Existence of solutions for nonlinear inequalities in G-convex spaces

PERGAMON

Applied Mathematics

Applied Mathematics Letters

Letters 15 (2002) 735-741

www.elsevier.com/locate/aml

Existence of Solutions for Nonlinear Inequalities in G-Convex XIE PING Department

of Mathematics,

Chengdu,

JONG

YEOUL

Department

610066,

PARK+

of Mathematics,

Kumjung,

Pusan

DING* Sichuan

Sichuan

Spaces

Normal P.R.

University

China

AND IL HYO Pusan

609-735,

National South

JUNG+ University

Korea

(Received July 2000; revised and accepted June 2001)

Abstract-By using a fixed-point theorem in G-convex spaces due to the first author, an existence result for abstract nonlinear inequalities without any monotonicity assumptions is established. As a consequence of our result, we obtain some further generalizations of recent known results. As application, an existence theorem for perturbed saddle point problems is obtained in noncompact G-convex spaces. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords-Fixed

1. Let

point, Abstract nonlinear inequality, Saddle point, G-convex space.

INTRODUCTION

E be a Hausdorff

convex

subset

topological

AND

PRELIMINARIES

vector

space with its dual space E* and X be a nonempty and f * E E*. Gwinner [l] and Ansari, and studied the following nonlinear inequality problem of finding

of E. Let 4 : X x X --+ R be a bifunction

Wong and Yao [2] introduced z E X such that:

where

(., .) denotes

Such types

the pairing

of nonlinear

between

inequalities

E* and E.

model some equilibrium

problems

that

arise from opera-

tions research, as well as some unilateral boundary value problems stemming from mathematical physics. Problem (1) also includes the classical variational inequality problems, variational-like inequality problems, and equilibrium problems studied by many authors as special cases, see [1,2] and the references

therein.

*The first author was supported by the NNSF of China (19871059), the KOSEF of Present address: Department of Mathematics, Education Department of China. Kumjung, Pusan 609-735, Korea. tThe second and third authors were supported by the Korea Research Foundation, D00021. The authors would like to thank the anonymous referees for their useful suggestions 0893-9659/02/$ - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. PII: SO893-9659(02)00035-S

Korea, and NSF of Sichuan Pusan National University, 1998, Project

No. 1998-15-

for improving this paper. Typeset

by d@-w

X. P. DING et al.

736

space and $J, 1c, : X x X -+ R be two bifunctions.

Let X be a topological following

abstract

nonlinear

inequality

problem

(in short,

ANIP($,$))

We consider

of finding

the

z E X such

that $(% Y) 2 Q(% Y), The ANIP(4,?I) (2) is also considered perturbed term, see [3]. Clearly,

if X is a convex subset

V’yEX.

as a perturbed

(2)

equilibrium

problem

inequality

problem

of this paper (2) in G-convex

may not have any monotonicity due to the first author,

is to establish

the existence

spaces without

assumptions.

an existence

theorem

$J is the

vector space E with its dual space E* and for

of a topological

some f E E*, let $(z, y) = (f, y - x ) for all (z, y) E X, then the abstract problem (2) reduces to problem (1). The main purpose

where

results

linear structure

By using a fixed-point of solutions

nonlinear

for the abstract

inequality nonlinear

where the bifunctions theorem

for problem

4 and II,

in G-convex

spaces

(2) is established.

As

consequences of our results, we obtain some further generalizations of the results of Ansari, Wong and Yao [2]. As application, an existence theorem for the perturbed saddle point problem is obtained in noncompact G-convex spaces. As we know, in game theory, the strategy set of each player may not be a convex set in topological vector spaces, see [4-61. Hence, it is absolutely necessary to study the saddle point problems and n-person game problems in G-convex spaces. Let X be a nonempty

set. 2x (respectively,

(X)) denotes

the family of all subsets

(respectively,

all nonempty finite subsets) of X. For any A E (X), IAl denotes the cardinality of A. Let A, be the standard n-dimensional simplex with the vertices ea, ei, . . . , e,. If J is a nonempty subset of (0, 1, . . . , n}, we denote by A J the convex hull of the vertices {ej : j E J}. If X is a topological space, a subset

D of X is said to be compactly

open in X if for each nonempty

K of X, D n K is open in K. Define the compact tint(D)

interior

of D, denoted

= U {B C X : B C D and B is compactly

compact

by tint(D),

subset

as in [7]

open in X} .

