EQUILIBRIUM PROBLEMS WITH LOWER AND UPPER BOUNDS IN TOPOLOGICAL SPACES

2005,25B(4):658-662

EQUILIBRIUM PROBLEMS WITH LOWER AND UPPER BOUNDS IN TOPOLOGICAL SPACES 1 Ding Xieping ( TtJIJ·-'f ) College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China

Abstract By employing a fixed point theorem due to Ding, Park and Jung, some existence theorems of solutions for equilibrium problems with lower and upper bounds are proved in noncompact topological spaces. These results further answer the open problem raised by Isac, Sehgal and Singh under much weaker assumptions. Key words Equilibrium problem, fixed point theorem, contractible space, acyclic space, topological space 2000 MR Subject Classification

1

49A27, 47HlO

Introduction

Let X be a topological space and f : X x X problem ( EP(X, 1) ) is to find x E X such that

f(x,y) 2: 0,

--->

R U{±oo} be a function. The equilibrium

I;j

Y E X.

(1)

The EP(X, 1) (1) includes many fundamental mathematical problems that arise from mechanics, engineering, economics, optimization, fixed point, saddle point, variational inequality and complementarity problem as special cases. See, for example, [1-4]. In 1999, Isac, Sehgal and Singh [5] raised the following open problem: given a nonempty closed subset K in a locally convex semireflexive topological vector space, a function

K

---> R

f :K

x

and two real numbers a < {3, it is interesting to know that under what conditions there

exists an

xE K

such that

a::; f(x, y) ::; {3,

I;j

y

E

K.

(2)

Recently, Li[6] and Chadli, Chiang and Yao l7] give some answers to the open problem by employing different methods under topological vector spaces. In this paper, we will consider the equilibrium problem (2) in general topological spaces without linear structure. 1 Received March 17, 2003. (2003A081)

This project was supported by the NSF of Sichuan Education Department

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Preliminaries

Let X be a topological space, 2x denotes the family of all subsets of X and F(X) denotes the family of all nonempty finite subsets of X. A subset A of X is said to be compactly open (resp., compactly closed) in X if for each nonempty compact subset K of X, An K is open (resp., closed) in K. It is clear that each open (resp., closed) subset of X is compactly open (resp., compactly closed) in X. Following Ding[8], define the compact closure and the compact interior of A, denoted by cclA and cintA, as cclA = niB ~ X : A ~ Band B is compactly closed in X}, cintA

= U{B

~ X: B ~ A and B is compacyly open in X},

respectively. It is easy to see that a subset A of X is compactly open (resp., compactly closed) if and only if cintA = A (resp., cclA = A). A topological space X is said to be contractible if the identity mapping Ix on X is homotopic to a constant function. In particular, any convex or star-shaped set in a topological vector space is contractible. We need the following fixed point theorem which is a special case of Theorem 2.2 of Ding, Park and Jung [9]. Theorem 2.1

Let X be a topological space and F, G : X

mappings such that (i) F(x) ~ G(x) for each x E X,

---->

2x be two set-valued

n

(ii) for each compact subset D of X, D = UyEx (cint F - 1 (y) D), (iii) there exists a nonempty subset X o of X such that for each N E F(X) there is a compact contractible subset L N of X containing X o UN and for any compactly open subset U of X, the set nXEU(G(x) LN) is empty or contractible, and the set n yEXO(cintF- 1 (y))Cis empty or compact where (cintF- 1 (y))Cdenotes the complement of cintF- 1 (y) in X.

n

Then there exists a point x E X such that x E G(x), i.e., x is a fixed point of G. Proof The conclusion of Theorem 2.1 holds from Theorem 2.2 with the index set I being a singleton in [9].

3

Existence of Solutions

In this section, by applying Theorem 2.1, we shall prove several existence theorems of solutions for the equilibrium problem with lower and upper bounds (2) in noncompact topological spaces. Theorem 3.1 Let X be a topological space and a, (3 be two real numbers with a ::; (3. Suppose I, 91,92 : X x X ----> R are three functions satisfying the following conditions:

(i) for each x

E

X, 91(X,X) 2: a and 92(X,X) ::; (3;

(ii) for each x E X,

{y EX: f(x,y) < a or f(x,y) > (3} C {y EX: 91(X,y) < a or 92(X,y) > (3}; (iii) for each nonempty compact subset D of X,

D=

U (cint{x EX: f(x, y) < a or f(x, y) > (3} nD);

VEX

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(iv) there exists a nonempty set X o in X such that for each N E F(X), there exists a compact contractible subset L N of X containing X o UN and for any nonempty open subset U of X, the set nXEU({y EX: gl(X,y) < a or g2(X,y) >,8} nLN) is empty or contractible, and the set nyEXo ccl{x EX: a:S; f(x,y):s;,8} is empty or compact. Then there exists an x E X such that a :s; f(x, y) :S;,8, lei y E X. Proof Define two mappings P, G : X ----; 2 x by

P(x)

=

{y

f(x,y) < a or f(x,y) > ,8}, lei x

E Y:

E X,

and

G(x) = {y

E

Y: gl(X,y) < a or g2(X,y) > ,8}, lei x E X.

