ỨNG DỤNG CỦA BÀI TOÁN TỰA CÂN BẰNG TỔNG QUÁT

Đồng thời nghiên cứu mối quan hệ giữa bài toán này với một số bài toán khác và ứng dụng bài toán này để chứng minh các bài toán như: bài toán tựa cân bằng vô hướng, bài toán quan hệ tựa[r]

APPLICATIONS OF GENERALIZED QUASI- EQUILIBRIUM PROBLEM

Nguyen Thi Hue *

College of Technology- TNU

SUMMARY

This article presents some sufficient conditions for the existence of solutions of the generalized quasi- equilibrium problem Simultaneously 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

xD such that f x x( , )  0 x D, in that

D is a convex closed set in topological space

X, f D:  D R is a function such that

( , ) 0

f x x  , 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 born 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,

DX, KZ are nonempty subsets Suppose

S D K  T D K 

F K   D D D

are multivalued mappings with nonemty

values

Find ( , )x y  D Ksuch that:

1/ xS x y( , );

2/ yT x y( , );

*

Email: hue.tnut@gmail.com

3/ 0F y x x z( , , , ), for all zS x y( , ) This problem is called the generalized quasi- equilibrium problem

The multivalued 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 Let DX, KZ are nonempty

subsets Assume that:

S D K  , T D K:  2K

F K   D D D

are multivalued mappings with noemty values Assume that:

(i) S is a compact continuos mutivalued mapping with closed values;

(ii) T is a compact acyclic multivalued mapping; (iii) For any fixed ( , )x y  D K, exists ( , )

tS x y such that 0F y x t z( , , , ) for all ( , )

zS x y ; (iv) For any ( , )y x  K D the set

 ( , ) | 0 ( , , , ), ( , )

A t S x y F x y t z for all z S x y

is acylic;

(v) F is a close multivalued mapping

Then, there exists ( , )x y  D Ksuch that: 1/ xS x y( , );

2/ yT x y( , ); 3/ 0F y x x z( , , , ), for all zS x y( , )

Proof We define the multivalued mapping

M D K  by

( , ) { ( , ) | 0 ( , , , ),

( , )}

M y x t S x y F y x t z

for all z S x y



( , )x y  D K

According to conditions (iii) and (iv),

( , )

M y x is a nonempty acyclic set

Now, we prove that M is a closed multivalued

mapping Indeed, assume that

x x y y tM y x t t, we

have to show tM y x( , ) Since

( , )

tS x y , the upper semicontinuity of S

with closed values implies tS x y( , ) For

( , )

tM y x , one can see

0F y( ,x t , , )z , for all zS x( ,y)

The lower semicontinuity of S and x x

follow that for any zS x y( , ) there exists

( , )

zS x y such that z z Thus,

0F y( ,x t , ,z) for all zS x( ,y)

Since (y,x t , ,z)( , , , )y x t z and the

multivalued mapping F is closed, then

0F y x t z( , , , ) for all zS x y( , ) This

means that tM y x( , ) and so M is a closed

multivalued mapping

Lastly, we define the multivalued mapping

P D K   by

( , ) ( , ) ( , )

P x y M y x T x y , ( , )x y  D K

We can see that M is a closed multivalued

mapping with nonempty compact acyclic

values and so M is a compact acyclic

multivalued mapping Therefore, the product

mapping P of two compact acyclic mappings

M and T is also compact acyclic Applying

Park’s fixed point Theorem in [5], we

conclude that there exists a point

( , )x y  D Kwith

( , )x y P x y( , )M x y( , )T x y( , ).This

follows:

1/ xS x y( , );

2/ yT x y( , );

3/ 0F y x x z( , , , ), for all zS x y( , )

The proof is complete

The Scalar quasi- equilibrium Problem

Let X, Y, Z are local convex topological

vector spaces Let DX, KZ are nonempty

subsets Assume that: S D K:  2D,

T D K  are multivalued mappings with noemty values Suppose Y=R,

    is a continuous function For any fixed ( , )y x  K D, the function ( , ,.) :y x D R

( , , )y x x 0

  Then, there exists ( , )x y  D K such that:

1/ xS x y( , ); 2/ yT x y( , ); 3/ ( , , ) y x z 0 for all zS x y( , )

Proof We define M K D:  2X,

F K D D D    by

( , ) ( , ) | ( , , ) ( , , ), ( , )

M y x  t S x y  y x z  y x t for all z S x y

, ( , )y x  K D ( , , , ) ( , ), ( , , , )

F y x t z  t M y x y x t z    K D D D

For each fixed ( , )y x  K D, S y x( , ) is a compact subset, ( , ,.)y x is a continuous function Therefore, there exists tS y x( , ) such that ( , , )y x t ( , , )y x z , for all ( , )

zS x y This follows that M y x( , ) is nonempty for each ( , )y x  K D Since for any fixed ( , )y x  K D, ( , ,.) y x is a quasi-convex function, it implies that M y x( , ) is convex Further, we can prove that M is a closed multivalued mapping with nonempty convex values, so is F Set

