数理解析研究所講究録 1207 巻 2001 年 55-66

55
集合値写像のスカラー化と Ky Fan の不等式 *Shogo Nishizawa (西澤正悟) \d,ag er Pando Gr. Georgiev\d,ag er and Tamaki Tanaka (田中 環)\S

Abstract:

ベクトル値関数のあるスカラー化手法を利用して, 集合値写像のスヵ
ラー化を考え, その性質を利用してベクトル値多価写像 (集合値写像) に対する 4 種類の Ky Fan の不等式を導き出す。 この際に, そのスカラー化関数の持っ遺伝性が大変有効に働く。 つまり, 元の集合値関数の持つある種の凸性と半連続性がそのままスヵラー化関数へも伝達される。 この遺伝性に対する詳細な報告は, もうひとっの論文に書かれている。 本論文では, その応用例として 4 タイプの Ky Fan の不等式を紹介する。

Key words:
Fan’s inequality, vector-va ued multifunctions, semicontinuous map
pings, quasiconvex functions.

1.

Introduction.

This paper is concerned with vector-valued variants of the following type of inequality:
if $f(x, x)\leq 0$ for all , then$x$$\min_{y\in X_{x}}\sup_{\in X}f(x, y)\leq 0$,

which is equivalent to the famous Fan’s minimax inequality (this equivalence was proved by
Takahashi [8, Lemma 1] firstly). This inequality is one of the main tools in the nonlinear andconvex analysis, equivalent to Brouwer’s fixed point theorem, Knaster-Kuratowski-Mazurkiewicz
theorem, and so on. As an analytical instrument, in many situations it is more appropriate
and applicable than other main theorems in nonlinear analysis. We refer to [2] for various typeequivalent theorems in nonlinear analysis.
In this paper we show four kinds of vector-valued Fan’s type inequality for multifunctions.
One of them (Theorem 3.1) generalizes the main result of Ansari-Yao in [1], namely, the existence
result in the s0-called Generalized Vector Equilibrium Problem. Any of our Theorems 3.1-3.4
implies the classical Fan inequality, while the main result in [1] does not imply it in its fullgenerality, but only for continuous functions. Our proofs are quite different from that in [1] andare based on the classical scalar Fan inequality. More precisely, in the proofs we use anew result(see Theorem 2.3) which follows from atw0-function result of Simons [7, Theorem 1.2] (used inwork is based on research 11740053 supported by Grant-in-Aid for Scientific Research from the Ministryof Education, Science, Sports and Culture of Japan. The first named author is supported by JSPS fellowshipand International Grant for Research in 1999 and 2000 at Hirosaki University. He is very grateful for the warm
hospitality of the University, during his stay as aVisiting Professor.
of Mathematical Science, Graduate School of Science and Technology, Niigata University, 8050,
Ikarashi 2-n0-ch0, Niigata 950-2181,-2181 新潟市五十嵐 2 の町 8050 新潟大学大学院自然科学研究科数$E$
理科学専攻)-mail:shogo@toki gs.niigata-u.ac.jp
of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1126 Sofia,$E$
Bulgaria(7) レガリア・ソフィア大学数理情報学部)-mad:pandogg@fnLum.-sofia bg
Current E-mail:georgiev@bsp brain.riken go
Information Science, Graduate School of Science and Technology, Niigata University, 8050, Ikarashi
2-n0-ch0, Niigata 950-2181, Japan (-2181 新潟市五十嵐 2 の町 8050 新潟大学大学院自然科学研究科情報理工学$E$
専攻)-mad:prtana@scux.sc.niigata-u.ac jp
2000 Mathematic Subject Classification. Primary:;Secondary:$\ovalbox{\tt\small REJECT} \mathrm{h}\mathrm{i}\mathrm{s}$$\mathfrak{s}_{\mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}$$\mathrm{J}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{n}(\overline{\mathrm{T}}950$$\mathrm{I}_{\mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}$$.\mathrm{j}\mathrm{p}$$\S_{\mathrm{D}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}}$$\mathrm{T}-950$$49\mathrm{J}53$$49\mathrm{J}35,47\mathrm{H}04$

56

Fig. 1:$f(x,y^{*})\not\subset \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$.

Fig.2:$f(x,y^{*})\cap \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)=\emptyset$.

