On Inherited Properties for Set-Valued Maps and Existence Theorems for Generalized Vector Equilibrium Problems (Nonlinear Analysis and Convex Analysis)

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On

Inherited

Properties

for Set-Valued

Maps and

Existence Theorems for

Generalized Vector

Equilibrium

Problems’

新潟大学大学院自然科学研究科西澤正悟 (Nishizawa, Shogo)\dagger

Graduate School of

Science

and Technology, Niigata University

新潟大学大学院自然科学研究科田中環 Tanaka $\mathrm{R}\mathrm{m}\mathrm{a}\mathrm{k}\mathrm{i})^{\mathrm{t}}$

Graduate School of Science and Technology, Niigata University

1. Introduction

This paper is concerned with

a

generalization of

an

existence theorem for the generalized

vector equilibrium problem in [1], in which

Ansari

and Yao proved

_an

_existence result

by using Fan-Browder type fixed point theorem. It is relative to

a

vector-valued Fan’s

inequality for set-valued

maps

in $[4, 5]$

.

In this paper,

we

consider the following two kinds ofgeneralized vector equilibrium

problems:

find $\overline{x}\in K$ such that $F(\overline{x},y)\not\subset$ -int$C(\overline{x})$ for every $y\in K$ (1.1)

and

find

$i\in K$ such that $F(\overline{x},y)$ rl (-int$C(\overline{x})$) $=\emptyset$ for

every

_{$y\in K$} (1.2)

where $E$ and $\mathrm{Y}$

are

two topological vector

spaces,

_$K$ is

a

nonempty

convex

subset of_$E$

,

$F:K\cross Karrow 2^{\mathrm{Y}}$ is

a

multifunction, $C$ : $Karrow 2^{\mathrm{Y}}$ $\mathrm{i}$

a

multifunction such that for each

$x\in K$, $C(x)$ is

a

closed

convex

cone

with int$C(x)\neq\emptyset$

.

We show existence theorems of

these problems by using Fan’s inequality. Our proofs ofTheorems 3.1 and

3.2 are

quite

different

from that in [1] and in the proofs

we use a

result of Georgiev and Tanaka [4,

Theorem 2.3] whichfollows from

a

twO-function result of

Simons

[11, Theorem 1.2].

By applying the twO-function result for special scalarizing functions possessing

qua-siconvexity and semicontinuity,

we

establish the proofsof the main theorems. For such

a

reason, it is

necessary

for those scalarizing functions to have suchconvexityand

semicon-tinuity. It is, therefore, important and

useful

to study what kindof scalarizing

functions

can

inherit

properties of

such kind of

convexity and semicontinuity

from

multifunctions.

’This work is basedon research 13640097 supported by Grants-in-Aidfor Scientific Research from

the Japan Societyfor thePromotion ofScience ofJapan.

$\uparrow E$-mail.$\cdot$ shogo@ekeland.

$\mathrm{g}\mathrm{s}$.niigata-u.ac.Jp

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213

Toshows

some

resultson theinherited properties,

we

considercertaingeneralizations

and modifications of convexity and semicontinuity for multifunctions in

a

topological vector space with respect to a

cone

preorder in the target space, which have motivated

by $[6, 7]$ and studied in [4] for generalizing theclassical Fan’s inequality. Convexity and

semicontinuity for multifunctions

are

inherited by the following scalarizing functions;

$\inf\{h_{C}(x,y;k) |y\in F(x)\}$ (1.3)

and

$\sup\{h_{C}(x,y;k)|y\in F(x)\}$ (1.4)

where $hc(x,y;k)= \inf\{t|y\in tk-C(x)\}$, $F$ : $E” \mathrm{p}$ $2^{\mathrm{Y}}$

is

a

multifunction, $C(x)$

a

closed

convex cone

with int$C(x)\neq/)$, $x$ and $y$

are

vectors in two topological vector spaces $E$

and $\mathrm{Y}$

,

respectively, and $k\in$ int$\mathrm{C}(\mathrm{x})$. Note that $h_{C}(x, \cdot;k)$ is positively homogeneous

and subadditive for every fixed $x\in E$ and $k\in$ int$\mathrm{C}(\mathrm{x})$, and that $h_{C}(x, y;k)\leq 0$

for $y\in-C(x)$, remark that $-h_{C}(x, -y;k)= \sup\{t|y\in tk+C(x)\}$

.

