For a nonempty set X, we denote the set of all subsets of X by 2X, and the set of all nonempty finite subsets of X by hXi. In addition, for any subset C of a topological space X, the interior of C is denoted by intXC, and the closure of C is denoted by clXC.
Let {ΓD} be a family of some nonempty contractible subsets of X indexed by A ∈ hXi such that ΓD ⊂ ΓD0 whenever D ⊂ D0. The pair (X, {ΓD}) is called an H-space. Given an H-space (X, {ΓD}), a nonempty subset C of X is called
(1) H-convex , if ΓD ⊆ C for all D ∈ hCi,
(2) weakly H-convex , if ΓD∩ C is nonempty contractible for each D ∈ hCi,
(3) H-compact , if for each D ∈ hXi, there exists a compact weakly H-convex subset K of X such that C ∪ D ⊂ K.
For a nonempty subset C of X, we define the H-convex hull of C as H-coC :=\
{K | K is H -convex in X and C ⊆ K}, and the closed H-convex hull of C as
H-coC :=\
{K | K is closed H-convex in X and C ⊆ K}.
Notice that the intersection of H-convex sets is also an H-convex set if the intersection is nonempty. Therefore, H-coC and H-coC are the smallest H-convex set and closed H-convex set containing C, respectively. If C is a finite subset of X, then H-coC is also called a polytope in X. Further, H-coC can be expressed as
H-coC =[
{H-coD | D ∈ hCi}.
An uniform structure U for a set X is a nonempty family of subsets of X × X such that the following conditions hold:
(1) for any U ∈ U , (x, x) ∈ U for each x ∈ X,
(2) if U ∈ U , then U−1 ∈ U , where U−1 := {(x, y) | (y, x) ∈ U }, (3) for any U ∈ U , there exists V ∈ U such that V ◦ V ⊆ U , where
V ◦ V := {(x, y) | there exists z ∈ X such that (x, z) ∈ V and (z, y) ∈ V },
(4) if U, V ∈ U , then U ∩ V ∈ U ,
(5) if U ⊆ V ⊆ X × X and U ∈ U , then V ∈ U .
In this event, the pair (X, U ) is called an uniform space, whose topology induced by U is the family of all subsets G of X such that for each x ∈ G, there is a V ∈ U such that V (x) ⊆ G, where
V (x) := {y ∈ X | (x, y) ∈ V }.
Every member V ∈ U is called an entourage. An entourage V is symmetric provided that (x, y) ∈ V implies (y, x) ∈ V . A subfamily B of an uniform structure U is called a base, if each member of U contains a member of B. In additions, for any subset C of X, its closure clXC can be expressed as
clXC = \
V ∈B
V (C).
For details on uniform spaces, we refer to [16, 39, 41].
An H-space (X, {ΓD}) is called an l.c.-space, if X is an uniform space whose topology is induced by its uniformity U , and there is a base B consisting of symmetric entourages in U such that for each V ∈ B, the set
V (E) := [
x∈E
V (x)
is H-convex whenever E is H-convex. We shall use the notation (X, U , B) to stand for an l.c.-space. Equivalently, l.c.-spaces can be defined as a milder condition: V (E) is H-convex whenever E is a polytope in X. Under this milder condition, we can prove that V (E) is H-convex whenever E is H-convex. Indeed, for any finite set D = {x1, x2, ..., xn}
This shows that V (E) is H-convex.
In an l.c.-space (X, U , B), a subset K of X is called precompact , if for any V ∈ U , there exists a finite set D such that K ⊆ V (D). An l.c.-space is called an l.c.-space with precompact polytopes if each polytope in X is precompact. For example, a locally convex topological vector space X is an l.c.-space with precompact polytopes, by setting ΓD = coD for all D ∈ hXi.
In 1990, Mehta [22] established a key theorem about condensing maps in a Banach space by using the Kuratowski’s measure of noncompactness. The result is very useful to prove the existence of fixed points for condensing maps. In 1993, Kim [17] generalized Mehta’s result to a locally convex Hausdorff topological vector space by using the measure of nonprecompactness due to Himmelberg et al. [11]. Recall that in a Banach space X, the Kuratowski’s measure α(S) of noncompactness of S is defined by
α(S) := inf{ε > 0 | S ⊆
n
[
i=1
Ki with diam(Ki) < ε for each i}.
