (6) Finally, putting ϕ = ϕ N we arrive at the conclusion Remark Theorem 3.1 improves Theorem 2.1 of Khanh et al (2011) since assumption (i) is weaker than the corresponding assumption (i) of that result Consequently, it also improves Theorem 2.1 of Ding (2007), Theorem 3.1 of Yannelis and Prabhakar (1983), and Theorem of Yu and Lin (2003) The next example gives a case where Theorem 3.1 is more convenient than Theorem 2.1 of Ding (2007) in terms of FC-spaces and Theorem 2.1 of Khanh et al (2011) with a GFC-space setting Recall that for a FC-space (X, ) and A, B ⊂ X , B is said to be a FC-subspace of X relative to A if, for each N = {x0 , , xn } ∈ X and {xi0 , , xik } ⊂ N ∩ A, ϕ N ( k ) ⊂ B Example 3.1 Let Z = [0, 1], X = R, and F, T : Z → X be defined by F(z) ≡ X , T (z) = [0, z] For the continuous functions ϕ N : n → X defined by, for each e ∈ n, ϕ N (e) = 0, if N ∈ N , √ 2, otherwise (we adopt that N contains also zero), (X, {ϕ N } N ∈ X ) is a FC-space We show that assumption (i) of Theorem 2.1 in Ding (2007) is not satisfied For N ∈ / N , z ∈ Z , and √ {xi0 , , xik } ⊂ N ∩ F(z) = N , one has ϕ N ( k ) = { 2} ⊂ T (z) = [0, z] ⊂ [0, 1] This means that T (z) is not a FC-subspace of X relative to F(z), as that assumption (i) requires Now, take the GFC-space (X, Y, {ϕ N } N ∈ Y ) with Y = Q We claim that assumption (i) of Theorem 2.1 in Khanh et al (2011) is not fulfilled, i.e., there does not exist any map H : Z → 2Y such that, for each z ∈ Z , N = {y0 , , yn } ∈ Y and 123 On topological existence theorems and applications {yi0 , , yik } ⊂ N ∩ H (z), one has ϕ N ( k ) ⊂ T (z) Indeed, suppose to the contrary that such a H exists We take z¯ ∈ Z and N = { y¯0 , , y¯n } ∈ Y√ such that N ∩ H (¯z ) = { y¯i0 , , y¯ik } = ∅ Set N ∗ = N ∪ {0.5} Then, ϕ N ∗ ( k ) = { 2} ⊂ T (¯z ) = [0, z¯ ], a contradiction To apply our Theorem 3.1, we take A = N, and H : Z → A defined by H (z) = F(z) ∩ N For each z ∈ Z and N A = {a0 , , an } ∈ A = N , we choose N ≡ N A ∈ Y to see that, for each {ai0 , , aik } ⊂ N A ∩ H (z) = N A , ϕ N ( k ) = {0} ⊂ T (z) = [0, z], i.e., (i) of Lemma 2.1, which is equivalent to (i) of Theorem 3.1, is fulfilled (ii) of Theorem 3.1 is satisfied because Z = H − (0) = intH − (0) By Theorem 3.1, T has a continuous selection We now apply the above result on continuous selections to prove the following five topological existence results We will first demonstrate a result on fixed points, and then show that it is equivalent to all the other four theorems Theorem 3.2 (fixed points) Let (X, Y, ) be a compact GFC-space and T : X → X Assume that there are a nonempty set A and H : X → A such that the following conditions hold (i) T has nonempty values and H ∗ is a g-T ∗ -KKM mapping; (ii) X = a∈A int H − (a) Then, T has a fixed point x¯ ∈ X , i.e., x¯ ∈ T (x) ¯ Proof According to Theorem 3.1, T has a continuous selection t = ϕ ◦ ψ, where ϕ : n → X and ψ : X → n are continuous Then, ψ ◦ ϕ : n → n is also continuous By virtue of the Tikhonov fixed-point theorem, there exists e¯ ∈ n such that ψ ◦ ϕ(e) ¯ = e ¯ Setting x¯ = ϕ(e), ¯ we have x¯ = ϕ(ψ(x)) ¯ = t (x) ¯ ∈ T (x) ¯ The proof is complete Remark Theorem 3.2 sharpens Corollary 3.1 (ii1 ) of Khanh et al (2011) since assumption (i) is weaker than the corresponding assumption (i) of that result Applied to the special case where X = Y = A is a nonempty compact convex subset of a topological vector space, T ≡ H , and ϕ N (·) = co(·), Theorem 3.2 generalizes Theorem of Browder (1968) Theorem 3.3 (sectional points) Let (X, Y, ) be a compact GFC-space and M be a subset of X × X Assume that there are a nonempty set A and H : X → A such that the following conditions hold (i) for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y / such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ X : (x, w) ∈ M}; (ii) X = a∈A int H − (a); (iii) (x, x) ∈ M for all x ∈ X Then, there exists x¯ ∈ X such that {x} ¯ × X ⊂ M 123 P Q Khanh et al Observe that, for a similar reason as in Remark 2, Theorem 3.3 generalizes Lemma of Fan (1961) Theorem 3.4 (maximal elements) Let (X, Y, ) be a compact GFC-space and T : X → X Assume that there are a nonempty set A and H : X → A such that the following conditions hold (i) H ∗ is a g-T ∗ -KKM mapping; (ii) w∈X T − (w) ⊂ a∈A int H − (a); (iii) x ∈ / T (x) for all x ∈ X Then, T has a maximal point x¯ ∈ X , i.e., T (x) ¯ = ∅ Theorem 3.5 (intersection points) Let (X, Y, ) be a compact GFC-space and G : X → X Assume that there are a nonempty set A and H : X → A such that the following conditions hold (i) H ∗ is a g-G-KKM mapping; (ii) x∈X X \ G(x) ⊂ a∈A int H − (a); (iii) x ∈ G(x) for all x ∈ X Then, x∈X G(x) = ∅ Theorem 3.6 (solutions of variational relations) Let (X, Y, ) be a compact GFCspace, Z a nonempty set, S : X → X , F : X × X → Z , and R(x, w, z) be a relation linking x ∈ X , w ∈ X and z ∈ Z , and i ∈ {1, 2} Assume that there are a nonempty set A and H : X → A such that the following conditions hold (i) for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold}; (ii) w∈X {x ∈ S − (w) : α3−i (z, F(x, w)), R(x, w, z) does not hold} ⊂ a∈A int H − (a); (iii) x ∈ / {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold} for all x ∈ X Then, there exists a solution x¯ ∈ X of (VRαi ), i.e., ∀w ∈ S(x), ¯ αi (z, F(x, ¯ w)), R(x, ¯ w, z) holds Now we will prove the equivalence of the above five theorems following the diagram Theorem 3.2 ⇒ Theorem 3.4 ⇒ Theorem 3.6 ⇒ Theorem 3.2 Theorem 3.3 Theorem 3.5 Theorem 3.2 ⇒ Theorem 3.3 Let T (x) = {w ∈ X : (x, w) ∈ / M} for x ∈ X If there is x¯ ∈ X such that T (x) ¯ = ∅, then {x} ¯ × X ⊂ M and we are done Suppose T (x) = ∅ for all x ∈ X and the conclusion of Theorem 3.3 is false By (i) of Theorem 3.3 and Lemma 2.1, (i) of Theorem 3.2 is satisfied Since the two assumptions (ii) are ¯ w) ∈ / M}, the same, by Theorem 3.2, x¯ ∈ T (x) ¯ for some x¯ ∈ X , i.e., x¯ ∈ {w ∈ X : (x, which contradicts (iii) of Theorem 3.3 123 On topological existence theorems and applications Theorem 3.3 ⇒ Theorem 3.2 Assume that all the assumptions of Theorem 3.2 are fulfilled and set M = {(x, w) ∈ X × X : w ∈ / T (x)} Suppose to the contrary that x ∈ / T (x) for all x ∈ X Then, (x, x) ∈ M for all x ∈ X , i.e., (iii) of Theorem 3.3 holds According to Lemma 2.1, for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), one / M It has ϕ N ( k ) ⊂ T (x) Hence, for each w ∈ ϕ N ( k ), w ∈ T (x), i.e., (x, w) ∈ / M}, i.e., (i) of Theorem 3.3 is satisfied follows that ϕ N ( k ) ⊂ {w ∈ X : (x, w) ∈ Since the two assumptions (ii) are the same, applying Theorem 3.3, one obtains x¯ ∈ X such that {x} ¯ × X ⊂ M It implies that w ∈ / T (x) ¯ for all w ∈ X , contradicting the assumption that T has the nonempty values Theorem 3.4 ⇒ Theorem 3.5 We set T (x) = X \ G − (x) for x ∈ X Then, − T (x) = X \G(x) and T ∗ (x) = G(x) It is not hard to see that, under the assumptions of Theorem 3.5, all assumptions of Theorem 3.4 are fulfilled Therefore, there exists ¯ = ∅ Hence, x¯ ∈ x∈X G(x) x¯ ∈ X such that T (x) ¯ = ∅, i.e., X \ G − (x) Theorem 3.5 ⇒ Theorem 3.4 Under the assumptions of Theorem 3.4, let G(x) = X \ T − (x) for x ∈ X Then, assumptions (i) and (ii) of Theorem 3.4 clearly imply the corresponding assumptions (i) and (ii) of Theorem 3.5 From (iii) of Theorem 3.4, one has x ∈ X \ T − (x) = G(x) for all x ∈ X , i.e., (iii) of Theorem 3.5 is satisfied