This function

$h_{C}(x,y;k)$ has been treated in

some

papers. Essentially, $h_{C}$(X, $\mathrm{j};k$) is equivalent to the

smallest strictly monotonic function

defined

by Luc [8]. For each $y\in \mathrm{Y}$, $hc\{x,$$y;k)k$

corresponds the mimimum vector of upper bounds

of

$y$ with respect to the

cone

$C(x)$

restricted

to the direction $k$

.

Similarly, $-h\mathrm{C}(\mathrm{x})-y;k)$$\cdot k$corresponds the maximum

vector

of lower bounds

of

$y$ with respect to the

cone

$C(x)$ restricted to the direction

$k$.

2. Inherited

Properties

of

Set-Valued

Maps

Further let $E$ and $\mathrm{Y}$ be topological vector spaces and $F$ and $C$ : $Earrow 2^{\mathrm{Y}}$ two

multifunc-tions. Denote $B(x)=$

co

((int$\mathrm{C}\{\mathrm{x})$) $\cap(2S ’ \overline{S}))$ (which plays

a

role of base for int$C(x)$

without uniqueness), where $S$ is

a

neighborhood of

0

in Y. We$\mathrm{o}\mathrm{b}\mathrm{s}\mathrm{e}_{l}\mathrm{r}\mathrm{v}\mathrm{e}$thefollowingfour

types ofscalarizing functions:

$\psi( (x;k):=\sup_{y\in F(x)}h_{C}(x,y;k)$, $/’ \mathrm{G}(x;k):=\inf_{y\in F(x)}h_{C}(X, \mathrm{j};k)$;

$- \varphi_{\overline{C}}^{F}(x;k)=\sup_{y\in F(x)}$-hc(x, $-y;k$), $-\psi_{\overline{C}}^{F}(x; k)$ _$=y\in$inf

$x$) $-h_{C}(x, -y;k)$

.

The first and fourth functions have symmetric properties and then results for the

fourth function $-\psi_{\overline{C}}^{F}(x; k)$

can

beeasily proved by those for the first function$\psi_{C}^{F}(x;k)$

.

Similarly, the results for the third function $-\varphi_{\overline{C}}^{F}(x;k)$

can

be deduced by those for the

second function $\varphi_{C}^{F}(x;k)$. By using these four functions

we measure

each image of

mul-function $F$ with respect to its 4-tuple ofscalars, which

can

be regarded

as

standpoints

for the evaluation of the image. To avoid

confusion

for properties ofconvexity,

we

con-sider the constant

case

of$C(x)=C$ (a

convex

cone) and $B(x)=B$ (a

convex

set),

and

$hc(x,y;k)=h_{C}(y;k):= \inf\{t|y\in tk-$ (J.

To

beginwith,

we

recall

some

kinds of

convexity

for multifunctions.

Definition 2.1, _{A multifunction} $F$ : $Earrow 2^{\mathrm{Y}}$ is called $C$-quasiconvex, if the

set

$\{x\in E|F(x)\cap(a-C)4\emptyset\}$is

convex

or

emptyfor every$a\in$ Y. If$-F$is C-quasiconvex,

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Remark 2.1. The above definition isexactly that of Ferro type (-1)- quasiconvex

map-pin.g in [7, Definition 3.5].

Definition 2.2. Amultifunction$F$ : $Earrow 2^{\mathrm{Y}}$ is called (in the

sense

of [7, Definition3.7])

(a) type-(iii) $C$-naturally quasiconvex if for every two points $x_{1}$,$x_{2}\in E$ and every

A $\in(0,1)$, there exists $\mu\in[0,1]$ such that

$\mu F(x_{1})+(1-\mu)F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})+C;$

(b) type-(v) $C$-narurally quasiconvex, if for

every

two points $x_{1}$,$x_{2}\in E$ and

every

A $\in(0,1)$, there exists $\mu\in[0,1]$ such that

$F$($\lambda x_{1}+$ (1-X)x2) $\subset$ $\mathrm{F}(\mathrm{x})+(1-\mathrm{n})\mathrm{F}(\mathrm{x}2)-C$.