The Himmelberg’s measure of nonprecompactness is defined on a locally convex topo-logical vector space with basis B consisting of convex open neighborhoods of 0, and the measure of a subset A is defined by
Q(A) := {V ∈ B | A ⊆ K + V for some precompact set K of X}.
Kim [17] established the following properties on Himmelberg’s measure of nonprecom-pactness:
(i) A is precompact iff Q(A) = B, (ii) if A ⊆ B, then Q(B) ⊆ Q(A), (iii) Q(clA) = Q(A),
(iv) Q(coA) = Q(A),
(v) Q(A ∪ B) = Q(A) ∩ Q(B).
As an extension of the above concepts on locally convex topological vector spaces, Huang, Kuo, and Jeng [15] extend to locally G-convex uniform spaces. Given a locally G-convex uniform spaces with precompact polytopes, they defined the measure of precompactness of A by
Ψ(A) := {V ∈ B | A ⊆ V [S] for some precompact set S}.
Unfortunately, they can’t obtain the property “Ψ(clA) = Ψ(A)” without the linearity.
So, they need some more conditions to derive their fixed point results. Here, we mildly change the definition of measure of precompactness in order to obtain this crucial property.
In an l.c.-space (X, U , B), we define the measure of precompactness of a subset A in X by
Q(A) := {V ∈ B | A ⊆ clXV (K) for some precompact set K of X}.
Note that the larger Q(A) is, the more nearly A is precompact. Under this terminology, we have:
Proposition 1.2.1. Let (X, U , B) be an l.c.-space with precompact polytope, and A, B ⊆ X. Then
(1) A is precompact iff Q(A) = B, (2) if A ⊆ B, then Q(B) ⊆ Q(A), (3) Q(clA) = Q(A),
(4) Q(H-coA) = Q(A),
(5) Q(A ∪ B) = Q(A) ∩ Q(B).
Proof. For (1), suppose that A is a precompact set. Then for any V ∈ B, we have (x, x) ∈ V for all x ∈ A. It follows that x ∈ V (x) ⊆ V (A). This result implies that A ⊆ V (A) ⊆ clX[V (A)]. This shows that V ∈ Q(A) for all V ∈ B. Thus, B ⊆ Q(A) and hence B = Q(A). Conversely, suppose Q(A) = B. Then for any U ∈ U , there exists a V ∈ U such that V ◦ V ⊆ U . Since B is a base, we can take this V ∈ B. Similarly, we have some V0 ∈ B such that V0 ◦ V0 ⊆ V . Since V0 ∈ B = Q(A), there exists a precompact set K such that A ⊆ clX[V0(K)]. Further, we have a finite set F such that K ⊆ V0(F ).
Thus,
A ⊆ clX[V0(K)]
⊆ clX[V0(V0(F ))] = clX[(V0◦ V0)(F )]
⊆ clX[V (F )]
⊆ V (V (F )) = (V ◦ V )(F )
⊆ U (F ).
Since U ∈ U is arbitrary, A is precompact.
For (2), if A ⊆ B, then for any V ∈ Q(B), B ⊆ clX[V (K)] for some precompact set K. So A ⊆ B ⊆ clX[V (K)], and hence V ∈ Q(A). This means that Q(B) ⊆ Q(A).
For (3), it is sufficient to prove that Q(A) ⊆ Q(clXA). Indeed, if V ∈ Q(A), then there is a precompact set K such that A ⊆ clX[V (K)]. It follows that clXA ⊆ clX[V (K)]
and hence V ∈ Q(clXA).
For (4), we have to show that Q(A) ⊆ Q(H-coA). If V ∈ Q(A), then A ⊆ clX[V (K)]
for some precompact set K. Consequently,
H-coA ⊆ H-co{clX[V (K)]} ⊆ clX[V (H-coK)],
since clX[V (H-coK)] is a H-convex set containing clX[V (K)]. Since H-coK is precom-pact, it follows that V ∈ Q(H-coA). This shows that Q(A) ⊆ Q(H-coA).