It is easy to see that D is compactly open in X if and only if tint(D) = D, and, for any nonempty compact subset K of X, we have tint(D) n K = intK(D fl K) where intK(D n K) denotes the interior of D n K in K. Let X and Y be two topological spaces and G : X + 2y be a set-valued mapping. G is said to be transfer compactly open-valued on X if, for each x E X and each nonempty compact subset K of Y, y E G(z) n K implies that there exists x’ E X, such that y E intK(G(x’) n K). Clearly, each mapping with open values is transfer open valued and is also transfer compactly open valued, and each mapping with transfer open values is transfer compactly open valued. In general, the inverse is not true. G is said to have the compactly local intersection property on X if, for each nonempty compact subset K of X and for each x E K with G(x) # 0, there exists an open neighborhood N(x) of x in X such that nzeN(ljnKG(z) # 0. It is clear that each set-valued mapping with local intersection property (see [7]) has the compactly local intersection property and the inverse is not true in general. A function f : X x Y -+ R is said to be O-transfer compactly upper (respectively, lower) semicontinuous in x if, for each nonempty compact subset K of X and for each x E K with {y E Y : f(x, y) < 0) # 0 (respectively, {y E Y : f(x, y) > 0) # 0), there exists an open neighborhood N(x) of x in X and a point y’ E Y such that f(z, y’) < 0 (respectively, f(z, y’) > 0) for all z E N(x) n K. If we define a set-valued mapping G : X 4 2y by G(x) = {y E Y : f(x, y) < 0) (respectively, G(x) = {y E Y : f(x, y) > 0)) for each z E X, then it is easy to see that G has the compactly local intersection property if and only if f(z, y) is O-transfer compactly upper (respectively, lower) semicontinuous in 2. The following concept of the G-convex space was introduced by Park [8,9] which mends the original definition of G-convex space in [lO,ll] by dropping the isotonic condition. Hence, a new G-convex space is essentially an L-convex space introduced by Ben-El-Mechaiekh et al. [12].

Existence of Solutions

737

Let X be a topological space and l? : (X) -+ 2x be a set-valued mapping satisfying the following condition. (1) For each A E (X) with IAl = n + 1, there exists a continuous mapping $A : A, -+ I’(A) such that, for each B E (A) with IBI = IJI + 1, PA

c l?(B) where A.I denotes the face

of A, corresponding B. Then the pair (X, I’) is called a G-convex space (or, an L-convex space called by Ben-El-Mechaiekh et al. [121). A subset D of a G-convex space (X,I’) is said to be G-convex if, for each A E (D), l?(A) c D. We define the G-convex hull of D denoted by G-co(D) as G-co(D)

=

n {B c X : D c B and B is G-convex} .

LEMMA 1.1. (See (131.) Let X and Y be two topological spaces, K be a nonempty compact subset of X, and G : X ---t 2y be a set-valued mapp’mg with nonempty values on each compact subset of X.

Then the following conditions are equivalent.

(I) G has the compactly local intersection property. (II) For each y E Y, there exists an open subset 0, 0, (III)

n K c G-‘(y)

and K = UyEy(O,

of X (which may be empty) such that

n K). is open

There exist a set-valued mapping F : X + 2’ such that for each y E Y, F-‘(y) in X, F-‘(y) n K c G-‘(y), and K = U,Ey(F-l(y) n K).

(IV) For each 5 E K, there exists y E Y such that z E cint(G-l(y)) K = U

(tint (G-‘(y))

n K) = U (G-‘(y)

n K) .

?JEY

?EY (V) G-l

n K and

: Y ---) 2x is transfer compactly open-vaJued on Y.

