李紫嫣,高大鵬,何中全

(西華師范大學(xué)數(shù)學(xué)與信息學(xué)院,四川南充 637002）

η-廣義混合向量平衡問題解的存在性*

李紫嫣,高大鵬,何中全

(西華師范大學(xué)數(shù)學(xué)與信息學(xué)院,四川南充 637002）

在Bana c h空間中,引進和研究了一類η-廣義混合向量平衡問題(η-GMVEP）,在適當(dāng)假設(shè)條件下證明了此類問題的等價性定理,并運用KKM定理得到其解的存在性定理.

η廣義混合向量平衡問題;偽單調(diào);Hausdorff度量;KKM映射;存在性

0 引言

最近幾年,向量變分不等式成為在自然科學(xué)和社會科學(xué)的很多研究領(lǐng)域的重要工具,并都得到了廣泛的應(yīng)用.向量變分不等式作為變分不等式的推廣,最早是由Giannessi在1980年引入到有限維歐式空間中并加以研究的[1].隨后,向量變分不等式、向量擬變分不等式和其補問題等在無限維空間中得到了廣泛研究和推廣[2～6].另一方面,向量平衡問題也得到了深入的研究[7～11].受文獻[4]、[12]、[13]的啟發(fā),本文引入了一類新的η-廣義混合向量平衡問題,并運用KKM定理,得到了此問題解的存在性定理.

1 預(yù)備知識

此問題在文獻[6]、[16]中被引入和研究.

此問題在文獻[18]中被引入和研究.

為了得到本文的結(jié)果,需要下面的定義和引理.

定義1[19]K是拓撲線性空間X中的非空子集.集值映射F∶K→2Y稱作KKM映射,如果對任意的非空有限子集{u1,u2,…,un}?K,總有：

2 主要結(jié)果與證明

定理1設(shè)X,Y為實的Banach空間,K?X為非空凸子集,{C(u）：u∈K}是Y中的真的閉凸錐,

余下仿定理1證明即可.

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[2]Chen G Y,Cheng G M.Vector variational inequality and vector optimization problem[M].Berlin：Springer Berlin Heidelberg,1987.

[3]Lee G M,Lee B S,Chang S.On vector quasivariational inequalities[J].Journal of Mathematical Analysis and Applications,1996,203(3）：626-638.

[4]Ansari Q H,Farajzadeh A P,Schaible S.Existence of solutions of a vector variational inequalities and vector complementary problem[J].Journal of Global Optimization,2009,45(2）：297-307.

[5]Lee G M,Kim D S,Lee B S,et al.Generalized vector variational inequality and fuzzy extension[J].Applied Mathematics Letters,1993,6(6）：47-51.

[6]Siddiqi A H,Ansari Q H,Khaliq A.On vector variational inequalities[J].Journal of Optimization Theory and Applications,1995,84(1）：171-180.

[7]Zeng L C,Yao J C.An existence result for generalized vector equilibrium problems without pseudomonotonicity[J].Applied Mathematics Letters,2006,19(12）：1-320-1-326.

[8]Huang N J,Li J,Thompson H B.Implicit vector equilibrium problems with applications[J].Mathematical and Computer Modelling,2003,37(12）：1-343-1-356.

[9]Konnov I V,Yao J C.Existence of solutions for generalized vector equilibrium problems[J].Journal of Mathematical A-nalysis and Applications,1999,233(1）：328-335.

[10]Ansari Q H,Schaible S,Yao J C.System of vector equilibrium problems and its applications[J].Journal of Optimization Theory and Applications,2000,107(3）：547-557.

[11]Zeng L C,Yao J C.Generalized Minty’s lemma for generalized vector equilibrium problems[J].Applied Mathematics Letters,2007,20(1）：32-37.

[12]Preda V,Beldiman M,Bǎtǎtorescu A.On variational-like inequalities with generalized monotone mappings[M].Ber-lin：Springer Berlin Heidelberg,2006.

[13]Li X,Kim J K,Huang N J.Existence of Solutions forηGeneralized Vector Variational Like Inequalities[J].Journal of Inequalities and Applications,2010(1）：1-13.

[14]Khan M F.On generalized vector variational-like inequalities[J].Nonlinear Analysis：Theory,Methods&Applications,2004,59(6）：879-889.

[15]Lee B S,Lee G M.A vector version of Minty’s lemma and application[J].Applied Mathematics Letters,1999,12(5）：43 -50.

[16]Jung D Y.Some generalization of Minty’s lemma[J].J Korea Soc Math Educ Ser B：Pure Appl.Math,1999,6(1）：33-37.

[17]Khaliq A,Rashid M.On generalized vector quasivariational-like inequality problems[J].Fixed Point Theory and Applications,2005,2005(3）：243-255.

[18]Chen G Y,Li S J.Existence of solutions for a generalized vector quasivariational inequality[J].Journal of Optimization Theory and Applications,1996,90(2）：321-334.

[19]Knaster B,Kuratowski C,Mazurkiewicz S.Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe[J].Fundamenta Mathematicae,1929,14(1）：132-137.

[20]Chen B,Huang N J.Vector variational like inequalities and vector optimization problems in Asplund spaces[J].Optimization Letters,2012,6(7）：1-513-1-525.

[21]Fan K.A generalization of Tychonoff’s fixed point theorem[J].Mathematische Annalen,1961,142(3）：305-310.

[22]Nadler Jr S B.Multi valued contraction mappings[J].Pacific J Math,1969,30(2）：475-488.

Existence of Solutions forηGeneralized Mixed Vector Equilibrium Problems

LI Ziyan,GAO Dapeng,HE Zhongquan
(School of Mathematics and Information,China West Normal University,Nanchong 637002,China）

This paper introduces some generalized mixed vector equilibrium problems in Banach spaces and proves the equivalence of the two classes ofηgeneralized mixed vector equilibrium problems under suitable assumptions.By using the equivalence theorem,we obtain some results on the existence of solutions forηgeneralized mixed vector equilibrium problems.

ηgeneralized mixed vector equilibrium problems;pseudo monotone;Hausdorff metric;KKM-mapping;the existence

O177.92

A

1009-1734(2014）02-0006-06 MSC(2000）：49J35,90C47

2013-12-10

教育部科學(xué)技術(shù)重點項目(211163）;西華師范大學(xué)基金項目(11A029,11A028）、(13D016）.

何中全,教授,研究方向：非線性分析.E-mail：sasaweb@sina.com

MSC 2000：49J35,90C47