By Theorem 3.5, there exists x¯ ∈ x∈X G(x) = X \ x∈X T − (x) It follows that ¯ = ∅ x¯ ∈ / x∈X T − (x), that is, T (x) Theorem 3.2 ⇒ Theorem 3.4 Suppose that T (x) = ∅ for each x ∈ X Then, (i) of Theorem 3.4 implies (i) of Theorem 3.2 Since X = w∈X T − (w), (ii) of Theorem 3.2 is satisfied along with (ii) of Theorem 3.4 By Theorem 3.2, T has a fixed point This contradicts (iii) of Theorem 3.4 and we are done Theorem 3.4 ⇒ Theorem 3.6 Let T : X → X be defined by T (x) = {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold} By (i) of Theorem 3.6 and Lemma 2.1, H ∗ is a g-T ∗ -KKM mapping, i.e., (i) of Theorem 3.4 is fulfilled It is not difficult to see that (ii) and (iii) of Theorem 3.6 imply the corresponding (ii) and (iii) of Theorem 3.4 Applying this theorem, we have x¯ ∈ X ¯ w)), R(x, ¯ w, z) holds such that T (x) ¯ = ∅ Consequently, ∀w ∈ S(x), ¯ αi (z, F(x, Theorem 3.6 ⇒ Theorem 3.2 Let the assumptions of Theorem 3.2 be satisfied We define two mappings S : X → X , F : X × X → Z and a relation R by, for x, w ∈ X , S(x) ≡ X, F(x, w) = {z } for an arbitrary z ∈ Z , αi (z, F(x, w)), R(x, w, z) holds ⇔ w ∈ / T (x) Then, one has, for all x ∈ X , {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold} = T (x) Suppose, for all x ∈ X , x ∈ / T (x) Then, (iii) of Theorem 3.6 is fulfilled Clearly, by (ii) of Theorem 3.2, (ii) of Theorem 3.6 is fulfilled Since H ∗ is a g-T ∗ -KKM mapping, by Lemma 2.1, (i) of Theorem 3.6 holds According to this theorem, x¯ ∈ X 123 P Q Khanh et al exists such that, for all w ∈ S(x) ¯ = X, αi (z, F(x, ¯ w)), R(x, ¯ w, z) holds This means that w ∈ / T (x) ¯ for all w ∈ X , contradicting the assumption (i) of Theorem 3.2 that T has nonempty values Remark The existence of the above-mentioned points has been obtained in a number of contributions, to various extents of generality and relaxation of assumptions, see, e.g., recent papers Hai et al (2009); Khanh et al (2011); Khanh and Quan (2010); Khanh et al (2009) Our assumption (i) here is of a very simple form and directly in terms of a general KKM mapping (the map T or H ∗ ) with respect to another map Example 2.1 shows that being such a general KKM mapping is properly weaker than being a usual KKM mapping following Definition 2.2, and hence assumption (i) is weaker than the existing corresponding conditions 3.2 Optimization-related problems A Quasivariational inclusion problems Now we consider the following quasivariational inclusion problem For any given sets U and V , we adopt the notations r1 (U, V ) means U ∩ V = ∅; r2 (U, V ) means U ⊆ V ; r3 (U, V ) means U ∩ V = ∅; r4 (U, V ) means U V, and the convention that r5 = r1 , r6 = r2 Let X, Z , Z be nonempty sets, S : X → X , F : X × X → Z , G : X × Z → Z and K : X × X × Z → Z For r ∈ {r1 , r2 , r3 , r4 } and α ∈ {α1 , α2 }, we consider the following quasivariational inclusion problem (QIPr α ) find x¯ ∈ X such that, ∀w ∈ S(x), ¯ α(z, F(x, ¯ w)), r (K (x, ¯ w, z), G(x, ¯ z)) This formulation was proposed in Hai et al (2009) and has been used in some papers, (e.g., Hai et al 2009; Khanh and Long 2013; Khanh et al 2011) It looks complicated, but the used notations make it include much more particular cases with similar proofs of the existence of solutions Theorem 3.7 For problem (QIPr j αi ), j = 1, , and i = 1, 2, assume that there are Y and such that (X, Y, ) is a compact GFC-space Assume, further that there are a nonempty set A and H : X → A such that the following conditions hold (i) for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ S(x) : α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))}; (ii) w∈X {x ∈ S − (w) : α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))} ⊂ a∈A int H − (a); (iii) x ∈ / {w ∈ S(x) : α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))} for all x ∈ X Then, problem (QIPr j αi ) has a solution 123 On topological existence theorems and applications Proof Employ Theorem 3.6 with the relation R defined by R(x, w, z) holds if and only if r j (F(x, w, z), G(x, z)) Under the assumptions of Theorem 3.7, the assumptions of Theorem 3.6 are easily seen to be satisfied Hence, x¯ ∈ X exists such that, ∀w ∈ S(x), ¯ αi (z, F(x, ¯ w)), R(x, ¯ w, z) holds Consequently, x¯ is a solution of (QIPri α j ) in this case B Stampacchia-type vector equilibrium problems Since problem (QIPr j αi ) includes most of optimization-related problems, sufficient conditions for the existence of their solutions can be derived directly from Theorem 3.7 Here, we mention only some important problems as examples, also for the sake of comparison with several recent existing results First, we discuss a relatively general model of Stampacchia-type vector equilibrium problems, and then apply the obtained result to other ones Let X, Z , S, F, and K be as for problem (QIPr α ) Let Z be a linear space and C : X → Z be nonempty-convex-cone-valued For r ∈ {r1 , r2 , r3 , r4 } and α ∈ {α1 , α2 }, we consider the following Stampacchia-type vector equilibrium problem (VEPr α ) find x¯ ∈ X such that, ∀w ∈ S(x), ¯ α(z, F(x, ¯ w)), r (K (x, ¯ w, z), (−C(x) ¯ \ {0})) Then, Theorem 3.7 becomes a sufficient condition for the existence of solutions to problem (VEPr α ) with the simple replacement of G(x, z) by −C(x) \ {0} Note that, here we not need Z to be equipped with a topology, and C to be closed-valued, as assumed in Lin (2012) The following example provides, for problem (VEPr α ), a case when Theorem 3.7 is applicable, while a recent existing result is not Example 3.2 Let X = Z = Z = R ∪ {±∞}, S : X → X , F : X × X → Z , K : X × X × Z → Z , and C : X :→ Z be given by S(x) = {0}, if x = 1, {1, − x}, if x = 1, F(x, w) ≡ {0}, K (x, w, z) = w2 (x + 1), C(x) ≡ (−∞, 0] Problem (VEPr4 α ) in this case is: find x¯ ∈ X such that, ∀w ∈ S(x), ¯ w (x + 1) ∈ / (0, +∞) To have a GFC-space, take Y = N and {ϕ N } defined by, for N ∈ Y and e ∈ ϕ N (e) = n, 1, if ∈ / N, 0, otherwise 123 P Q Khanh et al Then, (X, Y, {ϕ N }) is a GFC-space Next, we choose A = Y and H : X → A defined by H (x) = N, if x = 1, ∅, if x = To check assumption (i) of Theorem 3.7, consider x ∈ X and N A = {a0 , , an } ∈ A If x = we take N = {a0 + 2, , an + 2} ∈ Y to see that, for each {ai0 , , aik } ⊂ N A ∩ H (x) = N A , ϕN ( k) = {1} ⊂ {w ∈ S(x) : w (x + 1) ∈ (0, +∞)} = {1, − x} If x = then N A ∩ H (1) = ∅ Hence, assumption (i) of Theorem 3.7 is fulfilled Clearly, intH − (a) = int(X \ {1}) = X \ {1} for all a ∈ A, and so a∈A intH − (a)=X \ {1} For (ii) we only need to show that {x ∈ S − (w) : w (x + 1) ∈ (0, +∞)} 1∈ / w∈X Suppose ∈ S − (w) ¯ for some w¯ ∈ X such that w¯ (12 + 1) ∈ (0, +∞) By the / (0, +∞), a contradiction definition of S, w¯ = It implies that w¯ (12 + 1) = ∈ This means that assumption (ii) of Theorem 3.7 is satisfied For each x ∈ X , it is easy to see that x∈ / {w ∈ S(x) : w (x + 1) ∈ (0, +∞)} = {1, − x}, if x = if x = {w ∈ S(1) : w2 (12 + 1) ∈ (0, +∞)} = ∅, Thus, (iii) of Theorem 3.7 is checked We can also check directly that x¯ = is a solution However, Theorem 4.1 of Lin (2012) does not work, since, for each x ∈ X , / [0, +∞) for all w ∈ S(y), i.e., assumption there is no y ∈ X such that w2 (x + 1) ∈ (ii) of that theorem is not satisfied C Nash equilibria Let I = {1, , n} be a set of players and X , , X n nonempty sets A n-person non-cooperative game is a 2n-tuple (X , X , , X n , f , f , , f n ), where