Fig.3:$f(x,y^{*})\cap(-C(x))\neq\emptyset$.

Fig.4:$f(x,y^{*})\subset(-C(x))$.

[7] to derive Fan’s inequality), which we prove directly by Fan’s inequality. For asimple proof ofthe classical Fan inequality, based on Brouwer’s fixed point theorem, we refer to [3] and $??$ .
Our main tool in this paper (Theorem 2.3) is aslightly more general form of atw0-function
result of Simons [7, Corollary 1.6] and as aconsequence of our results, it implies the classical Fan
inequality.
The proofs of the main results (Theorems 4.1-4.4) use Theorem 2.3 for special scalar functions possessing semicontinuity and convexity properties, inherited by the semicontinuity and theconvexity properties of the multifunctions. The four types of Fan’s inequality can be regarded as
generahzations of the classical Fan’s inequality by substituting the nonpositivity of of the scalarfunction $(f(x,y)\leq 0)$ by various types of set relations between the images of multifunction andcone; see Figures 1-4.

Fan’s inequality and anew twxfunction result.

2.

Theorem 2.1 (Pan). Let $X$ be a nonempty compact convex subset of a topological vector space
be quasiconcave in its first variable and lower semicontinuous in its secondand :variable. Then$f$

$X\cross Xarrow \mathrm{R}$$\min_{y\in X_{x}}\sup_{\in X}f(x, y)\leq\sup_{x\in X}f(x, x)$.

Theorem 2.2 (Simons [7, Theorem 1.2]). Let be a nonempty compact convex subset oflower semicontinuous in its second variable,
topological vector space, :quasiconcave in its first variable, and $f\leq g$ on $Z\cross Z.$ Then
$Z$

$f$

$Z\cross Zarrow \mathrm{R}$$\min_{y\in Z_{x}}\sup_{\in Z}f(x,y)\leq\sup_{z\in Z}g(z, z)$

Proof. Define the function$\mathrm{c}\mathrm{o}f$$a$$g:Z\cross Zarrow \mathrm{R}$.

as aquasiconcave envelope of with respect to the first variable:$\mathrm{c}\mathrm{o}f(x,y):=\sup\{_{:\epsilon \mathrm{t}}\min_{1,\ldots,n\}}f(x_{i}, y) :$f$

x=\sum_{\dot{|}=1}^{n}\lambda_{\dot{\iota}^{Xj}}, x_{i}\in Z, \lambda_{i}\geq 0,\sum_{\dot{\iota}=1}^{n}\lambda j=1, n\in \mathrm{N}\}$,

function satisfies the conditions of Fan’s
where is the set of the natural numbers.
theresult.
latter,we
obtainapplying
theandNow we prove our main tool in this paper. Its proof is similar to that of [7, Corollary 1.6].
$\mathrm{N}$$\mathrm{s}\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{g}$$\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}_{1}1\mathrm{i}\mathrm{t}\mathrm{y}$Theorem 2.3. Let $X$ be a nonempty compact convex subset of a topological vector space, :quasiconvex in its
lower semicontinuous in its second variable, :second variable, and$a$$X\cross Xarrow \mathrm{R}$$b$$x,y\in X\mathrm{a}\mathrm{n}\mathrm{d}$$X\cross Xarrow \mathrm{R}$$a(x,y)>0\Rightarrow b(y, x)<0$.

Suppose that $\inf_{x\in X}b(x, x)\geq 0$ . Then there eists $z\in X$ such that $a(x, z)\leq 0$ for all $x\in X$ .
Proof. The proof is straightforward from Theorem 2.2 by defining $f(x, y)=1$ if $a(x, y)>0$
and $f(x, y)=0$ otherwise; $g(x, y)=1$ if $b(y, x)<0$ and $g(x, y)=0$ otherwise.

$\mathrm{I}$57
3.

Definitions and auxiliary results.