For (5), since Q(A ∪ B) ⊆ Q(A) and Q(A ∪ B) ⊆ Q(B), we have Q(A ∪ B) ⊆ Q(A) ∩ Q(B). Conversely, suppose V ∈ Q(A) ∩ Q(B). Then there exists precompact sets K1 and K2 such that A ⊆ clX[V (K1)] and B ⊆ clX[V (K2)]. Hence
A ∪ B ⊆ clX[V (K1)] ∪ clX[V (K2)] ⊆ clX[V (K1∪ K2)].
Since K1∪K2 is also precompact, it follows that V ∈ Q(A∪B), and hence Q(A)∩Q(B) ⊆
Q(A ∪ B). 2
Let (Xα, Uα, Bα)α∈I be a family of l.c.-spaces with precompact polytopes, where I is a finite or infinite index set, and let X =Q
β∈IXβ be the product H-space. For each α ∈ I,
let πα be the projection of X onto Xα, and Qα be a measure of precompactness in Xα. We say that a set-valued mapping Tα : X −→ 2Xα is Qα-condensing , if
Qα(πα(C)) ( Qα(Tα(C))
for every C satisfying πα(C) is a nonprecompact subset of Xα. It is easy to check that Tα : X −→ 2Xα is Qα-condensing whenever Xα is compact. Also, in case I = {1}, the projection πα is the identity on X. Thus, the above definition reduces to the usual Q-condensing mapping T : X −→ 2X; see for example [17, 22].
Mehta’s Theorem. [22] Let E be a Banach space, D a nonempty closed bounded convex subset of E. and T : D −→ 2D be a condensing mapping ( in Kuratowski’s sense ). Then there exists a nonempty compact convex subset K of D such that T (K) ⊆ K.
Kim’s Theorem. [17] Let E be a locally convex Hausdorff topological vector space, D be a nonempty closed bounded convex subset of E and T : D −→ 2DX be a condensing mapping ( in Himmelberg’s sense ). Then there exists a nonempty compact convex subset K of D such that T (K) ⊆ K.
Besides, we review some concepts about the product of H-space. Let (Xα, ΓDαα)α∈I be a family of H-spaces, where I is a finite or infinite index set. Tarafdar [36] has shown that the product space X =Q
β∈IXβ with product topology is also an H-space, together with the family {ΓD | D ∈ hXi}, which is defined by
ΓD =Y
α∈I
ΓDαα.
where Dα is the projection of D onto Xα. Under this terminology, we have the following propositions.
Proposition 1.2.2. [41] The projection of an H-convex set in the product H-space X =
The following fundamental theorem in [41], which generalizes Mehta’s and Kim’s result to the general setting of l.c.-spaces, shall play an important role in many theoretical applications. For the sake of completeness, we give the proof in detail.
Theorem 1.2.4. Let (Xα, Uα, Bα)α∈I be a family of l.c.-spaces with precompact polytopes, X := Q
β∈IXβ, and Tα : X −→ 2Xα be Qα-condensing. Then there exists a nonempty compact H-convex subset K :=Q
β∈IKβ of X such that Tα(K) ⊆ Kα.
Proof. Fix any x0 ∈ X. Let F be the family of all closed H-convex subsets C of X which contains x0 and satisfies the following conditions: C = Q
α∈ICα, where Cα are closed
Tα(x) ∈ Kα. That is, Tα(K) ⊆ Kα for each α ∈ I.
Applying Proposition 1.2.3, we can obtain a closed H-convex set K00, defined by K00 :=Y
Finally, by Proposition 1.2.3, we can easily check that
clXαKα0 = clXα[πα(K0)] = clXα[πα(H-co{x0} ∪ T (K))]
⊆ clXα[H-coπα({x0} ∪ T (K))]
= H-co(πα({x0} ∪ T (K)))
= H-co(πα({x0}) ∪ πα(T (K)))
= H-co(πα({x0}) ∪ Tα(K)).
To show that K is compact, it is sufficient to show that each Kα is compact by Tychonoff theorem. Assume that Kα is not compact for some α ∈ I. Thus, Kα is not precompact since Kα is closed. It follows that
Qα(Kα) = Qα(πα(K)) ( Qα(Tα(K)).
