| 摘要: | 假設XY 為實Hausdorff 拓樸向量空間。令K 為X 的非空集合，C K 2Y 為集合值映射滿足對於任意x K , C(x) 為真閉凸集及頂點在原點的點凸錐，且 intC(x) . 對任意x K , 我們定義“ C( x) " 與“ C( x) "關係如下： (1) C( x) z y y z C(x) ; 及 (2) C( x) z y y z C(x) . 並且，我們也使用下列符號： C( x) C( x) y z z y , C( x) C( x) y z z y . 類似地，我們可以定義“ intC( x) ", “ intC( x) ", “ C( x)\{0} "與“ C( x)\{0} "關係，如果我們將集合C(x) 以intC(x) 或C(x) \{0}取代。如果C(x) 為固定，則以C 取代。L(XY) 表所有從X 到Y 的連續線性映射。給定映射f Z K K Y 與T K 2L(X Y )。廣義向量值平衡問題(GVEP) Q 是找尋一個x K 使得 ( , ) 0 Q f sx y (1) 對所有y K 及某些s T(x)，其中Q是凸集及頂點在原點的點凸錐，且intQ . 這樣的解稱為(GVEP) Q 的弱解， Q 代表下列關係之一： intC( x) , C( x) , C( x)\{0} 。如s 不受y 影響，即，求一x K 伴隨某個s T(x) 使得 ( , ) 0 Q f sx y (2) 對所有y K ，這樣的解稱為(GVEP) Q 的強解。我們將所有(GVEP) Q 的弱解所成的集合記為w ，並且將所有(GVEP) Q 的強解所成的集合記為. 本計畫的第一年，我們將探討(GVEP) Q 的弱解集w 與強解集，具有何種拓樸性質？嘗試討論它們是否是閉集、連通集或是緊緻集？擾動廣義向量值平衡問題(PGVEP) Q 將於本計畫的第二年加以探討。令 XY , C 與T 如第一年所討論。令1 2 , 為兩參數集， 給定映射 1 f Z K K Y 及2 K : Y 。擾動廣義向量值平衡問題敘述如下：對每個1 2 (,) ，求x K() 使得 ( , , ) 0 Q f sx y 對所有的y K() 及某些s T(x) ，其中是凸集及頂點在原點的點凸錐，且intQ ， Q 代表下列關係之一： intC( x) , C( x) , C( x)\{0} 。我們將這樣的 (PGVEP) Q 之解集合記為( , ) w 。若找x K() 及某些s T(x) 使得 ( , , ) 0 Q f sx y 對所有y K() .我們將這樣的(PGVEP) Q 之解集合記為(,) . 本計畫的第二年的目的是求這兩個擾動廣義向量值平衡問題解集合( , ) w 與 (,) 的拓樸結構。並且嘗試尋找使它們為非空集合、閉集合、連通集或是緊緻集的充分條件。
Let XY be arbitrary real Hausdorff topological vector spaces. Let K be a nonempty set of X , C K 2Y set-valued mapping such that for each x K , C(x) is a proper closed convex and pointed cone with apex at the origin and intC(x) . For each x K , we can define relations “ C( x) " and “ C( x) " as follows: (1) C( x) z y y z C(x) ; and (2) C( x) z y y z C(x) . Furthermore, we use the following notations： C( x) C( x) y z z y , C( x) C( x) y z z y . Similarly, we can define the relations “ intC( x) ", “ intC( x) ", “ C( x)\{0} " and “ C( x)\{0} " if we replace the set C(x) by intC(x) or C(x) \{0} . If the mapping C(x) is constant, then we denote C(x) by C . L(XY) denotes the space of all continuous linear mappings from X to Y . The mappings f Z K K Y and T K 2L(X Y ) are given. The generalized vector equilibrium problem (GVEP) Q is to find an x K such that ( , ) 0 Q f sx y (1) for all y K and for some s T(x) , where Q is a convex and pointed cone with apex at the origin and intQ , Q denotes one of the following relations： intC( x) , C( x) , C( x)\{0} . Such solution is called a weak solution for (GVEP) Q . For the case that s does not depend on y , that is, to find an x K with some s T(x) such that f (sx, y)Q 0 (2) for all y K , we call such solution a strong solution of (GVEP) Q . Let us denote the set of weak solutions for (GVEP) Q by w , and the set of strong solutions for (GVEP) Q by . In the first year of this project, we will like to consider what topological structures do the sets, w and , of efficient solutions for generalized vector equilibrium problems have? And then we try to discuss whether those sets are closed, connect, compact ones or not? The perturbed generalized vector equilibrium problems (PGVEP) Q will be discussed in the second year of this project. That is, let XY , C and T be as above given. Let 1 2 , be two parametric sets, the mappings 1 f Z K K Y and 2 K : Y be given. The perturbed generalized vector equilibrium problem is as follows： For every 1 2 (,) , we will like to find an x K() such that ( , , ) 0 Q f sx y for all y K() and for some s T(x) , where Q is a convex and pointed cone with apex at the origin and intQ , Q denotes one of the following relations： intC( x) , C( x) , C( x)\{0} . Such set of efficient solutions for (PGVEP) Q is denoted by ( , ) w . If we find x K() and some s T(x) such that ( , , ) 0 Q f sx y for all y K() . Such set of efficient solutions for (PGVEP) Q is denoted by (,) . The purpose of the second year for the project is to find some topological structures for the two sets, ( , ) w and (,) , of efficient solutions of the perturbed generalized vector equilibrium problem. Furthermore, we try to find some sufficient conditions lead them to be nonempty or closed or connect or even compact sets. |