the ith player has the nonempty strategy set X i and the pay-off function f i : X := i∈I X i → R For a point x ∈ X , xiˆ stands for its projection onto X iˆ = j =i X j A point x¯ = (x¯1 , x¯2 , , x¯n ) ∈ X is said to be a Nash equilibrium point of if, for all i ∈ I and wi ∈ X i , ¯ ≥ f i (x¯iˆ , wi ) f i (x) n : X × X → R by (x, w) = i=1 ( f i (x) − f (xiˆ , wi )) Then, x¯ is a We define Nash equilibrium point of if and only if x¯ is a solution of the equilibrium problem (NEP) find x¯ ∈ X such that, for all w ∈ X, 123 (x, ¯ w) ≥ On topological existence theorems and applications Theorem 3.8 For the game , assume that there are Y and such that (X, Y, ) is a compact GFC-space Assume further that there are a nonempty set A and H : X → A such that the following conditions hold (i) for x ∈ X and each N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ X : (x, w) < 0}; (ii) w∈X {x ∈ X : (x, w) < 0} ⊂ a∈A int H − (a) Then, has a Nash equilibrium point Proof We simply apply Theorem 3.7 with r = r1 , S(x) ≡ X , Z = R, G(x) ≡ (x, w) (problem (NEP) does not include Z , α, and F) (−∞, 0), and K (x, w) = In this case, we note that (iii) of Theorem 3.7 is always satisfied Example 3.3 Consider a 2-person non-cooperative game with X = X = [0, 1] ∪ [2, 3], f (x1 , x2 ) = 2x1 − 3x2 , and f (x1 , x2 ) = x1 + 2x2 This game is equivalent to the equilibrium problem: find x¯ = (x¯1 , x¯2 ) ∈ X := X × X such that, for all w = (w1 , w2 ) ∈ X , (x, ¯ w) = x¯1 + x¯2 − w1 − w2 ≥ Then, for Y = X and ϕ N defined by, for N ∈ Y and e ∈ ϕ N (e) = n, (3, 3), if N ∈ [2, 3] × [2, 3] , (0, 0) otherwise, (X, Y, {ϕ N }) is a compact GFC-space We choose A = Y and H : X → A defined by H (x) = A, if x = (3, 3), ∅, if x = (3, 3) Consider x = (x1 , x2 ) ∈ X and N A = {a0 , , an } ∈ A If x = (3, 3), we take any N = {y0 , , yn } ∈ [2, 3] × [2, 3] ⊂ Y to see that, for each {ai0 , , aik } ⊂ N A ∩ H (x) = N A , one has ϕN ( k) = {(3, 3)} ⊂ {w ∈ X : (x, w) < 0} = {w ∈ X : x1 + x2 − w1 − w2 < 0} If x = (3, 3), then N A ∩ H (3, 3) = ∅ Hence, assumption (i) of Theorem 3.8 is fulfilled Clearly, intH − (a) = int(X \ {(3, 3)}) = X \ {(3, 3)} for all a ∈ A, and so − a∈A intH (a)=X \ {(3, 3)} (ii) of Theorem 3.8 is satisfied because {x ∈ X : (3, 3) ∈ / w∈X (x, w) < 0} = {x ∈ X : x1 + x2 − w1 − w2 < 0} w∈X Hence, all assumptions of Theorem 3.8 are checked A direct computation gives that the solution set is {(3, 3)} However, trying with Corollary 3.5 of Hai et al (2009), for x = (0, 0), we see that the set 123 P Q Khanh et al {y ∈ X : (x, y) < 0} = ((0, 1] ∪ [2, 3]) × ((0, 1] ∪ [2, 3]) is not convex, and hence assumption (i) of that result is not satisfied D Traffic networks Let a network consist of nodes and links (or arcs) Let Q = (Q , , Q l ) be the set of pairs called O/D pairs, each of which consists of an origin node and a destination one Assume that P j , j = 1, , l, is the set of paths connecting the pair Q j , and that P j includes r j ≥ paths Let m = r1 + + rl , and x = (x1 , , xm ) denote a path (vector) flow Assume that the constraint of the capacity of paths is of the form X = {x ∈ Rm : ≤ xs ≤ s, s = 1, , m} Let a vector cost T (x) = (T1 (x), , Tm (x)) be a multifunction of flow x Let the travel demand g j of the O/D pair Q j depend on equilibrium (vector) flow x¯ and denote the travel vector demand by g = (g1 , , gl ) Denote the Kronecker numbers by 1, if s ∈ P j , 0, if s ∈ / Pj , φ = (φ js ), j = 1, , l, s = 1, , m φ js = Then, the set of all feasible path flows is S(x) ¯ = {x ∈ X : φx = g(x)} ¯ For the case of multivalued costs, the following generalization of the Wardrop equilibrium was proposed in Khanh and Luu (2004) Definition 3.1 (i) A