行政院國家科學委員會補助專題研究計畫成果報告
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※ 拓樸擬單調性與向量值平衡態問題 ※
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Topological Pseudomonotonicity and ※
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Vector Valued Equilibrium Problems ※
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計畫類別：
5
個別型計畫 □整合型計畫
計畫編號：
NSC
90 – 2115 – M – 110 – 019
執行期間：
90 年 8 月 1 日 至 91 年 7 月 31 日
計畫主持人：
蔣 永 延
共同主持人：
計畫參與人員：
本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
執行單位：
國立中山大學應用數學系
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Topological Pseudomonotonicity and Vector
Valued Equilibrium Problems
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(Yungyen Chiang)
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[email protected]
Abstract. In this project, we intro-duce the topological pseudomonotonicity to vector valued bifunctions, and derive some existence results for vector equilib-rium problems with the corresponding bi-functions topologically pseudomonotone.
Key words : Vector equilibrium
problem, vector variational inequality, Fan-KKM Theorem, vector topological pseudomonotonicity.
1
Introduction
Throughout the report, let Z be a real topological vector space with an ordering cone C, that is, C is a closed convex cone
in Z with IntC 6= ∅ and C 6= Z, where IntC denotes the interior of C.
Let K be a nonempty subset of a real
topological vector space X. The
vec-tor equilibrium problem associated with a bifunction f from K × K into Z, (VEP(f, K) for short), is the problem to find bx ∈ K such that
f (bx, y) ∈ (−IntC)c
for all y ∈ K, where (−IntC)c is the
com-plement of −IntC in Z.
The main work of the project is to derive some existence results for the VEP(f, K) with f topologoically pseu-domonotone.
2
Preliminaries
For a given net {zα} in Z, let
Aα= {zβ : β α}
for every α. By use of the vector superior and inferior defined in [1], the limit supe-rior and limit infesupe-rior of the given net are respectively defined by Limsup zα = Inf ( [ α Sup Aα) ; Liminf zα = Sup ( [ α Inf Aα) .
The given function f is called vector topologically pseudomonotone if for every v ∈ IntC and for each net {xα}α∈I in K
satisfying
xα → x ∈ K and
Liminf f (xα, x) ∩ (−IntC) = ∅ ,
there is α0 in the index set I such that
{f (xβ, y) : β ≥ α}
⊂ f (x , y) + v − IntC for all α ≥ α0 and for all y ∈ K.
Recall that a function f of a topolog-ical space X into Z is called C-upper semicontinuous if for every z ∈ Z the set f−1(z − IntC) is open in X, (see [5]). Theorem 1 ([3], Theorem 2.4) Let X be a Hausdorff topological space, and let f be a function of X into Z. Then f is C-upper semicontinuous on X if and only if for every x ∈ X, for every
v ∈ IntC, and for any net {xα}α∈I in
X converging to x, there is an α0 in the
index set I such that
{f (xβ) : β ≥ α} ⊂ f (x) + v − IntC
for all α ≥ α0.
Corollary 2 ([3], Corollary 2.6) Let X be a Hausdorff topological space, let K be a nonempty subset of X, and let f : K × K −→ Z be a bifunction. If f is C-upper semicontinuous with respect to the first argument, then f is vector topologically pseudomonotone.
The bifunction f is said to satisfy the (L)-condition if for any x, y ∈ K and any net {xα} in K converging to x, one has
f (x , y) ∈ (−IntC)c whenever
f (xα, (1 − t)x + ty) ∈ (−IntC)c
for all α and 0 ≤ t ≤ 1.
Theorem 3 ([3], Theorem 2.7) Let
X be a Hausdorff topological space,
and let K ⊂ X be nonempty. If
f : K × K −→ Z is vector topologically pseudomonotone, then f satisfies the (L)-condition.
3
Existence Results
The main existence results for the VEP are derived by use of a consequence of the Fan-KKM Theorem [4] stated as Theo-rem 4.
Let K be a nonempty subset of a topo-logical vector space X. A set valued func-tion Φ form K into the family of sub-sets of X is a KKM mapping if for any nonempty finite set A ⊂ K, the convex
hull of A is contained in [
x∈A
Φ(x).
Theorem 4 Let K be a nonempty con-vex subset of a Hausdorff topological
vec-tor space X, and let Φ : K −→ 2X be a
KKM mapping. If Φ(x) is closed in X for every x, and if there is a nonempty compact convex set D ⊂ K such that
\
x∈D
Φ(x) is a compact subset of K, then \
x∈K
Φ(x) is nonempty.
To state the main results, we need some convexity for bifunctions. First, we generalize the notion of 0-diagonal con-vexity introduced by Zhow and Chen [6] to the vector case.
