行政院國家科學委員會專題研究計畫成果報告
計畫編號：NSC 90－2115－M－039－001－
執行期限：90 年 08 月 01 日至 91 年 07 月 31 日
主 持 人：林炎成
執行單位：中國醫藥學院通識教育中心
1. 摘要 在本研究報告中，我們在擬 H-空間及積 擬 H-空間上，使用全域交集定理，建立了集合 值映射的定點存在定理。並將此結果應用在抽 象經濟上， 建立了平衡點的存在理論。 最後， 我們也在擬 H-空間上，導出了樊璣型不等式成 立的結果。 關鍵詞：擬 H-空間, q-映射, 積空間, 定點定 理, 平衡點, 抽象經濟, Ky Fan 型不等式. Abstr actIn this projection report, we construct some fixed point results for set-valued maps on pseudo H-spaces and product spaces of pseudo H-spaces by using the whole intersection theorem (Theorem 1 in section 3). As applications, we establish the existence result of the equilibrium point in the abstract economies. We also derive a Ky Fan Type minimax inequality on pseudo H-spaces.
Keywor ds: Pseudo H-spaces, q-maps, Product Space, Fixed Point Theorem, Equilibrium Point, Abstract Economy, Ky Fan type inequality. 2. Pr elimilies
Let I be an index set and for each c, let
Ei be a topological space. Let E=
I i∈
ΠEi. For
each i∈I, let Ti : E → 2Ei be a set-valued
map. A pointx∈E is called a fixed point of T(x)=
I i∈
Π Ti(x) if x∈T(x ). That is, xi∈
Ti ( x ) for each i ∈I, where x is the i
projection of x onto Ei.
In 1992, E. Tarafdar[T92] established
the model of the abstract economies in product H-spaces [H87], and discussed the existence of equilibrium points of the abstract economies by using the fixed point theorem on the product H-spaces. E. Marchi et al [ML93] used the Peleg’s theorems to establish the fixed point theorem in the product H-spaces, and deduce the Ky Fan type inequalities and present a generalization of Ky Fan’s intersection theorem for sets with convex section.
In the recent years, the fixed point theorems on the product space of various generalized spaces have been discussed by many authors. Lin et al [LP98] established the theorems on G-convex spaces. Ansari el al [ALY00] discussed the theorem on the product space of Hausdorff topological
spaces with applications to abstract
economies. D. O’Regan [O98] discussed the theorem on Hausdorff topological space by using the DKT mapping, and also discussed the existence theorem of equilibrium point in abstract economies.
The main results of this projection report is to derive more general fixed point theorems on the product space of pseudo H-spaces. From this direction, we can
establish the existence theorems of
equilibrium point of abstract economies. Now, we first define the pseudo H-space and q-map as follows.
2 Definition 1. Let X be a topological space.
The pair (X, q) is said to be a pseudo H-space if for each nonempty finite subset A of X, the
mapping q : △|A|-1 → 2X is continuous
with nonempty compact values.
Lemma 1. Any finite product space of
pseudo H-spaces is also a pseudo H-space.
Definition 2. Let (X, q) be a pseudo H-space.
A mapping F : X → 2X is a q-map if For
each nonempty finite subset A of X, q(△
|A|-1 )⊂ F(x) A x∪∈ and q(△ |J|-1 )⊂ F(x) J x∪∈ for
all nonempty finite subset J of A.
3. An Inter section Theor em.
We first discuss the following intersection theorem.
Theor em 1. Let (X, q) be a pseudo H-space.
A mapping F : X → 2X is a q-map with
compactly closed (or compactly open) values.
Then ∩ ≠φ
∈LF(x)
x for all nonempty finite
subset L of X. Furthermore, if F(x)
N
x∩∈ is
compact for some nonempty finite subset N
of X, then ∩ ≠φ
∈XF(x)
x .
4. Some Fixed Point Results.
We shall use Theorem 1 to derive the following fixed point theorems.
Theor em 2. Let (X, q) be a pseudo H-space.
The mapping O : X → 2X has open values.
f : X → 2X. Suppose that
(1) for each y∈X, for each nonempty finite
subset A of X\(Oc)-1(y), q( △|A|-1) ⊂ X\(Oc)-1(y),
(2) f-1 (x) contains at least one O(x) for each
x∈X, (3) X= O(x) X x∪∈ ; and (4) Oc(x) N
x∩∈ is compact for some nonempty
finite subset N of X. Then f has fixed point.
The condition (1) in Theorem 2 can be replace by (1’) as follows, the conclusion still hold.
Theor em 3. Let (X, q) be a pseudo H-space.