If $-F$ is type-(iii) [resp., type-(v)] $C$-naturally quasiconvex, then $F$ is said

to

be

type-(iii) [resp., type-(v)] $C$-naturally quasiconcave, which is equivalent to

a

type-(iii) [resp.,

type (v)$]$ $(-C)$-naturally quasiconvexmapping.

However, there is

no

relationship

between

those for types (iii) and (v) in general.

Proposition 2.1.

See

[7, Theorem 3.1]. For a

_{multifunction}

$F$ : $Earrow 2^{\mathrm{Y}}$

,

type-(iii) $C$-naturally quasiconvexity implies C-quasiconvexity.

Proposition 2.2. For each$x\in E$ and a

multifunction

$F:Earrow 2^{\mathrm{Y}}$,

(i) $\psi_{C}^{F}(x;k)$ is

convex

with respect to variable $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$;

(ii) $f_{C}^{F}$

Cx;

$k$) is

convex

with respect to variable $k\in$ int$C$,

if

_$F(x)$ is

a convex

set.

Now,

we

show

some

inherited

properties ofconvexity for

multifunctions.

Lemma

2.1.

_If

$F$

:

$Earrow 2^{\mathrm{Y}}$ is _{$typearrow(\mathrm{v})C$}-narurally quasiconvex, then _{$\psi^{F}(x):=$}

inf$k\in B\psi_{C}^{F}(x;k)$ is quasiconvex, and especially$\psi_{C}^{F}(x;k)$ is $quas\acute{\iota}convex$with respectto

vari-able$x$ where $k\in$ int$C$

.

(ii) $\varphi_{C}^{F}(x;k)$ is

convex

with respect to variable $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$,

if

$F(x)$ is

a convex

set.

Now,

we

show

some

inherited

properties ofconvexity for

multifunctions.

Lemma

2.1.

_If

$F$

:

$Earrow 2^{\mathrm{Y}}$ is _{$typearrow(\mathrm{v})C$}-naturally quasiconvex, then _{$\psi^{F}(x):=$}

$\inf_{k\in B}\psi_{C}^{F}(x;k)$ is quasiconvex, and especially$\psi_{C}^{F}(x;k)$ is $quas\acute{\iota}convex$with respectto

vari-able$x$ where $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$

.

Lemma 2.2.

_If

$F$ : $Earrow 2^{\mathrm{Y}}$ is convex-valued and _$C$ quasiconvex, then _{$\varphi^{F}(x):=$}

$\inf_{k\in B\mathrm{j})}\mathrm{r}(x;k)$ is quasiconvex, and especially$\varphi_{C}^{F}(x;k)$ is quasiconvex with respect to

vari-able $x$ where $k\in$ int$C$.

Remark 2.2. When

we

replace $F$ by _$-F$ in the two lemmas above, it leads to the

quasiconcavity ofscalarizing functions $-p^{-F}$ and $-\varphi^{-F}$ By Proposition2.1, if$F$ : $Earrow$

$2^{\mathrm{Y}}$ is convex-valued and type-(iii)

$C$-naturally quasiconvex,

then

$\varphi^{F}(x)$ is quasiconvex.

we

show

some

inherited

properties

from

some

kinds of

semicontinuity.

We

introduce two types of cone-semicontinuity for multifunctions, which

are

regarded

as

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215

Definition 2.3. Let $\hat{x}\in E.$ A multifunction $F$ is called $C(\hat{x})$-upper semicontinuousat

$x_{0}$, if for every $y\in C(\hat{x}> )\cup(-C(\hat{x}))$ satisfying with $F(x_{0})\subset y+$ int$C(\hat{x})$ , there exists

an

open $U\ni x_{0}$ such that $F(x)\subset y+$int$C(\hat{x})$ for every $x\in U.$

Definition 2.4. Let $\hat{x}\in E.$ A multifunction $F$ is called $C(\hat{x})$-lorner semicontinuousat

$x_{0}$, if for every open $V$ such that $F(x_{0})\cap V4$

$\emptyset$, there exists

an

open $U\ni x_{0}$ such that

$F(x)\cap$ ($V+$ int$C(\hat{x})$) $4$ $\emptyset$ for every $x\in U.$

Remark

2.3.