Let X be a topological vector space, and let K be a nonempty convex subset of X. A bifunction f : K × K −→ Z will be called vector 0-diagonally convex if for any finite set {y1, ... , yn} ⊂ K,
n X j=1 λjf (x , yj) ∈ (−IntC)c whenever x = n X j=1
λjyj with λj ≥ 0 for all
j and
n
X
j=1
λj = 1.
We also consider C-quasiconvex-like bi-functions introduced by Ansari and Yao
[2]. The bifunction f : K × K −→ Z is called C-quasiconvex-like if
f (x , ty1+ (1 − t)y2) ∈ f (x , y1) − C
or
f (x , ty1+ (1 − t)y2) ∈ f (x , y2) − C
for all x, y1, y2 ∈ K and for 0 ≤ t ≤ 1.
We now state the main results. In
the rest of the report, let K denote a nonempty and convex subset of a a Haus-dorff topological vector space X, and let f be a bifunction from K × K into Z
Theorem 5 ([3], Theorem 3.3) Assume that f satisfies the following conditions.
(i) f is vector 0-diagonally convex.
(ii) f is vector topologically
pseu-domonotone.
(iii) For every y ∈ K the function x 7−→ f (x , y) is C-upper semicontinuous on the convex hull of every nonempty finite subset of K.
(iv) There is a nonempty compact set A ⊂ K, and there is a nonempty compact convex set B ⊂ K such that if x ∈ K ∩ Ac, then f (x , y
x) ∈
−IntC for some yx ∈ B.
Then the VEP(f, K) has a solution.
Theorem 6 ([3], Theorem 3.4) Assume that f satisfies the following conditions.
(i) f (x , x) ∈ (−IntC)c for all x ∈ K. (ii) f is C-quasiconvex-like
(iii) f is vector topologically
pseu-domonotone.
(iv) For every y ∈ K the function x 7−→ f (x , y) is C-upper semicontinuous on the convex hull of every nonempty finite subset of K.
(v) There is a nonempty compact set A ⊂ K, and there is a nonempty compact convex set B ⊂ K such that if x ∈ K ∩ Ac, then f (x , yx) ∈
−IntC for some yx ∈ B.
Then the VEP(f, K) has a solution.
Theorem 5 follows from Theorem 3 and the following result.
Lemma 7 ([3], Lemma 3.5) Assume that f satisfies the following conditions.
(i) f is vector 0-diagonally convex. (ii) f satisfies the (L)-condition.
(iii) For every y ∈ K the function x 7−→ f (x , y) is C-upper semicontinuous on the convex hull of every nonempty finite subset of K.
(iv) There is a nonempty compact set A ⊂ K, and there is a nonempty compact convex set B ⊂ K such that if x ∈ K ∩ Ac, then f (x , yx) ∈
−IntC for some yx ∈ B.
Then the VEP(f, K) has a solution.
Theorem 6 is proved by the following two lemmas.
Lemma 8 ([3], Lemma 3.8) Let x ∈ K, and let E be a nonempty finite subset of K. Assume that f is C-quasiconvex-like. If f (x , y) ∈ (−IntC)c for some y in the convex hull of E, then there exists y ∈ E such that f (x , y) ∈ (−IntC)c.
Lemma 9 ([3], Lemma 3.11)
Assume that f satisfies the
follow-ing conditions.
(i) f (x , x) ∈ (−IntC)c for all x ∈ K.
(ii) f is C-quasiconvex-like.
(iii) For every y ∈ K the function x 7−→ f (x , y) is C-upper semicontinuous on the convex hull of every nonempty finite subset of K.
(iv) f satisfies the (L)-condition.
(v) There is a nonempty compact set A ⊂ K, and there is a nonempty compact convex set B ⊂ K such that if x ∈ K ∩ Ac, then f (x , y
x) ∈
−IntC for some yx ∈ B.
Then the VEP(f, K) has a solution.
References
[1] Q.H. Ansari, X.C. Yang and J.C. Yao : Existence and Duality of Im-plicit Vector Variational Problems. Preprint.
[2] Q.H. Ansari and J.C. Yao : An existence result for the generalized vector equilibrium problem, Appl. Math. Lett., 12(8), 53 - 56 (1999). [3] O. Chadli, Y. Chiang and S. Huang
: Topological pseudomonotonicity
and vector equilibrium problems, J. Math. Anal. Appl., 270, pp. 435-450, 2002.
[4] K. Fan : A Generalization of
Tychonoff’s Fixed-Point Theorem, Math. Ann., Vol.142, pp. 305 - 310, 1961.
[5] T. Tanaka : Generalized Semicon-tinuity and Existence Theorems for Cone Saddle Points, Appl. Math. Optim., 36 (1997), 313 - 322.
[6] J. X. Zhow and G. Chen : Diag-onal convexity conditions for prob-lems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl., Vol.132 (1988), 213 -225.