Further let $E$ and $Y$ be topological vector spaces and $F$, $C:Earrow 2^{Y}$ two multivalued mappingsand let for every $x\in E$ , $C(x)$ be aclosed convex cone with nonempty interior. We introduce
two types of cone-semicontinuity for set-valued mappings, which are regarded as extensions of
the ordinary lower semicontinuity for real-valued functions; see [5].
(an open base of int $C(x)$ ), where $S$ is aneighborhood
Denote$Y$ ,of 0inand define the function $h(k, x, y)= \inf\{t : y\in tk-C(x)\}$ , Note that $h(k, x, \cdot)$ is
positively homogeneous and subadditive for every fixed $x\in E$ and. Moreover, we$B=C\cap(2S\backslash\circ
s$ , where $C$ is
use the following notations $h(k,y)= \inf\{t : y\in tk-C\}$ , andaconvex closed cone and is aneighborhood of 0in $Y$ . Note again that $h(k$ , is positively$C$ .
homogeneous and subadditive for every fixedFirstly, we prove some inherited properties from cone-semicontinuity.
$B(x)=(\mathrm{i}\mathrm{n}\mathrm{t}C(x))\cap(2S\backslash \overline{S})$$k\in \mathrm{i}\mathrm{n}\mathrm{t}C(x)$$S$$\cdot$$)$$k\in \mathrm{i}\mathrm{n}\mathrm{t}$. The multifunction $F$ isDefinition 3.1. Let-upper semicontinuous at , if forevery $y\in C(\hat{x})\cup(-C(\hat{x}))$ such that, there exists an open $U\ni x_{0}$ such thatfor every $x\in U$ . If $Y$ is aBanach space, we shall say that $F$ is $(-C)^{c}$ upper, if for any
semicontinuous at
and $k\in C$ such that $(k+\epsilon B_{Y}-C)\cap F(xo)=\emptyset$ , there
exists $\delta>0$ such that $(k+\epsilon B_{Y}-C)\cap F(x)=\emptyset$ for every $x\in B(x0;\delta)$ .
$\hat{x}\in E$$C(\hat{x})$$x_{0}$$F(x_{0})\subset y+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})$$F(x)\subset y+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})$$\epsilon>0$$x_{0}$. The multifunction $F$ isDefinition 3.2. Let-lower semicontinuous at , if for$V$every open
such that $F(x_{0})\cap V\neq\emptyset$ , there exists an open $U\ni x_{0}$ such that $F(x)\cap(V+$int
for every $x\in U$ . If $Y$ is aBanach space, we shall say that $F$ islowersemicontinuous at $x0$ , if for any
and $y0\in F(xo)$ there exists an open $U\ni x_{0}$ such that
$F(x)\cap(y0+\epsilon B_{Y}+C(\hat{x}))\neq\emptyset$ for every $x\in U$ , where
denotes the open unit ball in Y.
$\hat{x}\in E$$C(\hat{x})$$x_{0}$$C(\hat{x}))\neq\emptyset$$C(\hat{x})$$\epsilon>0$$B_{Y}$

Remark 3.1. In the two definitions above, the corresponding notions for single-valued function
are equivalent to the ordinary one of lower semicontinuity for real-valued function whenever$Y=\mathrm{R}$and $C=[0, \infty)$ . When the coneconsists only of the zero of the space, the notion
in Definition 3.2 coincides with that of lower semicontinuous set-valued mapping. Moreover it
is equivalent to the cone lower semicontinuity defined in [5], based on the fact of$V+C(\hat{x})$ ;see [9, Theorem 2.2].
$C(\hat{x})$$V+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})=$

Proposition 3.1 If for some $x_{0}\in E$ ,is a compact subset and multivalued map$W$
$:=Y$
()
ping
(int
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.
$A\subset \mathrm{i}\mathrm{n}\mathrm{t}$$\langle$$\cdot$$C(x\mathrm{o})$$C(\cdot)\}$Proof. Assume the contrary. Then there exists anet
converging tosuch that for every$A$there exists. Since is compact, we may assume that $a_{i}arrow a\in A$ . Since $W$ has
aclosed graph, it follows that $a\in W(x\mathrm{o})$ , which is acontradiction.
$\{x_{i}\}$$i$$x_{0}$$a_{i}\in A\backslash C(x_{i})$$\mathrm{I}$