feasible path flow x¯ is said to be a weak equilibrium flow if, ∀Q j , ¯ ∀q, s ∈ P j , ∃z ∈ T (x), [z q < z s ] ⇒ [x¯q = q or x¯s = 0], for j = 1, , l and q, s = 1, , r j (ii) A feasible path flow x¯ is called a strong equilibrium flow if (i) is satisfied with ∃z ∈ T (x) ¯ replaced by ∀z ∈ T (x) ¯ In Khanh and Luu (2004), it is proved that a feasible path flow x¯ is a strong (weak) equilibrium flow if and only if x¯ is a solution of the quasivariational inequality, with α = α1 (α = α2 , respectively), ¯ such that, ∀w ∈ S(x), ¯ α(z, T (x)), ¯ (QVIα ) find x¯ ∈ S(x) z, w − x¯ ≥ Theorem 3.9 For our traffic network problem, with α ∈ {α1 , α2 }, assume that there are Y and such that (X, Y, ) is a compact GFC-space Assume further that there are a nonempty set A and H : X → A such that the following conditions hold 123 On topological existence theorems and applications (i) for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ X : α3−i (z, T (x)), z, x − w < 0}; (ii) w∈X {x ∈ X : α3−i (z, T (x)), z, x − w < 0} ⊂ a∈A int H − (a) Then, the traffic problem has a solution Proof Apply Theorem 3.7 with r = r1 , X = Rm , Z = (Rm )∗ (the dual of X ), Z = R, F(x, w) = T (x) for all w ∈ X , G(x, w) ≡ (−∞, 0) and K (x, w, z) = z, w − x E Saddle points Let B, D be topological spaces and f a real function on B × D We consider the saddle-point problem ¯ d) ¯ ∈ B × D such that, for all (b, d) ∈ B × D, (SPP) find (b, ¯ ¯ f (b, d) ≤ f (b, d) Theorem 3.10 For problem (SPP), assume that there are Y and such that (B × D, Y, ) is a compact GFC-space, a nonempty set A, and H : B × D → A such that the following conditions hold (i) for each (b, d) ∈ B × D and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (b, d), ϕ N ( k ) ⊂ {(b , d ) ∈ B × D : f (b, d ) − f (b , d) < 0}; (ii) (b ,d )∈B×D {(b, d) ∈ B × D : f (b, d ) − f (b , d) < 0} ⊂ a∈A int H − (a) Then, (SSP) has a solution Proof We simply apply Theorem 3.7 with X = B × D, Z = R, r = r1 , α = α1 , S(b, d) ≡ B × D, G(b, d) ≡ (−∞, 0) and K ((b, d), (b , d )) = f (b, d ) − f (b , d) F Constrained minimization problems Let X be a topological space, f : X → R and g : X × X → R Consider the following constrained minimization problem (MP) find x¯ ∈ X such that f (w) ≥ f (x) ¯ for all w satisfying g(x, ¯ w) ≤ Theorem 3.11 For problem (MP), assume that there are Y, such that (X, Y, ) is a compact GFC-space, a nonempty set A, and H : X → A such that the following conditions hold (i) for each x ∈ X and N A = {a0 , , an } ∈ A , there exists N = {y0 , , yn } ∈ Y such that, for each {ai0 , , aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ X : g(x, w) > 0, f (w) − f (x) < 0}; (ii) w∈X {x ∈ X : g(x, w) > 0, f (w) − f (x) < 0} ⊂ a∈A int H − (a) Then, (MP) has a solution Proof To apply Theorem 3.7, we set Z = R, r = r1 , S(x) = {w ∈ X : g(x, w) ≤ 0}, G(x) ≡ (−∞, 0), and K (x, w) = f (w) − f (x) 123 P Q Khanh et al Existence theorems on product GFC-spaces and applications 4.1 Existence theorems on product GFC-spaces Theorem 4.1 (collective fixed points) Let I be an index set, {(X i , Yi , {ϕ Ni })}i∈I be a family of GFC-spaces, Ti : X → X i for i ∈ I and X = i∈I X i be a compact space Assume that there exist a nonempty set Ai and Hi : X → Ai , for i ∈ I , such that the following conditions hold (i) for each i ∈ I , Ti has nonempty values and Hi∗ is a g-Ti∗ -KKM mapping; (ii) X = a i ∈Ai intHi− (a i ) Then, there exists x¯ ∈ X such that x¯i ∈ Ti (x) ¯ for all i ∈ I Proof For each i ∈ I , by Theorems 3.1, Ti has a continuous selection ti = ϕi ◦ ψi , where m i ∈ N and ϕi : m i → X i , and ψi : X → m i are continuous maps Let → m i be the canonical projection of onto m i We = i∈I m i and pi : define