The following result is a special case of Theorem 2.3 with X = D of Ding [7] where the G-convex space (X, I’) can be considered as the isotonic condition has been removed. THEOREM 1.1. Let (X, I’) be a G-convex space, K be a nonempty compact subset of X, and x be two set-vaJued mappings such that we have the following. G,T:X-+2 (i) G satisfies one of the Conditions (I)-(V)

in Lemma 1.1.

(ii) For each z E X, N E ((tint G-‘)-‘(z)) implies l?(N) C T(z). (iii) For each 2 E K, G(z) # 8. (iv) For each N E (X), there exist a nonempty compact G-convex subset LN of X containing N such that LN \ K C

U

tint. (G-‘(y)).

l!ELN

Then there exists a point 2 E X such that 4 E T(g). COROLLARY 1 .l.

If Condition (ii) of Theorem 1.1 is replaced by the following condition:

(ii)’ for each 2 E X, G-COG(~) C T(z). Then the conclusion of Theorem 1.1 still holds. PROOF. It, is enough to show Condition that 2 E X and N for each 2 E X, we l?(N) c G - co(z) c Corollary 1.1 follows

(ii)’ implies Condition

(ii) of Theorem

1.1.

Suppose

E ((cintG-‘)-l(z)). Note that (cintG-l)-‘(z) c (G-‘)-l(z) = G(z) have N C G(z) C G-COG(Z). Since G - COG(Z) is G-convex, we obtain T(z). Hence, Condition (ii) of Theorem 1.1 is satisfied. The conclusion of from Theorem 1.1.

COROLLARY 1.2. Let (X,I’) mapping such that

and K be as in Theorem 1.1 and G : X --+ 2x be a set-valued

(i) G has nonempty G-convex values; (ii) Conditions (i) and (iv) of Theorem 1 .l hold. Then G has a fixed point in X.

X. P. DING etal.

738

PROOF. It is easy to check that all conditions of Corollary 1.1 with T = G are satisfied. The conclusion follows from Corollary 1.1. REMARK 1.1. If X is a compact G-convex space, then Condition (iv) of Theorem 1.1 is satisfied automatically. Theorem 1.1, Corollary 1.1, and Corollary 1.2 improve and generalize Theorem 2 of (91to noncompact setting, and in turn generalize many Fan-Browder type fixed-point theorems and have numerous applications, see [7-111 and references therein.

2. EXISTENCE

THEOREMS

THEOREM 2.1. Let (X, I’) be a G-convex space, K be a nonempty compact 4, $I : X x X -t R be two bifunctions such that we have the following.

subset

of X, and

(i) For each z E X, c$(z, z) = $J(z, z) = 0 and the set {y E X : 4(x, y) < +(s, y)} is G-convex. (ii) The function

f(z, y) = c$(z, y) - $(z, y) is O-transfer

(iii) For each N E (X),

there exists a nonempty

compactly

compact

G-convex

upper semicontinuous subset

in x.

LN of X containing

N such that LN \ K c

tint ({x E X : 4(x, y) < ti(x, Y)))

u

.

YELN

Then

there exists a point

C?E X such that

PROOF. Define a set-valued mapping F : X -+ 2x by F(y) = {x E X : 4(x, y) L $(x, y)} for each y E X. Then the solution set of the ANIP(4,$) (2) is S = ngEx F(y). Now we show S # 0. Assume the contrary, S = 0, then for each x E X, there exists a y E X such that 4(x, y) < +(x, Y). Now, define a set-valued mapping G : X -+ 2x by

G(x) = {Y E X : 4(x:,Y) < $J(x,Y))

1

VXEX,

then G(x) # 0 for each x E X. By (i), G( x ) is nonempty G-convex for each x E X. By (ii), the mapping G has the compactly local intersection property. Condition (iii) implies that Condition (iv) of Theorem 1.1 is satisfied. By Corollary 1.2,there exists Z E X such that n: E G(Z). It follows from (i) that 0 = ~(z,z) is a point 2 E X such that

< Q(Z,Z) = 0, w h’Kh is impossible. Hence, S # 8, and there

4(% Y) L ?f%%Y),

vy E x.