The mapping O : X → 2X has open values.
f : X → 2X. Suppose that
(1’) for each w∈X, for each nonempty finite
subset A of f(w), q(△|A|-1)⊂f(w),
(2) f-1 (x) contains at least one O(x) for each
x∈X, (3) X= O(x) X x∪∈ ; and (4) Oc(x) N
x∩∈ is compact for some nonempty
finite subset N of X. Then f has fixed point.
By using Theorem 3, we can discuss the following fixed point theorem in the product space of pseudo H-spaces.
Theor em 4. Let I be a finite index. For each
i∈I, (Xi, qi) is a pseudo H-space. X=
I iΠ∈ Xi.
Oi : Xi → 2X has open values. fi : X → 2Xi.
Suppose that
(1) for each i∈I and w∈X, for each
nonempty finite subset Ai of f i(w), qi (△
|Ai|-1
)⊂fi (w),
(2) for each i∈I , fi-1(xi) contains at least one
(3) for each i∈I , X= i( i) iO x X xi∪∈ ; and (4) for each i∈I, ic( i) Ni xi∩∈ O x is compact for
some nonempty finite subset Ni of Xi.
Then f=
I i∈
Πfi has fixed point.
5. Application to Abstr act Economies.
The abstract economic model introduced by Tarafdar [T92] is described on H-space. By adapt the technique of Tarafdar, we can discuss the abstract economies on pseudo H-space. Now, we state the model as follows. Let {(Xi, qi) : i∈I} be a family of pseudo
H-spaces, where I is an index. For each i∈I,
Ti: X=
I i∈
ΠXi→2Xi is the constraint set-valued
mapping and Ui: X→R is the utility or pay
off function. For each i∈I, the preference
set-valued mapping Pi : X→2Xi is defined by
Pi (x) = {yi∈Xi : Ui(yi,xi)>Ui(x)}, where xi∈Xi and Xi = i j I j ≠ ∈ Π Xj for each i∈I. An
abstract economy is defined by E={(Xi, qi, Ti,
Pi): i∈ I}. A point x ∈ X is called an
equilibrium point or a generalized Nash equilibrium point[N50] of the economy E if Ui( x )=Ui( x i, x i)= ) ( sup x T zi∈i Ui(zi, x i) for
each i∈I. Hence, an equilibrium point x∈X
of the economy E is given by xi∈ Ti(x )
and Pi (x )∩Ti (x )=φ for each i∈I.
Theor em 5. Let I be a finite index. For each
i∈I, (Xi, qi) is a pseudo H-space. X=
I i∈
ΠXi.
Oi : Xi → 2X has open values. Pi , Ti : X →
2Xi. Suppose that
(1) for each i∈I and w∈X, for each
nonempty finite subset Ai of T i(w), qi (△
|Ai|-1
)⊂Ti (w), and for each nonempty
finite subset Bi of P i(w), qi (△|Bi|-1)⊂Pi
(w),
(2) for each i∈I , Ti-1 (xi)∩(Gic ∪Pi-1(xi))
contains at least one Oi (xi) for each
xi∈Xi, (3) for each i∈I , X= i( i) iO x X xi∪∈ , (4) for each i∈I, ic( i) Ni xi∩∈ O x is compact for
some nonempty finite subset Ni of Xi;
and
(5) for each i∈I and for each x∈X, xi∈Pi(x)
and Ti(x)≠φ.
Then the abstract economies E={(Xi, qi, Ti,
Pi): i∈I} has an equilibrium point.
6. The Ky Fan type minimax inequality.
Finally, we derive the Ky Fan type minimax inequality[F72] on pseudo H-space.
Theor em 6. Let (X, q) be a pseudo H-space.
The mapping f : X×X → R. Given any ì∈R.
Suppose that
(1’) for each y∈X, for each nonempty finite
subset A of { x∈X : f(x,y) > ì}, q(△
|A|-1
)⊂ { x∈X : f(x,y) > ì},
(2) for each for each nonempty finite subset A of X and z∈△|A|-1
, f(w,w)≤ì for all
w∈q(z).
Then for any given nonempty finite subset L
of X, there is a y0∈X such that f(x,y0)≤ì for
all x∈L. In additional, if
M
x∩∈ { y∈X :
f(x,y)≤ì} is contained in a compact subset of
X for some nonempty finite subset M of X, then there is a y0∈X such that f(x,y0)≤ì for
all x∈X.
We note that under the hypothesis of Theorem 6, if ì=
X x∈
sup f(x,x), then the last assertion can deduce that the following inequality hold : X x X y∈ sup∈ inf f(x,y)≤ X x∈ sup f(x,x).
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