In the two definitions above, the

notions

for single-valued functions

are

equivalent to the ordinary notion oflower semicontinuity of real-valued ones, whenever

$\mathrm{Y}=\mathrm{R}$ and $C(x)=[0, \infty)$. Usual upper semicontinuous

multifunction

is also

(cone-$)$ upper semicontinuous. When the

cone

$C(\hat{x})$ consists only of the

zero

of the space,

the notion in Definition 2.4 coincides with that oflower

semicontinuous

multifunction.

Moreover, it is equivalent to the cone-lower semicontinuity defined in [6], based

on

the

fact that $V+$ int$C(\hat{x})=V+C(\hat{x})$;

see

[13, Theorem 2.2].

Proposition 2.3. See [10, Proposition 2]. Assume that there exists

a

compact subset

$D\subset \mathrm{Y}$ satisfying (i) $A\subset$ cone/} where coneD $:=\{\lambda x|\lambda\geq 0,x\in D\}$ and (ii) $D\subset$

int$C(x_{0})$

for

some

$x_{0}\in E$

.

If

$W(\cdot):=\mathrm{Y}\backslash$

{int

$C(\cdot)$

}

has a closed graph, then there

exists

an

open set

$U\ni x_{0}$ such that $A\subset C(x)$

for

every

$x\in U$

.

In particular$C$ is lower

semicontinuous.

Lemma 2.3. Suppose that $W$ ; $Earrow 2^{\mathrm{Y}}$

defined

as

$W(x)=\mathrm{Y}\backslash$ int$C(x)$ has

a

closed

graph.

_If

$F$ is $(-C(x))$-upper semicontinuous at $x$

for

each $x\in E$ and there exists $a$

compact-valued

_{multifunction}

$D$ : $Earrow 2^{\mathrm{Y}}sat\dot{u}$fying

for

each $x_{0}\in E,$ (i) $D(x_{0})\subset$

int$C(x_{0})$ and (ii)

for

every $t\in \mathrm{R}$, _{$k\in B(x_{0})$} and $x\in E$ satisfying with $tk$ – $\mathrm{F}(\mathrm{x})\subset$

int$\mathrm{C}(\mathrm{x})$, tk-F(x)\subset coneD(#o), then

$\psi^{F}(x):=$ inf $\sup h_{C}(x, y;k)$

$k\in B(x)_{y}\in F(x)$

is upper semicontinuous.

_If

the mapping $C$ is constant-valued, then $p^{F}$ is upper

semi-continuous.

Lemma 2.4. Suppose that $W$ : $Earrow 2^{\mathrm{Y}}$

defined

as

$W(x)=\mathrm{Y}\backslash$int$C(x)$ has

a

closed

graph.

_If

$F$ is $(-C(x))$ lowersemicontinuotts

for

each $x\in E$ and there eists

a

compact

valued

_{multifunction}

$D:Earrow 2^{\mathrm{Y}}$ satisfying

for

each $x_{0}\in E,$ $(\mathrm{i})D(x_{0})\subset$ int$C(x_{0})$ and

(ii)

_for

ever

$ryt<t^{*}\in \mathrm{R}_{f}k\in E$ $\mathrm{B}(\mathrm{x}\mathrm{o})$ $x\in E$ and$y\in F(x_{0})$ satisfying with $F(x)\cap[y+$

$tk-$int$\mathrm{C}(\mathrm{x}\mathrm{Q})]4$ $\emptyset$, _{$\mathrm{F}(\mathrm{x})\cap[y+t^{*}k-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}D(x_{0})]\neq$}$\emptyset_{f}$ then $\varphi^{F}(x):=$ inf inf $h_{C}(x,y;k|$

$k\in B(x)y\in F(x)$

is upper semicontinuous.

_If

the mapping $C$ is constant-valued, then $\varphi^{F}$ is upper

semi-continuous.