REMARK 2.1. Theorem 2.1 is a new result. If X is a compact G-convex space, then by letting X = K = LN for each N E (X), Condition (iii) is satisfied trivially. If Q(x,Y) = 0 for all x, y E X, then Theorem 2.1 reduces to an equilibrium existence theorem. THEOREM 2.2. Let (X,I’) be a G-convex space, K be a nonempty compact 4, $I : X x X + R be two bifunctions such that we have the following.

subset

of X, and

(i) For each x E X, 4(x,x) = +(x, z) = 0 and the set {y E X : 4(x, y) < Q(z, Y)} is G-convex. (ii) For each y E X, lim inf,,,. 4(x, y) 5 4(x*, y) w h enever x + x* and x I+ +(z, y) is lower semicontinuous. (iii) For each N E (X), there exists a nonempty compact G-convex subset LN of X containing N such that LN \K

C

u YELN

{X EX:d'(X, Y) < $(X,Y)).

Existence of Solutions

739

Then there exists f E X such that

For each y E X, let F(y)

PROOF.

= {x E X : 4(x, y) > $J(z,y)}.

Then the solution set of

the ANIP(4, $J) is S = nycx F(y). We show that for each y E X, F(y) is closed. Indeed, let {x~}~~l\ c F(y) be a net such that 2~ -+ x*, then, by (ii) and the definition F(y), we have

Hence, x* E F(y) and F(y) is closed for each y E X. ANIP($,$J)

Now we prove that the solution set S of

is nonempty. If S = 0, then for each x E X, there exists a point y E X such that

4(x, y) < $(x, y). Define a set-valued G : X 4 2x by

G(x) =

{Y E

X : dG, Y) < $J(x,Y/))

vx E x.

1

Then for each x E X, G(x) # 8, and by (i), G has nonempty G-convex values. For each y E X, we have

G-‘(y) =

=

{x E

X :y E

G(x)} =

{xE X

: ~(x,Y) < Q(x,Y>)

x \ {x E x : 4(x, Y) L dJ(x7 Y)) = x \ F(Y)

is open in X, and hence, G-l is transfer compactly open-valued. Note that for each y E Y, we Condition (iv) implies Condition (iv) of Theorem 1.1 have G-‘(y) = int G-‘(y) = tint G-‘(y), holds, and hence, all conditions of Corollary 1.2 are satisfied; there exists a point z E X such that z E G(Z). It follows that 0 = $J(%,?) < +!J(?,?) = 0, which is a contradiction. Hence, S # (b and there exists a point i E X such that

4 (2,Y) 2 1c, 6 Y) )

vy E

x.

REMARK 2.2. If X is a compact G-convex space, by letting X = K = LN for each N E (X), then Condition (iv) of Theorem 2.1 is satisfied trivially. Theorem 2.2 generalizes Theorem 2 of Ansari, Wong and Yao [2] from nonlinear inequality problem (1) in Hausdorff topological vector spaces to the ANIP(4, G) in G- convex spaces without linear structure. COROLLARY 2.1. Let X be a nonempty convex subset of a real topological vector space E and f be a nonzero continuous linear functional on X. Let 4 : X x X + R be a bifunction such that we have the following. (i) (ii) (iii) (iv)

+(x,x) = 0 for all 2 E X. For each 2 E X, y H $(x, y) is convex. For each y E X, liminf,,,. +(x,y) 5 4(x*,y) whenever x -+ x*. There exist a nonempty compact subset K of X and a nonempty compact convex subset X0 of X such that for each x E X \ K, there is a y E co(Xo U {x}) satisfying @(X,Y)

< (f,Y

-4.

Then there exists a point f E X such that d&Y)

t/y E

L (f,Y-4,

x.