{ | 0 ( , , , ), z S(x,y)}

A t D F y x t z for all 

={tD t| M y x( , )}M y x( , )

We can see that A is a convex set and therefore

acyclic Applying Exists theorem of solutions,

there exists ( , )x y  D K such that:

1/ xS x y( , ) 2/ yT x y( , ) 3/ 0F y x x z( , , , ), for all zS x y( , ) This follows that ( , , )y x z 0 for all ( , )

zS x y

Quasivariational relation Problem

Let X, Y, Z are local convex topological

vector spaces Let DX, KZ are nonempty

subsets Assume that: S D K:  2D,

T D K  are multivalued mappings

with noemty values Let ( , , , )y x t z be a

relation linking yK x t z, , , D  is a

relation, often given by equality, inequality of

real functions, or by inclusions, intersections

of multivalued mappings Assume that

(i) S is a compact continuous multivalued

mapping with nonempty closed values;

(ii) T is a compact acyclic multivalued

mapping with nonempty values;

(iii) For any ( , )x y  D K, there exists

( , )

tS x y such that ( , , , )y x t z holds for all

( , )

zS x y ;

(iv) For any fixed ( , )y x  K D the set

{ ( , ) | ( , , , ) ( , )}

A t S x y  y x t z holds for all zS x y

is acyclic;

(v) The relation  is closed

Then, there exists ( , )x y  D K such that:

1/ xS x y( , );

2/ yT x y( , );

3/ ( , , , )y x x z holds for all zS x y( , )

Proof We defined multivalued mappings

M K D  , F K D D D:    2X by

( , ) ( , ) | ( , , , ) ( , )

M y x  t S x y y x t z holds for all z S x y

, ( , , )y x z   K D D

( , , , ) ( , ),( , , , )

F y x t z  t M y x y x t z    K D D D

According to condition (iii) there exists

( , )

tM y x , for all zS x y( , ) This implies

that 0F y x t z( , , , ), for all zS x y( , )

Further, we can see that the set

{ | 0 ( , , , ), z S(x,y)}

A t D F y x t z for all 

={tS x y( , ) |tM y x( , ),for all zS x y( , )}

is acyclic

We prove that M is a closed multivalued

mapping Indeed, assume that

x x y y tM y x t t, we

have to show tM y x( , )

Since tS x( ,y), the upper semicontinuity of S with closed values yields ( , )

tS x y For tM y( ,x), We can see that (y,x t , , )z

 holds, for all zS x( ,y) The lower semicontinuity of S and

x x , y y follow that for any ( , )

zS x y there exists zS x( ,y) such

that z z Thus, (y,x t , ,z) holds for all zS x( ,y)

Since (y,x t , ,z)( , , , )y x t z and the relation  is closed, we conclude that ( , , , )y x t z

 holds for all zS x y( , ) This means that tM y x( , ) and so M is a closed multivalued mapping This follow that F is a closed multivalued mapping

Applying Exists theorem of solutions, there

exists ( , )x y  D K such that:

1/ xS x y( , ); 2/ yT x y( , ); 3/ 0F y x x z( , , , ), for all zS x y( , ) This follow that ( , , , )y x x z holds for all ( , )

zS x y

REFERENCES

1 Guerraggio, A and Tan, N.X (2002), “ On General Vector Quasi- optimization Problem”,

Mathematical Methods Of operation Re- seach,

Vol.55, 347-358

2 Park, S (1993), “Admissible classes of

multifuntions on generalized convex spaces” Proc coll Nat Sci, 1-21

3 Lin, L.J, and Tan, N.X (2007), “ On Inclution

Problems of type I and Related Problems”, J

Global Option, Vol.39, no.3, 393-407

4 Luc, D.T (2008) “ An Abstract problem in

variational Analysis”, J optim Theory Appl,

Vol.138, no.1, 65-76

5 Park, S (2000), “ Fixed points and Quasi- Equilibrium Problems Non- linear Operation

Theory”, Math ematical and computer Modelling,

Vol 32, 1297-1304

TÓM TẮT

ỨNG DỤNG CỦA BÀI TOÁN TỰA CÂN BẰNG TỔNG QUÁT

Nguyễn Thị Huệ *

Trường Đại học Kỹ thuật Công nghiệp - ĐH Thái Nguyên

Bài báo này trình bày bài toán tựa cân bằng tổng quát và điều kiện đủ cho sự tồn tại nghiệm của

nó Đồng thời nghiên cứu mối quan hệ giữa bài toán này với một số bài toán khác và ứng dụng bài toán này để chứng minh các bài toán như: bài toán tựa cân bằng vô hướng, bài toán quan hệ tựa biến phân

Từ khóa: cân bằng, tựa cân bằng, tựa cân bằng vô hướng, quan hệ tựa biến phân, ánh xạ đa trị

*

Email: hue.tnut@gmail.com