Remark

2.4. When

we

replace $F$by $-F$ inthe two lemmas above, it leads tothelower

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3. Existence Results

Firstly,

we

introduce

our

main tool, which is presented in [4, Theorem 2.3], for proving

the main results in this paper.

Lemma 3.1. See [4, Theorem 2.3]. Let $X$ be

a

nonempty compact

convex

subset

of

$a$

topological vector space, $a$ : $X\cross Xarrow \mathrm{R}$ lower semicontinuous in its second variable, $b:X\cross Xarrow \mathrm{R}$ quasiconvex in its second variable, and

$x$,$y\in X$ and $a(x,y)>0\Rightarrow b(y,x)<0.$

If

$\inf_{x\in X}b(x, x)\geq 0,$ then there exists $z$ $\in X$ such that$a(x, z)\leq 0$

for

every $x\in$

.X.

Now

we

present two existence results for generalized vector equilibrium problems.

If

$\inf_{x\in X}b(x, x)\geq 0,$ then there exists $z$ $\in X$ such that$a(x, z)\leq 0$

for

every $x\in X.$

Now

we

present two existence results for generalized vector equilibrium problems.

Theorem 3.1. Let$K$ be

a

nonempty

convex

subset

of

a

topological vector space $E$

,

$\mathrm{Y}a$

topological vector space. Let $F:K\mathrm{x}Karrow 2^{\mathrm{Y}}$ be

a

multifunction.

Assume

that

(i) $C:Karrow 2^{\mathrm{Y}}$ is a

multifunction

such that

for

every _{$x\in K,$ $C(x)$} _is a closed

convex

cone

in $\mathrm{Y}$ with int_{$C(x)\neq\emptyset$};

(ii) $W:K-2^{\mathrm{Y}}$ is

a

multifunction defined

as

$W(x)=\mathrm{Y}\backslash$

-int

$\mathrm{C}(\mathrm{x}))$

, and the

graph

of

$W$ is closed in $K\mathrm{x}\mathrm{Y}$;

(iii)

_for

every $x$,$y\in K$, $F(\cdot, y)$ is $(-C(x))$-upper semicontinuous at $x$;

(iv) there eists

a

_{multifunction}

$G:K\mathrm{x}Karrow 2^{\mathrm{Y}}$ such that

(iv) there exists

a

_{multifunction}

$G:K\mathrm{x}Karrow 2^{\mathrm{Y}}$ such that

(a)

_for

every $x\in K$, $G$(x,$x$) $\not\subset$ -int$\mathrm{C}(\mathrm{x})$,

(b)

_for

every $x,y\in K$, $F(x, y)\subset$ -int$C(x)$ implies $G(x, y)\subset$ -int$\mathrm{C}(\mathrm{x})$,

(c) $G(x, \cdot)$ is type-(v) $C(x)$-naturally quasiconvex

on

$K$

for

every_{$x\in K,$}

(d) $G(x,y)$ is compact,

if

$G(x,y)\subset$ -int$C(x)$

(v)

there

eists

a

nonempty compact

convex

subset$P$

of

$K$

such

that

for

every

$x\in K\backslash P$

,

there

eists $y\in P$ with $F(x,y)\subset$ -int$\mathrm{C}(\mathrm{x})$;

(vi) there exists

a

compact-valued

_{multifunction}

$D$ : $Karrow 2^{\mathrm{Y}}$ such that

for

each$x_{0}\in E,$

(a) $D(x_{0})\subset$ int$C(x_{0})$

,

(b)

_for

every $t\in \mathrm{R}$, _{$k\in B(x_{0})$} and_{$x\in E$} satisfying withtk-F(x)\subset int_{$C(x_{0})$},

tk-F(x)\subset coneD$(\mathrm{x}\mathrm{Q})$

.

Then, the solutions

set

$S=$

{

$x\in K|F(x,y)\not\subset$ -int$\mathrm{C}(\mathrm{x})$, for

every

$y\in K$

}

is

a

nonempty

and

compact subset

_of

$P$

.

(b)

_for

every $t\in \mathrm{R}$, _{$k\in B(x_{0})$} and_{$x\in E$} satisfying withtk-F(x)\subset int_{$C(x_{0})$}, $tk-F(x)\subset \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}D(x_{0})$

.