Lemma 3.1. Suppose that multifunction $W$ : $Earrow 2^{Y}$ defined as
closed graph. If the multifunction $F$ is $(-C(x))$ -upper semicontinuous at(the restriction of
the function

has$x\in E$ , then
each
for

$W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$

$x$

$a$

$\varphi_{1}|x$$\varphi_{1}(x):=\inf_{k\in B(x)}\sup_{y\in F(x)}h(k, x, y)$to the set $X$ ) is upper semicontinuous, if $(F, X)$ satisfies the property (P);(P) for every $x\in X$ there exists an open $U\ni x$ such that the set $F(U\cap X)$ is precompact in$Y$ , that is,
is compact.
$\overline{F(U\cap X)}$

If the mapping$C$

is constant-valued, then$\varphi_{1}$is upper semicontinuous,

58
Proof. Assume that (F, X) has property (P). Letofthere exists
such that$>0$ and $x_{0}\in X$ be given.

$\epsilon$

By the definition$k_{0}\in B(x\mathrm{o})$$\varphi_{1}$$h(k_{0}, x_{0}, y)<\varphi_{1}(x_{0})+\epsilon$$\sup$.

$y\in F(x_{0})$Since $\sup_{y\in F(x_{0})}h(k_{0}, x0, y)=\inf\{t : F(x\mathrm{o})\subset tk_{0}-C(xo)\}$ , we can take$\inf\{t :Since $F$ is$(-C(x\mathrm{o}))$F(x_{0})\subset tk_{0}-C(x\mathrm{o})\}<t_{0}<\varphi_{1}(x\mathrm{o})+\epsilon$-upper semicontinuous at$F(x)\subset t_{0}k_{0}$$x_{0}$.

, there exists an open $U\ni x_{0}$ such that

-int $C(xo)$ for every $x\in U$.

By Proposition 3.1 and property (P), forthat$F(x)\subset t’k0-intC(x)$and$t_{0}<t’<\varphi_{1}(x\mathrm{o})+\epsilon$, there exists an open

$U_{1}\subset U$suchfor every $x\in U_{1}\cap X$ .

$k_{0}\in B(x)$

Then

inf

$=$

$\varphi_{1}(x)$$\sup h(k, x, y)$$k\in B(x)_{y\in F(x)}$$h(k_{0}, x, y)$$\sup$$\leq$$y\in t’k_{0}-C(x)$$=$$t’h(k_{0}, x, k\mathrm{o})+$$h(k_{0}, x, y)$$\sup$$y\in-C(x)$$\leq$$t’$$\leq$$\varphi_{1}(x\mathrm{o})+\epsilon$.

The proof of the second statement (when is constant-valued) is similar, but in this case there
is no need to use PrOpOsitiOn3.1 and property (P).
$C$$\mathrm{I}$

Lemma 3.2. Suppose that the multifunction $Fis-C(x)$ -loetter semicontinuous for each $x\in E$
and the multifunction $W:Earrow 2^{Y}$ defined by
has a closed graph. Then the(therestriction
functionof$W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$$\varphi_{2}|x$$\varphi 2$$(x):=$

inf

inf$h(k, x, y)$$k\in B(x)y\in F(x)$to the set $X$ ) is upper semicontinuous, if $(F, X)$ satisfies the property (P). If the mapping $C$ isconstant-valued, thenis upper semicontinuous.
$\varphi 2$

Proof. Let
and $x_{0}\in E$ be given. By the definition of , for, and $z_{0}\in F(xo)$ such that $z_{0}-t_{0}k_{0}\in$ -int $C(xo)$ . By
there exists $k_{0}\in B(xo)$ ,Proposition 3.1, there exists an open set $U_{1}\ni x0$ such that$\epsilon>0$$\varphi_{2}$$k_{0}\in \mathrm{i}\mathrm{n}\mathrm{t}$$z\mathit{0}-t_{0}k_{0}\in$$C(x\mathrm{o})$-int $C(x)$andTherefore$h(k\circ, x, z\mathrm{o})\leq t0$

Letsuch that$\gamma<\epsilon/2$$U_{2}$. By$(-C(x\mathrm{o}))$$t0\in(\varphi_{2}(x\mathrm{o}), \varphi_{2}(xo)+\in)$$k_{0}\in \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$

for every $x\in U_{1}$ .

for every $x\in U_{1}$ .

-lower semicontinuity of $F$ , there exists an open set$G(x):=F(x)\cap$ [ $z_{0}+\gamma k0$ -int $C(xo)$ ]$\neq\emptyset$

for every $x\in U_{2}$ .

