广义的Fan-Ha截口定理

来源期刊：中南大学学报(自然科学版)2001年第4期

论文作者：刘心歌 刘心笔 唐美兰

文章页码：437 - 440

关键词：H-空间;局部交性质;极小极大定理;集值映射

Key words：H-space; local intersection property; minimax theorem; set-valued mapping

摘    要：

为了进一步研究极小极大不等式,首先引进了H-空间,将极小极大定理中的闭性条件与凸性条件进一步削弱,利用反证法与有限交性质将Fan-Ha截口定理以及极小极大定理推广为非线性H-空间上更一般的形式:设(X,{Γ_A}),(Y,{Γ_D})为2个HausdorffH-空间,B C X×Y,且满足如下条件:

a·对每个x∈X,{y∈Y,(x,y) B}为H-凸集或空集.

b·对每个y∈Y,{x∈X,(x,y)∈C}为X中的紧闭集.

c·对每个x∈X,存在A_x X×Y,A_x=P_x×Q_x.其中P_x为X中的紧闭集,Qx为Y中的紧集.

d·又假设存在X的非空紧集K,对每个X的有限子集N,存在X的紧子集L_N,L_N N,使得

①对每个y∈Y,L_N∩{x∈X,(x,y)∈A_z,对所有z∈L_N }是零调的;

②对每个x∈L_N\K,{y∈Y,(x,y)∈A_z,对所有z∈L_N } {y∈Y,(x,y)∈B};

e·对每个x∈K,{y∈Y,(x,y)∈Az,对所有z∈X}=Φ. 则存在x₀∈X,使得{x₀}×Y C.利用广义的Fan-Ha截口定理,容易将参考文献[1]中的所有结论推广到H-空间上.

Abstract:

In order to study the Fan-Ha section theorem,H-space and local intersection property are introduced. Relaxing convexity and closedness of some sets, Fan-Ha section theoremandminimaxtheoremare generalized to H-space, that is, let ({X{Γ_A}), (Y,{Γ_D}) be two Hausd or ff H-spaces,B C X×Ysuch as follows:

a·for each x∈X, {y∈Y, (x,y) B} is H-convex or empty;

b·for each y∈Y, {x∈X, (x,y)∈C}is compactly closed inX;

c·for each x∈X, there exists a nonempty setAx X×Y, A_x=P_x×Q_x, P_x is a compactly closed subset in X,Q_x is a compact subset of Y.

d·Further, suppose that there exists a nonempty compact subset K of X and for each finite subset N of X, there exists a compact subset L_N of X containing N such that

①for each y∈Y, LN∩{x∈X, (x,y)∈A_z for all z∈LN} is acyclic;

②for each x∈LN\K, {y∈Y,(x,y)∈A_z for all z∈LN} {y∈Y, (x,y)∈B};

e·for each x∈K, {y∈Y, (x,y)∈A_z for all z∈X}=Φ. Then there exists a point x₀∈X such that{x₀}×Y C.