Then, the solutions

set

$S=$

{

$x\in K|F(x,y)\not\subset-\mathrm{i}\mathrm{n}\mathrm{t}C(x)$, for

every

_{$y\in K$}

}

(6)

217

Proof. Put

$a(x,y):=-$ inf $\sup$ $C(y)z;k)$, $b(x, y):=$ inf $\sup$ $C(x)z;k)$

.

$k\in B(y)_{z}\in F(y,x)$ $k\in \mathrm{f}1(x)$

$z\mathrm{e}\mathrm{c}\mathrm{y}(x,y)$

It is easy to check that

$a(x,y)>0$ if and only if $F(y, x)\subset$ -int$C(y)$

by using condition (vi), and also

$b(y, x)<0$ if and only

if

$G(y, x)\subset$ -int$C(y)$

by using condition (vi), and also

$b(y, x)<0$ if and only

if

$G(y, x)\subset$ -int$C(y)$

by using (d) of the condition (iv), and then $a(x, x)\leq 0$ and $b(x, x)\geq 0.$ Denote

$S_{y}:=$

{

_{$x\in P|F$}($x$,$y)\not\subset$ -int$C(x)$

}

$=\{x\in P|a(y, x)\leq 0\}$ . (3.1)

Since

$a(y, \cdot)$ is lower semicontinuous (by Lemma 2.3), the set $S_{y}$ is closed. Let $\mathrm{Y}_{0}$ be

a

finitesubset of$K$

.

Denoteby $Z$the closed

convex

hull of$\mathrm{Y}_{0}\cup P$. Obviously $Z$ is compact

and

convex.

Lemmas 2.1,

2.3

and (b)

of

the condition (iv) show that the conditions of

Lemma

3.1 are

satisfied.

Now

we

apply

Lemma

3.1

and obtain

a

point $z\in Z$ such that $a(y, z)\leq 0$ for

every

$y\in Z,$ which

means

$F(z, y)\not\subset$ -int$C(z)$ for every $y\in Z$. (3.2) The conditions (v) and (3.2) imply that $z\in P.$ Relation (3.1) implies that

$\cap\{S_{y}|y\in \mathrm{Y}_{0}\}\neq\emptyset$.

So

we

proved that the family $\{S_{y}|y\in K\}$ has finite intersection property. Since $P$ is

compact,

$\cap\{S_{y}|y\in K\}$ ’ $\emptyset$,

which

means

that there exists $x_{0}\in K$ such that

$F(x_{0},y)\langle?-\mathrm{i}\mathrm{n}\mathrm{t}C(x_{0})$ for every $y\in K.$

So

we

proved that $S$ is nonempty, and since $S$ is

a

closed subset of $P$, the proof is

completed. I

So

we

proved that $S$ is nonempty, and since $S$ is aclosed subset of $P$, the proof is

completed.

I

Remark 3.1. The above theorem is

a

generalizationof the theorem that it is replaced

$F$ and $G$ in [4, Theorem 4.1] by $-F$ and $-G$, respectively. Themain difference between

our

result and [4, Theorem 4.1] is (c) of the condition (iv), which is

more

generalized

with respect to convexity.

Theorem 3.2. Let$K$ be

a

nonempty

convex

subset

of

a

topological

vector

space$E$, $\mathrm{Y}a$

topological

vector

space. Let $F:K\mathrm{x}Karrow 2^{\mathrm{Y}}$ $be$

a

multifunction.

Assume

that

(i) $C:Karrow 2^{\mathrm{Y}}$ _is

a

multifunction

such that

for

every _{$x\in K,$ $C(x)$} is

a

closed

convex

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(ii) $W$ : $Karrow 2^{\mathrm{Y}}$ is

a

multifunction

defined

as

_{$W(x)=\mathrm{Y}\mathrm{z}$} (-int$\mathrm{W}(\mathrm{x})$), and

the

graph

of

$W$ is closed in $K\cross \mathrm{Y}j$

(iii)

_for

every $x,y\mathrm{E}$ $K$, $\mathrm{F}\{\mathrm{x},$

$y$) is $\mathrm{C}(\mathrm{x}))$-lorner semicontinuous at $x$;