Applying Proposition 1.2.1, we have
Qα(clXαKα0) ⊇ Qα(H-co(Pα({x0}) ∪ (Tα(K))))
= Qα(πα({x0}) ∪ (Tα(K)))
= Qα(πα({x0}) ∩ Qα(Tα(K)))
= B ∩ Qα(Tα(K))
= Qα(Tα(K)) ) Qα(Kα),
which contradicts with the fact Kα = clXαKα0. Therefore, Kα is compact for each α ∈ I
and the proof is complete. 2
At the end of this section, we list some basic known definitions and propositions, which will be used in this thesis. Let X be a topological space, (X, U , B) an l.c.-space, θ : X −→ Y a single-valued map, and S, T : X −→ 2Y be two set-valued mappings.
(1) T is said to be compactly open lower section (resp. open lower section ), if for each y ∈ Y , T−(y) is compactly open (resp. open) in X .
(2) T is said to be upper semicontinuous (u.s.c.), if for each x ∈ X and each open subset V of Y with T (x) ⊆ V , there exists a neighborhood Nx of x such that T (z) ⊆ V for all z ∈ Nx.
(3) T is said to be lower semicontinuous (l.s.c.), if for each x ∈ X, and any open subset V of Y with T (x) ∩ V 6= ∅, there exists a neighborhood Nx of x such that T (z) ∩ V 6= ∅ for all z ∈ Nx.
(4) T is said to be almost upper semicontinuous (a.u.s.c.), if for each x ∈ X and each open subset V of Y with T (x) ⊆ V , there exists a neighborhood Nx of x such that T (z) ⊆ clYV for all z ∈ Nx.
(5) T is said to be almost lower semicontinuous (a.l.s.c.), if for each x ∈ X and each V ∈ U , there exists a neighborhood Nx of x such that
\
z∈Nx
V (T (z)) 6= ∅.
(6) The set-valued mappings S ∩ T : X −→ 2Y, H-coS : X −→ 2Y, and clT : X −→ 2Y are defined by
(S ∩ T )(x) := S(x) ∩ T (x), H-coS(x) := H-co[S(x)], and
clT (x) := clYT (x) for all x ∈ X, respectively.
Proposition 1.2.5. [44] Let X, Y be two topological spaces and A be a closed (resp.
open) subset of X. If F1 : X −→ 2Y and F2 : X −→ 2Y are lower semicontinuous (reps.
upper semicontinuous) such that F1(x) ⊆ F2(x) for all x ∈ A, then the set-valued mapping F : X −→ 2Y defined by
F (x) := F1(x) , if x ∈ A, F2(x) , if x ∈ X \ A.
is also lower semicontinuous ( reps. upper semicontinuous ).
Proposition 1.2.6. Let (Xα, Uα, Bα)α∈I be a family of l.c.-spaces, X = Q
β∈IXβ, and U =Q
β∈IUβ be the product uniformity on X. Then there is a base B such that (X, U , B) forms an l.c.-space.
Proof. It suffices to show that the product uniformity U has a base B consisting of sym-metric entourages such that, for each V ∈ B and for each H-convex set E, V (E) is H-convex. For each Vα ∈ Bα, Vα(πα(E)) is H-convex by Proposition 1.2.2. Now, we
so that each Vj(E) is H-convex. Therefore,
V (E) = Y
Chapter 2
Maximal Elements for L θ -majorized Mappings
In 1976, Borglin and Keiding [3] used new concepts of K.F.correspondences and KF -majorized correspondences for their existence results of maximal elements. The second notion was extended by Yannelis and Prabhakar [42] to L-majorized correspondences. In this chapter, we consider a more general setting of Lπα-majorized Qα-condensing map-pings Tα : X −→ 2Xα, where each Xα is an l.c.-space with precompact polytopes, and X := Q
β∈IXβ. Without any linear structure and compactness, we shall prove that the family {Tα | α ∈ I} admits a common maximal element under the mild condition that each {x | Tα(x) 6= ∅} is compactly open. As an application, we derive a new existence theorem of equilibria for noncompact abstract economies. Finally, the existence of solu-tions to a system of generalized quasi-variational inequalities in l.c.-spaces are also derived.
