In this section, we research the existence problem of solutions to a system of generalized quasi-variational inequalities by using Theorem 2.3.2. Let I be an index set, (Xα)α∈I, (Yα)α∈I two families of topological spaces, and X := Q
β∈IXβ, Y := Q
β∈IYβ. For each α ∈ I, let Tα : X −→ 2Yα, Aα : X −→ 2Xα be two set-valued mappings, and φα : X × Yα× Xα −→R a real-valued function. The system of generalized quasi-variational inequalities (in short, SGQVI) is defined as follow:
(SGQVI) Find (ˆx, ˆy) ∈ X × Y such that for each α ∈ I, ˆ
xα ∈ clAα(ˆx), ˆyα ∈ Tα(ˆx), and φα(ˆx, ˆyα, zα) ≥ 0 for all zα ∈ Aα(ˆx).
If the index set I = {1}, then (SGQVI) reduces to the usual quasi-variational inequality:
Find (ˆx, ˆy) ∈ X × Y such that ˆx ∈ clA(ˆx), ˆy ∈ T (ˆx), and φ(ˆx, ˆy, z) ≥ 0 for all z ∈ A(ˆx).
Recall that a topological space X is called acyclic, if all of its reduced ˇCech homology groups over rationals vanish. In particular, any contractible space is acyclic, and thus any convex or star-shaped set is acyclic. In an H-space X, a function f : X −→ R ∪ {±∞}
is called H-quasiconvex, provided that for each r ∈ R, the set {x ∈ X | f(x) < r} is H-convex. Next result provides an existence theorem of solutions to SGQVI.
Theorem 2.4.1. Let (Xα, Uα, Bα)α∈I be a family of l.c.-spaces with precompact polytopes, (Yα)α∈I a family of topological space, and X :=Q
β∈IXβ, Y :=Q
β∈IYβ. Suppose that for each α ∈ I, Aα : X −→ 2Xα is an almost upper semicontinuous Qα-condensing set-valued mapping with nonempty H-convex values, satisfying A−1α (xα) is compactly open for each xα ∈ Xα, and Tα : X −→ 2Yα is upper semicontinuous with nonempty compact values. If φα : X × Yα× Xα −→R is upper semicontinuous such that
(1) for each (x, yα) ∈ X × Yα, zα 7→ φα(x, yα, zα) is H-quasiconvex, (2) for each x ∈ X, there exists yα ∈ Tα(x) such that φα(x, yα, xα) ≥ 0,
(3) for each (x, zα) ∈ X × Xα, the set {yα ∈ Tα(x) | φα(x, yα, zα) ≥ 0} is acyclic, then there is a solution to SGQVI.
Proof. For each α ∈ I, we define a set-valued mapping Pα : X −→ 2Xα by semi-continuous and Tα : X −→ 2Yα is upper semicontinuous with nonempty compact values, the function x 7→ supyα∈Tα(x)φα(x, yα, zα) is upper semicontinuous, by Proposition 21 of
For any fixed α ∈ I, since φα is upper semicontinuous and Tα(ˆx) is compact, for each zα ∈ Aα(ˆx), there exists yα(zα) ∈ Tα(ˆx) such that φα(ˆx, yα(zα), zα) ≥ 0. This leads us to define a set-valued mapping Gα : Aα(ˆx) −→ 2Tα(ˆx) by
Gα(xα) := {yα ∈ Tα(ˆx) | φα(ˆx, yα, xα) ≥ 0} for all xα∈ Aα(ˆx).
Then the graph of Gα is closed, by the upper semicontinuity of φα. Moreover, since Tα(ˆx) is compact, Gα is upper semicontinuous with nonempty acyclic values by (3).
If the conclusion of Theorem 2.4.1 is false, then there exists β ∈ I such that for each yβ ∈ Tβ(ˆx), there exists a point zβ ∈ Aβ(ˆx) satisfying φβ(ˆx, yβ, zβ) < 0. Let the set-valued mapping Eβ : Tβ(ˆx) −→ 2Aβ(ˆx) be defined by
Eβ(yβ) := {xβ ∈ Aβ(ˆx) | φβ(ˆx, yβ, xβ) < 0} for all yβ ∈ Tβ(ˆx).
Then, by (1) and the H-convexity of Aβ(ˆx), Eβ has nonempty H-convex values. For each xβ ∈ Aβ(ˆx), the set
Eβ−1(zβ) = {yβ ∈ Tβ(ˆx) | zβ ∈ Eβ(yβ)} = {yβ ∈ Tβ(ˆx) | φβ(ˆx, yβ, zβ) < 0}
is open in Tβ(ˆx). By [6, Theorem 3.1], there exist (¯xβ, ¯yβ) ∈ Aβ(ˆx) × Tβ(ˆx) such that
¯
xβ ∈ Eβ(¯yβ) and ¯yβ ∈ Gβ(¯xβ), i.e., φβ(ˆx, ¯yβ, ¯xβ) < 0 and φβ(ˆx, ¯yβ, ¯xβ) ≥ 0, which is a
trivial contradiction. This completes the proof. 2
Remark that Theorem 2.4.1 generalizes and improves a result of Wu and Shen [39, Theorem 8] as follows:
(1) The space X need not be perfectly normal.
(2) Theorem 2.4.1 is concerning a system of generalized quasivariational inequalities.