(iv) there exists a

_{multifunction}

$G:K\cross Karrow 2^{\mathrm{Y}}$ such that

(a)

_for

every $x\in K,$ $\mathrm{G}(\mathrm{z}, x)\cap$ (-int$\mathrm{W}(\mathrm{x})$) $=\emptyset$,

(b)

_for

every$x$,$y\in K$, $\mathrm{F}\{\mathrm{x},$

$y$)$\cap$(-int$\mathrm{W}(\mathrm{x})$)

$\mathit{1}$ $\emptyset$ implies

$\mathrm{W}(\mathrm{x})y)\cap$(-int$\mathrm{W}(\mathrm{x})$) $\neq\emptyset_{f}$

(c) $\mathrm{G}(\mathrm{z}, \cdot)$ is $C(x)$-quasiconvet

on

$K$

for

every

$x\in K,$

(d) $G$ is convex-valued;

(v)

there

eists

a

nonemptycompact

convex

subset$P$

of

$K$ such that

for

every$x\in K\backslash P$

,

there eists $y\in P$ with $F(x, y)$ _rl (-int$C(x)$) $\neq\emptyset$;

(vi) there eists a compact-valued

_{multifunction}

$D$ : $Karrow 2^{\mathrm{Y}}$ such that

for

each _{$x_{0}\in E,$}

(a) $\mathrm{W}(\mathrm{x})\subset$ int$\mathrm{C}(\mathrm{x}0)]$

(b)

_for

every $t<t^{*}\in \mathrm{R}$, _{$k\in B(x_{0})$}, _{$x\in E$} and _{$y\in F(x_{0})$} satisfying $with$

$\mathrm{F}\{\mathrm{x}$) $\cap$[_$y$$+tk-$ int$C(x_{0})$] $\neq\emptyset$

,

$\mathrm{F}\{\mathrm{x}$)rl $[y+t^{*}k-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}D(x_{0})]\neq\emptyset$

.

Then, the solutions

set

$S=$

{

$x\in K|$b(x,$\mathrm{y})\cap$ (-int$C(x))=\emptyset)$, for every $y\in K$

}

is a nonempty and compact subset

_of

$P$

.

Proof. Put

$a(x,y):=- \inf_{k\in B(y)z\in}\mathrm{i}\mathrm{f}y$$\mathrm{f},x)G(y, z;k)$, $b(x, y):= \inf_{k\in B(x)}\inf_{z\in G(x,y)}W(x)z;k)$

.

It is

easy

to check that

$a(x, y)>0$ if and only if $F(y, x)\cap$(-int$C(y)$) $\mathrm{z}$ $\emptyset$,

$b(y, x)<0$ ifand only if $G(y, x)\cap$ (-int$C(y)$) $\neq\emptyset$,

$\mathrm{a}(\mathrm{x}, x)\leq 0,$ $\mathrm{W}(\mathrm{x})x)\geq 0.$

$b(y,x)<0$ ifand only if $G(y, x)\cap$ (-int$C(y)$) $\neq\emptyset$,

$a(x, x)\leq 0,$ $b(x, x)\geq 0.$

Lemmas

2.2, 2.4 and (b)ofthe condition (iv) show that the conditions of Lemma 3.1

are

satisfied.

Further the proof is the

same

as

that of Theorem 3.1, but in this

$\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}$

$S_{y}:=$

{

$x\in P|$$F$($x$,$y$) rl (-int$C(x))=\emptyset$

}.

Remark 3.2. The above theorem is

a

improvement of the theorem that it is replaced

$F$ and $G$ in [4, Theorem4.2] by $-F$ and _$-G$, respectively. However, (d) of the condition

(iv) is addedin comparison with [4, Theorem4.2], because

we

wantto

use

Lemma 2.2 in

(8)

219

4. Conclusions

We have established new inherited properties of convexity for set-valuedmaps. By using

one

ofthose

new

inheritedproperties and applyingto set-valued Fan’sinequalityin $[4, 5]$,

we

have generalized the existence theorem in [1]. We have also presented

an

existence

theorem for

a

different type of the generalized vector equilibrium problem in [1].

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