Then the condition (ii) implies that the condition (i) of Theorem 2.1 holds and for each y E Y,

P-1(y) = {x EX: f(x, y) < a or f(x, y) > ,8}. Suppose on the contrary that the conclusion of Theorem 3.1 is not true, then for each x P(x) is nonempty. By (iii), we have that for each compact subset D of X, D

=

E

X,

U (cintp-1(y) nD),

yEX

i.e., the condition (ii) of Theorem 2.1 holds. By (iv), there exists a nonempty set X o of X such that for each N E F(X) , there is a compact contractible LN of X containing X o UN and for any open subset U of X, the set nXEU(G(x) LN) is empty or contractible, and the set

n

yEXo

(cintp-l(y))c = =

n

n

(X \ cint{x EX: f(x,y) <.a or f(x,y) > ,8}) yEXo

n

yEXo

ccl{x EX: a

:s; f(x, y) < ,8}

is empty or compact. The condition (iii) of Theorem 2.1 is satisfied. By Theorem 2.1, there exists a point x E X such that x E G(x). It follows that from the definition of G that either gl(X, x) < a or g2(X, x) > ,8, which contradicts the condition (i). Hence the conclusion is true. This completes the proof. Remark 3.1 Theorem 3.1 further answers the open problem raised by Isac, Sehgal and Singh [5] in general setting of noncompact topological spaces and generalizes the corresponding results of Li [6J and Chadli, Chiang and Yao [7J to general topological spaces.

When f(x, y) following result.

= gl(X, y) = g2(X, y)

for all (x, y) E X x X, from Theorem 3.1 we obtain the

Theorem 3.2 Let X be a topological space and a,,8 be two real numbers with a Suppose f : X x X ----; R is a function satisfying the following conditions:

(i) for each x E X, a:S; f(x,x) :S;,8; (ii) for each compact subset D of X,

D

=

U (cint{x EX: f(x,y) < a

yEX

or f(x,y) > ,8}nD);

:s; ,8.

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(iii) there exists a nonempty set X o in X such that for each N E F(X), there exists a compact contractible subset L N of X containing X o UN and for any nonempty open subset U of X, the set {y EX: f(x, y) < 0: or f(x, y) > ,8} LN) is empty or contractible, and the set nyEX O ccl{x EX: 0: :::; f(x, y) :::; ,8} is empty or compact. Then there exists an x E X such that 0: :::; f(x, y) :::;,8, V Y E X. Theorem 3.3 Let X be a nonempty convex subset of a topological vector space E, and 0:, {3 be two real numbers with 0: :::; (3. Suppose f, gl, g2 : X x X -+ R be three functions such that

nxEU(

n

(i) gl(X,X) ~ 0: and g2(X,X) :::;,8 for all x E (ii) for each x E X and for any N E F(X),

x,

N c {y EX: f(x,y) <

0:

or f(x,y) >,8}

=> co(N) c {y EX: gl(X,y) < 0: or g2(X,y) > {3}; (iii) for each each nonempty compact subset D of X,

U (cint{x EX: f(x,y) <

D =

0:

or f(x,y) > ,8}nD);

yEX

(iv) there exists a nonempty subset X o of X and for each N E F(X) there exists a compact convex subset L N of X containing X o UN such that the set K = nyEX O ccl{ x EX: 0: :::; f(x, y) :::; ,8} is empty or compact. Then there exists an x E X such that 0: :::; f(x, y) :::;,8, V Y EX. Proof Define two mappings F, G : X -+ 2x by

F(x) = {y E Y : f(x, y) <

0:

or f(x, y) > {3}

and

G(x) = coF(x) V x E X.

