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Nguy€n Thj Hu§ T?p chf KHOA Hpc & CONG NGHB 162(02): 155-158

APPLICATIONS OF GENERALIZED QUASI- EQUILIBRIUM PROBLEM

Nguyen Thi Hue'

College ofTechnology- TNU

SUMMARY

This article presents some sufficient conditions for the existence of solutions of the generalized

quasi- equilibrium problem. Sunultaneously studying the relationship between this problem with

some other problems and applying this problem to prove some problems such as the Scalar quasi￾equilibrium Problem , Quasivariational relation Problem.

Keywords: Equilibrium,Quasi- equilibrium. Scalar quasi- equilibrium, Quasivariational relation,

multivalued mapping.

INTRODUCTION

The equilibrium problem Blum- Oettli: find

3c e D such that /(;c,S)^0 Vj:ei?, in that

D is a convex closed set in topological space

X, f:Dx.D—>R is a fimction such that

f{x,x) = 0, was extended for problems in

infinite dimensional spaces with any cones.

The introduction of the concept and prove the

existence of effective points of a set in space

order be bom by cone led to the study of the

various optimization problems. Later this

theory was developed for the problems related

to multivalued mappings in infinite

dimensional space. And continue to expand

for the quasi problems such as: Quasi

optimazation problem, quasi- equilibrium

problem.This article give some sufficient

conditions for the existence of the generalized

quasi- equilibrium problem. And it is applied

to prove some other problems.

APPLICATION OF GENERALIZED

QUASI- EQUILIBRIUM PROBLEM

Problem. Let X. Y, Z are nonempty sets,

DcX iCcZ are nonempty subsets. Suppose

S:DxK^2^ ,T:DxK^2'^

F:KxDxDxD-*2^

are multivalued mappings with nonemty

values.

Find {x,y)eDxKsuch that:

l/xeS{x,y);

2lyeT{x,y);

' Email, [email protected]

3/ 0BF{y,x,x,z), for a\i zeS{x,y).

This problem is called the generalized quasi￾equilibrium problem.

The muhivalued mappings S, T are constraint

and F is an objective multivalued mapping

that are often determined by equalities and

inequalities or by inclusions and intersections

of multivalued mappings.

Exists theorem of solutions

Let X, Y, Z are local convex topological

vector spaces. LetD^X, /iTc Z are nonempty

subsets. Assume that:

S:DxK->2'^, T:DxK^2^

F:KxDxDxD^2^

are multivalued mappings with noemty

values. Assume that:

(i) S is a compact continues mutivalued

mapping with closed values;

(ii) T is a compact acyclic multivalued mapping;

(iii) For any fixed {x,y)&DxK, exists

t'^S{x,y) such that fi€F{y,x,t,z) for all

.ze5(3c,7);

(iv) For any {y,x)eKxD the set

A = [teS{x,y)\O^F{x,y,t,z),for allzeS{x,y)]

is acylic;

(v) F is a close multivalued mapping.

Then, there exists (J,J')eZ>xA: such that:

\lxeS{x,y);

2/yeT{x,y);

3fOeF{y,x,x,z),fora\\ zeS{x,y).

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