樸勇杰,姜今錫,鐘立楠

(延邊大學(xué)理學(xué)院數(shù)學(xué)系,吉林延吉 133002)

關(guān)于一般化凸乘積空間上的極大元和平衡點(diǎn)

樸勇杰,姜今錫,鐘立楠

(延邊大學(xué)理學(xué)院數(shù)學(xué)系,吉林延吉 133002)

利用一般化凸乘積空間上的Fan-Browder型不動(dòng)點(diǎn)定理給出了新的極大元存在定理,然后定義了兩個(gè)概念:“類(lèi)Uθ”和“類(lèi)V”,并討論了在抽象經(jīng)濟(jì)中平衡點(diǎn)的存在性問(wèn)題.文中所得結(jié)論改進(jìn)和推廣了文獻(xiàn)中的相應(yīng)結(jié)果.

一般化凸空間;Γ-凸;極大元;平衡點(diǎn);類(lèi)Uθ;類(lèi)V

1 引言與預(yù)備知識(shí)

在過(guò)去的30多年里,已經(jīng)出現(xiàn)了很多關(guān)于Warlasian平衡點(diǎn)存在問(wèn)題的古典的A rrow-Debreu定理[1]的一般化形式.首先,文[2]利用極大元方法給出了競(jìng)爭(zhēng)性平衡的存在性的證明.另一方面,文[3]在局部凸拓?fù)渚€性空間上通過(guò)引進(jìn)具有類(lèi)Lθ的優(yōu)先映射給出了一個(gè)新的存在定理.

為了給出極大元或平衡點(diǎn)的存在定理,我們可以把平衡點(diǎn)的存在問(wèn)題轉(zhuǎn)化成與之等價(jià)的相應(yīng)映射的不動(dòng)點(diǎn)的存在問(wèn)題.

我們首先給出一些概念和注記.

一般化凸空間（簡(jiǎn)稱(chēng)G-凸空間）(X,D;Γ)是由一個(gè)拓?fù)淇臻gX和一個(gè)非空集合D組成，并滿(mǎn)足對(duì)任何A={a0,a1,…,an}∈〈D〉,存在一個(gè)X的子集Γ(A)和一個(gè)連續(xù)函數(shù)使得由J?{0,1,2,…,n}推出φn(?J)?Γ({aj:j∈J}).這里〈D〉表示D的所有非空子集全體，?n表示頂點(diǎn)為v0,v1,…,vn的n-單形，而?J=co{vj:j∈J}表示相應(yīng)于J的?n的面.我們用ΓA表示Γ(A),對(duì)任何A∈〈D〉.

已有許多G-凸空間的例子[1112].其典型例子是拓?fù)渚€性空間中的任何凸子集,Lassonde[13]意義下的凸空間以及Hovarth[14]的C-空間(也稱(chēng)H-空間).

本文假設(shè)D?X,并且如果D=X,則用(X;Γ)表示(X,D;Γ).

對(duì)G-凸空間(X,D;Γ),稱(chēng)非空子集Y?X為Γ-凸的,如果每個(gè)N∈〈D〉且N?Y推出ΓN?Y.并Y的Γ-凸包按下列方式給出:

設(shè)X和Y是兩個(gè)拓?fù)淇臻g.一個(gè)集值映射(簡(jiǎn)稱(chēng),映射)T:X—Y是指X到Y(jié)的冪集2Y的映射.稱(chēng)T(x)為T(mén)在x的值,并記T(A)=∪{T(x):x∈A},對(duì)A?X.

根據(jù)T,我們可定義如下一些映射:

2 極大元與平衡點(diǎn)

[1]A rrow K J,Debreu D.Existence of an equilibrium for a com petitive economy[J].Econometrica,1954,22:265-290.

[2]Gale D,M asColell A.On the Role of Com p lete,Transitive Preference in Equilibrium Theory[M]//Schw¨diauer G.Equilibrium and D isequilibriumin Econom ic Theory.Dordrecht:Reidel,1978.

[3]M etha G,Tan K K,Yuan X Z.Fixed points,m axim al elem ents and equilibria of generalized gam es[J]. Nonlinear Anal.,1997,28:689-699.

[4]Berge C.Sur une convexit′e r′egu li`ere et ses app lications`a la th′eorie des jeux[J].Bull.Soc.M ath.France, 1954,82:301-319.

[5]Ding X P.Equilibria of noncom pact generalized gam eswith U-m a jorized preference correspondences[J].App l. Math.Lett.,1998,11:115-119.

[6]Ding X P,Tan K K.On equilibria of non-com pact generalized gam es[J].J.M ath.Anal.App l.,1993,177:226-238.

[7]Ding X P,Yuan G X Z.The study ofexistence ofequilibria for generalized gam eswithout lower sem icontinuity in locally topological vector spaces[J].J.M ath.Anal.App l.,1998,227:420-438.

[8]Lin L J,Park SH.On some generalized quasi-equilibrium prob lem s[J].J.Math.Anal.App l.,1998,224:167-181.

[9]Tan K K,Wu X.A note on abstract econom ieswith upper sim icontinuous correspondence[J].App l.Math. Lett.,1998,11:21-22.

[10]Yuan X Z,Tarafdar E.M aximal elements and equilibria of generalized games for condensing correspondences[J].J.M ath.Anal.App l.,1996,203:13-30.

[11]Park SH.New Susclasses of Generalized Concev Spaces[M]//NY.Fixed Point Theory and App lications (Chinju,1998),91-98.Huntington:Nova Sci.Publ.,2000.

[12]Park SH.Elem ents of the KKM theory for generalized convex spaces[J].Korean J.Com p.App l.M ath., 2000,7:1-28.

[13]Lassonde M.On the use of KKM multim aps in fixed point theory and related topics[J].J.M ath.Anal.App l., 1983,97:151-201.

[14]Horvath C D.Contractibility and generalized convexity[J].J.M ath.Anal.App l.,1991,156:341-357.

[15]樸勇杰.Fan-Browder type fixed point and equilibrium point on generalized convex product spaces[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2004,20(3):197-203.

[16]樸勇杰.一般化凸空間上的一些基本性質(zhì)[J].延邊大學(xué)學(xué)報(bào):自然科學(xué)版,2002,28(3):157-159.

Onmaximal elements and equilib rium points on generalized convex product spaces

PIAO Yong-jie,JIANG Jin-xi,ZHONG Li-nan

(Departm ent of Mathem atics,College of Science,Yanbian University,Yanji 133002,China)

NeWexistence theorems of m axim al elem ent are given by using the Fan-Brow der type fixed point theorems on generalized convex product spaces,and then two concepts:“of class Uθ”and“of class V”are introduced and existence problems of equilibrium point in an abstract economy are discussed.Ourmain results im prove and generalize the corresponding results in recent literatures.

generalized convex space,Γ-convex,maximal element,equilibrium point,of class Uθ,of class V 2000M SC:47H 05,47H 10

O189.1;O 177.91;

A

1008-5513(2009)02-0217-06

2007-07-05.

國(guó)家自然科學(xué)基金(10361005),延邊大學(xué)科研項(xiàng)目(2004[8]).

樸勇杰(1962-),博士,教授,研究方向：非線性分析.