2.1 Preliminary
In order to establish our main results of this chapter, we give basic definitions and list some properties. Let X be a topological space, Y an H-space, T : X −→ 2Y a set-valued mapping, and θ : X −→ Y be a single-valued map.
(1) T is said to be of class Lθ, if
(a) for each x ∈ X, θ(x) /∈ H-coT (x),
(b) for each y ∈ Y , T−1(y) is compactly open in X.
(2) A set-valued mapping Tx : X −→ 2Y is an Lθ-majorant of T at x , if there exists an open neighborhood Nx of x in X such that
(a) for each z ∈ Nx, T (z) ⊆ Tx(z) and θ(z) /∈ H-coTx(z), (b) for each y ∈ Y , Tx−1(y) is compactly open in X.
(3) T is said to be Lθ-majorized , if for each x ∈ X with T (x) 6= ∅, there exists an Lθ-majorant of T at x.
Here, we say that a nonempty subset D of a topological space X is compactly open, if D ∩ K is open for all compact subsets K of X. In case θ : X −→ X is the identity map on X, with Y = X, all notations above are simplified to be of class L, L-majorant, and L-majorized, respectively. For more details, see [7, 19, 42].
Proposition 2.1.1. Let X be a topological space, (Y, ΓD) an H-space, and T : X −→ 2Y be a set-valued mapping. If for each y ∈ Y , T−1(y) is compactly open in X, then so is (H-coT )−1(y).
Proof. For each y0 ∈ Y and any given nonempty compact subset K of X, we fix arbitrary x0 ∈ K ∩ [(H-coT )−1(y0)]. Since y0 ∈ H-coT (x0), there exist y1, y2, . . . , yn ∈ T (x0) such that
y0 ∈ H-co{y1, y2, . . . , yn}.
For each i = 1, 2, · · · , n, the set T−1(yi) is compactly open in X; i.e., K ∩ T−1(yi) is open in K and x0 ∈ T−1(yi). We define
U :=
n
\
i=1
(K ∩ T−1(yi)).
Then U is open in K and x0 ∈ U . To complete the proof, we must show that U ⊆ K ∩ [(H-coT )−1(y0)]. For any x ∈ U , we have x ∈ K and x ∈ T−1(yi) or yi ∈ T (x) for all i = 1, 2, · · · , n. Hence
y0 ∈ H-co{y1, y2, . . . , yn} ⊆ H-coT (x),
i.e., x ∈ (H-coT )−1(y0). Consequently, x0 ∈ U ⊆ K ∩ [(H-coT )−1(y0)]. 2
In 1992, Tarafdar [34] proved the following fixed point theorem:
Theorem A. Let X = Q
α∈IXα be the product space of compact H-spaces Xα, α ∈ I.
Suppose that Tα : X → 2Xα satisfies the following conditions for each α ∈ I.
(1) For each x ∈ X, Tα(x) is a nonempty H-convex subset of Xα for each α ∈ I.
(2) For each xα ∈ Xα, Tα−1(xα) contains an open subset Oxα (may be empty) of X such that S
xα∈XαOxα = X.
Then T :=Q
α∈ITα has a fixed point.
Based on the above results, we are able to show a generalized fixed point theorem as follows.
Theorem 2.1.2. Let (Xα, Uα, Bα)α∈I be a family of l.c.-spaces with precompact polytopes, and X :=Q
β∈IXβ. For each α ∈ I, let Tα: X −→ 2Xα be Qα-condensing such that (1) for each x ∈ X, Tα(x) is a nonempty H-convex subset of Xα,
(2) for each xα ∈ Xα, Tα−1(xα) contains a compactly open subset Oxα of X such that S
xα∈XαOxα = X (where Oxα may be empty for some xα).
Then T :=Q
α∈ITα has a fixed point.
Proof. By Theorem 1.2.4, there exists a nonempty compact H-convex subset K of X with
Corollary 2.1.3. Let (X, U , B) be an l.c.-space with precompact polytopes. and T : X −→
2X be Q-condensing such that