PROOF. Define a set-valued mapping r : (X) + 2x by l?(N) = co(N),

V’N E (W,

X. P. DING etal.

740

the convex hull of N. Then (X, I’) is a G-convex space. Define 1L : X x X + R by ti(z,y) = (.f,v - 4, f or all (z, y) E X x X, then $(z,Y) continuous function and $(z,z) = 0 for all 5 E X. It follows from (ii) that, for the set {Y E X : $(T y) < $J(G Y)} is convex, and hence, it is G-convex. For each where co(N)

LN = co(Xo u N); Condition

a bifunction

denotes

then

LN is a nonempty

compact

(iv), for each 2 E LN \ K C X \ K, there

G-convex

subset

is a bilinear each z E X, N E (X), let

of X containing

N.

By

exists y E co(Xo U {z}) c LN such that

#(z, y) < (f, y - z) = $(z, y). It follows that

By Theorem

2.2, the conclusion

REMARK 2.3. in [2, Theorem

of Corollary

It is easy to see that 21. Hence,

2.1 holds.

Condition

Corollary

(iv) of Corollary

2.1 is also a slight

2.1 is weaker than

improvement

of Theorem

Condition

3

2 of Ansari,

Wang and Yao [2].

3. APPLICATION As we know, in game theory, the strategy set of each player may not be a compact convex subset in a topological vector space, see [4-61. Hence, it is absolutely necessary to consider saddle point problems and n-person game problems in noncompact G-convex spaces without linear structure. As space is limited, we only give an application of our Theorem 2.1 to the existence of perturbed saddle

points

where the strategy

set of players

is a noncompact

G-convex

space.

compact THEOREM 3.1. Let (X,I’) b e a G-convex space, K be a nonempty q5,?+!J : X x X + R be two bifunctions such that we have the following.

subset

of X, and

(i) For each 5, y E X, q5(z, x) = $(z, z) = 0, the set {y E X : +(z, y) < $.J(z, y)} and the set {z E X : $J(z,y) < I$(z, y)} are both G-convex. upper semicontinuous in (ii) The function f(z, y) = $J(z, y) - $(z, y) is O-t ransfer compactly X and -f (x, y) is O-transfer compactly lower semicontinuous in y. (iii) For each N E (X), there exists a nonempty

compact

G-convex

subset

LN of X containing

N such that LN \ K c

u tint ({x E X : 4(x:, Y) < @(x7 Y)}) YELN

and

LN \ K c XELN Then q5 has a perturbed

saddle point (2, ij) E X x X such that we have the following.

(a) 4(&c) = +(%8). (b) 4(x, 9) + I$(& 8) - 1cl(x, 121 I 4(%6) I 46%Y) + [1L(%5) - $J(% Y)], v (~7 Y) E X x X. (c) inf,Ex supzEx [4(x, Y) - +(x7 Y)l = suPzE~ inf,Ex M~7 Y) $(x, Y)l= 0. PROOF. By y E X. Let Theorem 2.1 i.e., 4(x, jj) -

Theorem 2.1, there exists a point 2 E X such that q5(?, y) - $(Z, y) 2 0, for all q5l(y,x) = -4(x, y) and $~l(y,z) = -$(z,y), for all (x, y) E X x X. Then, by again, there exists a point jj E X such that $1($,x) - $1($,x) > 0, for all z E X, $(x, 9) 5 0, for all x E X. It follows that 4 (x, 6) - $ (2, ti) I 0 I 4 (23Y)

- $ (2,Y)

7

V(x,y)

E

x xx.

Existence of Solutions

This implies

741

$(?, jj) = $J(c?, C) and

4 (z, 9) - $ (z, Q) 5 4 (% 9) - 1c,(% fi) L 4 (% Y) - II, (2, Y) > From

(3),

we obtain

that

conclusion

(2) holds and

Since infzEX supVEx [4(x, y) - $,(2, y)] L SUP,,X infzEx [4(x, y) - $(x, y)] is always have that conclusion (3) holds. REMARK 3.1. If $J(z,y) = 0, for all (z, y) E X x X, then Theorem theorem and a minimax theorem in noncompact G-convex spaces.

3.1

reduces

true,

to a saddle

so we point

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