Then for each x E X, F(x) C G(x). The condition (i) of Theorem 2.1 is satisfied. Suppose on the contrary that the conclusion of Theorem 3.3 is not true, then for each x E X, F(x) i=- 0. For each y E X, we have F-1(y) = {x EX: f(x,y) < 0: or f(x,y) > ,8}. Clearly, the condition (iii) implies the condition (ii) of Theorem 2.1 holds. By the condition (iv), there exists a nonempty subset X o of X and for each N E F(X), there is a compact convex subset L N of X containing X o UN such that the set K

=

n

n

(cintF-1(y))C

=

yEXo

=

n

(X \ cintF- 1 (y))

yEXo

ccl{x EX:

0::::;

f(x,y) :::;,8}

yEXo

n

is empty or compact. For any open subset U of X, we have that the set nXEU(G(x) L N ) = nXEu(coF(x) L N ) is empty or convex. Note that each nonempty convex set in a topological vector space is contractible, the condition (iii) of Theorem 2.1 is also satisfied. By Theorem 2.1, there exists a point x E X such that x E G(x) = coF(x). It follows from the condition (ii) that x E coF(x) c {y EX: gl(X, y) < 0: or g2(X, y) > ,8} and hence we have that either gl (x, x) < 0: or g2(X, x) > ,8 which contradicts the condition (i). Therefore the conclusion of Theorem 3.3 must holds.

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Remark 3.2 If for each y E X, the set {x EX: 0: ::; f(x, y) ::; ,8} is compact closed in X, it is easy to see that the condition (iii) of Theorem 3.3 is satisfied. The coercive condition (iv) is weaker than the coercive condition (iv) of Theorem 2.2 of Chadli, Chiang and Yao [7J. Hence

Theorem 3.1 and Theorem 3.3 generalize Theorem 2.2 in [7] under much weaker assumptions. Corollary 3.1 Let X be nonempty convex subset of a topological vector space E, and 0:,,8 be two real numbers with 0: ::; ,8. Suppose f : X x X ---+ R be a function such that (i) for each N E F(X), co(N) c UyEN{X EX: 0: ::; f(x, y) ::; ,8}, (ii) for each y E X, the set {x EX: 0:::; f(x,y) ::;,8} is compactly closed in X, (iii) there exists a nonempty subset X o of X which is contained in a compact convex subset Xl of X such that the set K = nyEXO {x EX: 0: ::; f(x, y) ::; ,8} is empty or compact. Then there exists an x E X such that 0: ::; f(x, y) ::;,8, \j Y E X. Proof For each N E F(X), let L N =co(XI UN), then LN is a compact convex subset of X containing X o UN. The conclusion of Corollary 3.1 follows from Theorem 3.2. Remark 3.3 If for each x E X, 0: ::; f(x, x) ::; ,8 and {y EX: 0: ::; f(x, y) ::; ,8} is an extremal subset of X (see Li [6J), then the condition (i) of Corollary 3.1 holds. In fact, if the condition (i) of Corollary 3.1 does not hold, then there exist N E F(X) and x E co(N) such that x tf:- UyEN{X EX: 0: ::; f(x, y) ::; ,8}, i.e., N C {y EX: f(x, y) < 0: or f(x, y) > ,8}. Since {y EX: 0: ::; f(x, y) ::; ,8} is an extremal subset of X, by Lemma 2.4 of Li [6J, {y E: f(x, y) < 0: or f(x, y) > jJ} is a convex subset of X. It follows that

x E co(N)

c {y

EX: f(x,y) < 0: or f(x,y) >,8}

which contradicts that 0: ::; f(x, x) ::; ,8 for all x E X. It is easy to see that the condition (iii) is weaker than the condition (iii) of Theorem 3.1 of Li [6J. Hence Theorem 3.2 and Corollary 3.1 improve and generalize Theorem 3.1 of Li [6J. References 1 Blum E, Oettli W. From optimization and variational inequalities to equilibriun problems. Math Students, 1994, 63: 123-145 2 Oettli W, Schlager D. Existence of equilibria for g-monotone mappings. In: Nonlinear Analysis and Convex Analysis. Singapore: World Scientific, 1999. 26-33 3 Bianchi M, Schaible S. Generalized monotone bifunctions and equilibrium problems. J Optim Theory Appl, 1996, 90(1): 31-43 4 Ding X P, Park J Y, Jung I H. Existence of solutions for nonlinear inequalities in G-convex spaces. Appl Math Lett, 2002, 15: 735-741 5 Isac G, Sehgal V M, Singh S P. An alternate version of a variational inequality. Indian J Math, 1999, 41(1): 25-31 6 Li J. A lower and upper bounds of a variational inequality. Appl Math Lett, 2000, 13(5): 47-51 7 Chadli 0, Chiang Y, Yao J C. Equilibrium problems with lower and upper bounds. Appl Math Lett, 2002, 15: 327-331 8 Ding X P. New H - K K M theorems and their applications to geometric property, coincidence theorems, minimax inequality and maximal elements. Indian J Pure Appl Math, 1995, 26(1): 1-19 9 Ding X P, Park J Y, Jung I H. Fixed point theorems on product topological spaces and applications. Positivity, 2004, 8: 315-326