two mappings : → X and : X → by, for e ∈ and x ∈ X , (e) = i∈I ϕi ( pi (e)), (x) = i∈I ψi (x) Then, and are continuous and so is ◦ : → By virtue of the Tikhonov fixed-point theorem, there exists e¯ ∈ such that ( ◦ )(e) ¯ = e ¯ Setting x¯ = (e), ¯ we have x¯ = ( (x)) ¯ = ( ¯ i∈I ψi ( x)) = i∈I ϕi ( pi ( ¯ i∈I ψi ( x))) = i∈I (ϕi ◦ ψi )(x) ¯ It follows that x¯i = (ϕi ◦ ψi )(x) ¯ = ti (x) ¯ ∈ Ti (x) ¯ for all i ∈ I Note that Theorem 4.1 sharpens Theorem 3.1 of Khanh et al (2011), and hence also improves Theorem 2.2 of Ding (2007), since assumption (i) here is weaker than the corresponding ones in those results, as illustrated by the following Example 4.1 Let I = {1, 2}, X = X = R, Y1 = Q and Y2 = N For each N1 ∈ Y1 and N2 ∈ Y2 , let continuous functions ϕ N1 : n → X and ϕ N2 : n → X be defined by, for e1 ∈ n and e2 ∈ n , ϕ N1 (e1 ) = 0, if N1 ∈ N , √ 2, otherwise and ϕ N2 (e2 ) = Then, (X , Y1 , {ϕ N1 }) and (X , Y2 , {ϕ N2 }) are GFC-spaces Let T1 : X := X × X → X and T2 : X → X be given by T1 (x) = {0} and T2 (x) = {1} Assumption (i) of Theorem 3.1 in Khanh et al (2011) is not fulfilled, since there does not exist any map H1 : X → 2Y1 such that for each x ∈ X , N1 = {y01 , , yn11 } ∈ Y1 123 On topological existence theorems and applications and {yi10 , , yi1k } ⊂ N1 ∩ H1 (x), one has ϕ N1 ( k1 ) ⊂ T1 (x) Indeed, suppose to the exists We take x ∗ ∈ X and N = { y¯01 , , y¯n11 } ∈ Y1 such that contrary that such a H1 √ N ∩ H1 (x ∗ ) = { y¯i10 , , y¯i1k } = ∅ For N1∗ = N ∪ {0.5}, one has ϕ N1∗ ( k1 ) = ⊂ T1 (x ∗ ) = {0}, a contradiction To apply our Theorem 4.1, we take A1 = A2 = N, H1 : X → A1 and H2 : X → A2 defined by H1 (x) = H2 (x) = N For each x ∈ X and N A1 = {a01 , , an11 } ∈ A1 = N , we choose N1 ≡ N A1 ∈ Y1 to see that, for each {ai10 , , ai1k } ⊂ N A1 ∩ H1 (x) = N A1 , ϕ N1 ( k1 ) = {0} ⊂ T1 (x) = {0}, i.e., (i) of Lemma 2.1, which is equivalent to (i) of Theorem 4.1, is fulfilled It is not difficult to see that the other assumptions of Theorem 4.1 are satisfied On the other hand, we easily check directly that x¯ = (0, 1) is a (collective) fixed point, i.e., ∈ T1 ((0, 1)) and ∈ T2 ((0, 1)) Now, we pass to systems of coincidence points Theorem 4.2 (systems of coincidence points) Let I and J be index sets, {(X i , Yi , {ϕ Ni })}i∈I and {(X j , Y j , {ϕ N })} j∈J be families of GFC-spaces, T j : X → X j for j j ∈ J , Fi : X → X i for i ∈ I , X := i∈I X i be a compact space and X := Aj j∈J X j Assume that there exist nonempty sets Ai , A j and maps H j : X → , G i : X → Ai , for i ∈ I , j ∈ J , such that the following conditions hold (i) for each j ∈ J , T j has nonempty values and H j∗ is a g-T j∗ -KKM mapping; (ii) for each i ∈ I , Fi has nonempty values and G i∗ is a g-Fi∗ -KKM mapping; (iii) X = a j ∈A intH j− (a j ) and X = a i ∈Ai intG i− (a i ) j ¯ for all Then, there exists (x, ¯ x¯ ) ∈ X × X such that x¯i ∈ Fi (x¯ ) and x¯ j ∈ T j (x) (i, j) ∈ I × J Proof For each j ∈ J , by (i), (iii) and Theorem 3.1, T j has a continuous selection t j : X → X j and hence we obtain a continuous map t : X → X defined by t (x) = j∈J t j (x) For each i ∈ I , define two new set-valued maps Pi : X → Ai and Q i : X → X i by Pi (x) = G i t (x) andQ i (x) = Fi t (x) We see from (ii) that, for each i ∈ I and NiA = {a0i , , ani i } ⊂ Ai , there exists Ni = {y0i , , yni i } ⊂ Yi such that, for each {l0 , , lki } ⊂ {0, , n i }, one has the following string of equivalent statements Fi∗ (ϕ Ni ( ki ki )) ⊂ G i∗ (alih ) ⇔ X \ Fi− (ϕ Ni ( ki ki )) ⊂ h=0 X \ G i− (alih ) h=0 ⇔ t − X \ Fi− (ϕ Ni ( ki ki )) ⊂ t − X \ G i− (alih ) h=0 123 P Q Khanh et al ki ⇔ X \ t − Fi− (ϕ Ni ( ki )) ⊂ X \ t − G i− (alih ) h=0 ⇔ Q i∗ (ϕ Ni ( ki