(3) Theorem 2.4.1 need not have an extra acyclic condition [39, Theorem 8 (iii)].
As a consequence, the following Corollary also improves [39, Corollary 9].
Corollary 2.4.2. Let (Xα)α∈I be a family of locally convex topological vector spaces, Xα∗ the conjugate space with respect to Xα, and X := Q
β∈IXβ, X∗ := Q
β∈IXβ∗. For each α ∈ I, let Aα : X −→ 2Xα be an almost upper semicontinuous Qα-condensing set-valued mapping with nonempty convex values, satisfying A−1α (xα) is compactly open for each xα ∈ Xα, and Tα : X −→ 2Xα∗ be upper semicontinuous with nonempty compact convex values. Then there exists (ˆx, ˆy) ∈ X × X∗ such that for each α ∈ I,
ˆ
xα ∈ clAα(ˆx), ˆyα ∈ Tα(ˆx), and Rehˆyα, zα− ˆxαi ≥ 0 for all zα ∈ Aα(ˆx).
Proof. Let Nα be the family of all neighborhoods of zero in Xα. For each Nα ∈ Nα, let
Uα := {(xα, yα) ∈ Xα× Xα | xα− yα ∈ Nα}.
Then (Xα, {coDα}) is an l.c.-space with precompact polytopes, whose uniformity is Uα := { Uα | Nα ∈ Nα}.
For each (x, yα, zα) ∈ X × Xα∗× Xα, we define
φα(x, yα, zα) := Rehyα, zα− πα(x)i.
Then φα is continuous and satisfies all the conditions of Theorem 2.4.1. Consequently,
the conclusion follows from Theorem 2.4.1. 2
Chapter 3
Maximal Elements For Φ θ -majorized Mappings
In the previous chapter, our existence theorems of maximal elements concern prefer-ence correspondprefer-ences which are majorized by correspondprefer-ences with lower open sections.
It is well known that if a correspondence has open lower sections, then it must be lower semicontinuous. However, a continuous correspondence need not have open lower or upper sections in general. So, it is important for us to consider different types of preferences and obtain some existence results without open lower sections. In 1999, Yuan and Tarafdar [44] derived their existence theorems of maximal elements and equilibria for generalized games and qualitative games, in which the preferences are majorized by upper semicon-tinuous correspondences, instead of being majorized by correspondences which have lower open sections. For details, see [42, 44] and the references wherein.
In this chapter, we first give an existence theorem of maximal elements for a new type of preference correspondences which are Φ-majorized. Then some existence theo-rems for generalized games are obtained in l.c.-spaces (resp., l.c.-space with precompact polytopes) in which the constraint or preference correspondences are Φ-majorized (resp., Qα-condensing).
3.1 Preliminary
Recall that in an H-space (X, {ΓD}), a set-valued mapping T : X −→ 2X is called a KKM mapping, if ΓA ⊆ T (A) for each A ∈ hXi. The following KKM principle is a well-known extension, due to [26]:
Theorem 3.1.1. Let (X, {ΓD}) be an H-space. If F : X −→ 2X is a KKM mapping with closed (resp. open) values, then the family {F (a) | a ∈ X} has the finite intersection property.
By using the above KKM principle, we first establish a general fixed point theorem.
Theorem 3.1.2. Let (X, U , B) be an l.c.-space. If T : X −→ 2X is a compact upper semicontinuous set-valued mapping with nonempty closed H-convex values, then T has a fixed point.
Proof. We may assume that V ∈ B is always closed. Let V ∈ B. Since all open entourages of U form a basis, there exists an open W ∈ U such that W ⊆ V . Note that for each x ∈ X, W (x) is an open neighborhood of x. Since
K := clXT (X)
is compact, there exists a finite set
M := {y1, . . . , yn} ∈ hXi
such that K ⊆Sn
i=1W (yi).
For each yi ∈ M , let
F (yi) := {x ∈ X | T (x) ∩ V (yi) = ∅}.
Since T is upper semicontinuous, each F (yi) is open. Moreover, since T (X) ⊆ K ⊆
Since the conclusion of Theorem 3.1.1 does not hold, F : M −→ 2X cannot be a KKM mapping; that is, there exist N ∈ hM i and xV ∈ ΓN such that semicontinuous with closed values, the graph of T is closed in X × clXT (X), and hence
we have x0 ∈ T (x0). This completes our proof. 2
At the end of this section, we list some propositions which is useful in the following section for proving our main results.
Proposition 3.1.3. [44] Let X be a topological space and Y be a normal space. If F, G : X −→ 2Y have closed values and are upper semicontinuous at x ∈ X, then F ∩ G is also upper semicontinuous at x.
Proposition 3.1.4. [9] Let X be a topological spaces and (Yα)α∈I be a family of compact spaces, Y := Q
β∈IYβ. If each Tα : X −→ 2Yα is an upper semicontinuous set-valued
mapping with closed values, then the set-valued mapping T : X −→ 2Y defined by
T (x) =Y
α∈I
Tα(x), for all x ∈ X,
is also upper semicontinuous with closed values.
Proposition 3.1.5. [36] Let X be a topological space and (Y, U , B) be a compact l.c.-space. For any upper semicontinuous set-valued mapping T : X −→ 2Y, the mapping x 7→ cl[H-coT (x)] is upper semicontinuous with compact H-convex values.