ki )) Pi∗ (alih ), ⊂ h=0 i.e., Pi∗ is a g-Q i∗ -KKM mapping While from (iii), we have intG i− (a i ) X = a i ∈Ai It follows that intG i− (a i ) = X = t − (X ) = t − a i ∈Ai a i ∈Ai intt − G i− (a i ) = = a i ∈Ai t − intG i− (a i ) int Pi− (a i ) a i ∈Ai Now that the assumptions of Theorem 4.1 hold for Pi and Q i , it yields a x¯ ∈ X such that, for all i ∈ I , ¯ = Fi (t (x)) ¯ x¯i ∈ Q i (x) ¯ one sees that x¯i ∈ Fi (x¯ ) and x¯ j ∈ T j (x) ¯ for all (i, j) ∈ I × J Setting x¯ = t (x) 4.2 Applications A Systems of variational relations Let I be an index set, {X i }i∈I a family of sets, and X = i∈I X i For i ∈ I , let Si : X ⇒ X i and Ri (x, xi ) be a relation linking x ∈ X and xi ∈ X i We consider the following system of variational relations (SVR) ¯ find x¯ = (x¯i )i∈I ∈ X such that, for all i ∈ I, x¯i ∈ Si (x) ¯ x¯i ) holds and Ri (x, For problem (SVR), we set E i = {x = (xi )i∈I ∈ X : xi ∈ Si (x)} Theorem 4.3 For problem (SVR), assume that X := i∈I X i is a compact space and there are {Yi }i∈I and {ϕ Ni }i∈I such that {(X i , Yi , {ϕ Ni })}i∈I is a family of GFC-spaces, a nonempty set Ai and Hi : X → Ai for i ∈ I such that the following conditions hold (i) for each i ∈ I , x ∈ X , and N Ai = {a0i , , ani i } ∈ Ai , there exists Ni = {y0i , , yni i } ∈ Yi such that, for each {a ij0 , , a ijk } ⊂ N Ai ∩ Hi (x), i 123 On topological existence theorems and applications ϕ Ni ( (ii) X = a i ∈Ai ki ) ⊂ {xi ∈ X i : Ri (x, xi ) holds}, if x ∈ E i , if x ∈ / Ei ; Si (x), int Hi− (a i ) Then, (SVR) admits a solution Proof For i ∈ I , we define Ti : X → X i by Ti (x) = {xi ∈ X i : Ri (x, xi ) holds}, if x ∈ E i , if x ∈ / Ei Si (x), By (i) of Theorem 4.2, it is not hard to see that (i) of Lemma 2.1, which is equivalent to (i) of Theorem 4.1, is satisfied Clearly, (ii) of Theorem 4.2 is just (ii) of Theorem 4.1 According to Theorem 4.1, there exists x¯ = (x¯i )i∈I ∈ X such that x¯i ∈ Ti (x) ¯ for all ¯ a contradiction i ∈ I Suppose that x¯ ∈ / E i Then, by the definition of Ti , x¯i ∈ Si (x), ¯ x¯i ) holds for all i ∈ I Hence, x¯ ∈ E i , i.e., x¯i ∈ Si (x), and Ri (x, B An abstract economy Let I be a finite or infinite set of agents For each i ∈ I , let X i be a nonempty set An abstract economy, see Yannelis and Prabhakar (1983), is a family of triples ϒ = (X i , Si , Pi )i∈I , where Si : X = i∈I X i → X i and Pi : X = i∈I X i → X i are correspondences A solution of ϒ is a point x¯ ∈ X ¯ and Si (x) ¯ ∩ Pi (x) ¯ = ∅ for each i ∈ I For a point x ∈ X , xiˆ satisfying x¯i ∈ Si (x) stands for its projection onto X iˆ = j =i X j For abstract economy ϒ, we set E i = {x = (xi )i∈I ∈ X : xi ∈ Si (x)} Theorem 4.4 For abstract economy ϒ, assume that X := i∈I X i is a compact topological space and there are {Yi }i∈I and {ϕ Ni }i∈I such that {(X i , Yi , {ϕ Ni })}i∈I is a family of GFC-spaces, a nonempty set Ai , and Hi : X → Ai for i ∈ I such that the following conditions hold (i) for each x ∈ X and N Ai = {a0i , , ani i } ∈ Ai , there exists Ni = {y0i , , yni i } ∈ Yi such that, for each {a ij0 , , a ijk } ⊂ N Ai ∩ Hi (x), i ϕ Ni ( (ii) X = a i ∈Ai ki ) ⊂ {xi ∈ X i : Si (x) ∩ Pi (xiˆ , xi ) = ∅}, if x ∈ E i , Si (x), if x ∈ / Ei ; int Hi− (a i ) Then, ϒ has a solution Proof Apply Theorem 4.2 with the relation Ri defined by, for i ∈ I , ˆ Ri (x, xi )holds ⇔ Si (x) ∩ Pi (x i , xi ) = ∅ 123 P Q Khanh et 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variational relation problem For a set U and a point x under consideration,... theorems and applications to optimization-related problems 3.1 Topological existence theorems In this subsection, we establish the existence of important objects in applied analysis in pure topological