Hence$G(U_{2}\cap X)\subset z0+\gamma k_{0}$-int$C(x\mathrm{o})$(3.1)$U_{2}\subset U_{1}$, $x0\in$
(3.2)

59
$\overline{G(U_{2}\cap X)}\subset z_{0}+2\gamma k_{0}$By Proposition 3.1 there exists an open

$U_{3}\subset U_{2}$

,

-int

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

such that$U_{3}\ni x\mathit{0}$for every $x\in U_{3}$ .

$\overline{G(U_{2}\cap X)}\subset z_{0}+2\gamma k_{0}-\mathrm{i}\mathrm{n}\mathrm{t}C(x)$This implies$F(x)\cap$($z_{0}+2\gamma k_{0}$Take $x\in U_{3}\cap X$ and $y_{x}\in F(x)\cap$ (-int $C(x)$ . We obtain$-$

int $C(x)$ )$z_{0}+2\gamma k_{0}$$\neq\emptyset$

for every $x\in Us\cap X$ .

-int $C(x)$ ). Therefore $y_{x}=z_{0}+2\gamma k_{0}+c_{x}$ , where$c_{x}\in$.

$\varphi_{2}(x_{0})+c$$\geq$$t_{0}$$\geq$$h(k\circ, x, zo)$$=$$h(k_{0}, x, y-2\gamma k_{0}-c_{x})$$\geq$$h(k0\cdot, x, y)-h(k_{0}, x, 2\gamma k\mathrm{o})-h(k_{0}, x_{j}c_{x})$(by (3.1))$\geq$$h(k_{0}, x, y)-2\gamma$$\geq$$\varphi_{2}(x)-\epsilon$(by subadditivity of ($h$$k_{0}$, , ))$x$$\cdot$.

Hence$\varphi 2(x\mathrm{o})+2\epsilon$$\geq\varphi 2(x)$for every $x\in U_{3}\cap X$ .

The proof of the second statement (when $C$ is constant-valued) is similar, but in this case there
is no need to use Proposition 3.1 and property (P).

$\mathrm{I}$

Lemma 3.3. Suppose that $Y$ is a Banach space and the multifunction $F:Earrow 2^{Y}$ is $(-C)^{c_{-}}$
upper semicontinuous and locally bounded (it means that for every point $x0\in E$ there exists anopen set $U\ni x_{0}$ and $p>0$ such that $F(x)\subset pB_{Y}$ for every $x\in U$ , where
denotes theopen unit ball in $Y$ ). Suppose that the multifunction $C$ has a closed graph and the cone $C(x)$has a compact base $B(x)=(2\overline{B_{Y}}\backslash B_{Y})\cap C(x)$ for every . Then the function
is lower
semicontinuous.
$B_{Y}$$x$$\varphi_{2}$Proof. Firstly we shall prove that the function $g(k, x):= \inf_{y\in F(x)}h(k, x, y)$ is lower semicontinuous. It is easy to see that$g(k, x)= \inf\{t :(tk-C(x))\cap F(x)\neq\phi\}$(if $(tk-C(x))\cap F(x)=\emptyset$ for every , we put $g(k,$ $x)=+\infty$ ). Takeand letbe sequences such thatand
. Let $\lim\inf h(k_{i}, x_{i})=l$ . There exists a
subsequenceofsuch that $k_{i_{n}}arrow k_{0}\in B(x_{0})$ and. Assumethat $l<g(k_{0}, x_{0})$ . Then there exists
such that$t$$\{x_{i}\}\backslash \{k_{i}\}$$x_{i}arrow x_{0}$$\{(k_{i_{n}}, x_{i_{n}})\}$$(k\mathit{0}, x\mathrm{o})\in Y\cross E$$k_{i}arrow k_{0}$$\{(k_{i}, x_{i})\}$$\mathit{1}=\lim g(k_{i_{n}}, x_{i_{n}})$$\epsilon>0$$l+\epsilon$$<g(k_{0}, x_{0})-\epsilon$

.

(3.3)

By the definition of , there exists$g$$y_{i}\in F(x_{i})\cap[(g(k_{i}, x_{i})+\epsilon)k_{i}-C(x_{i})]$$\forall i\in \mathrm{N}$

.

Hence

(3.4)

$y_{i}=[g(k_{i}, x_{i})+\epsilon]k_{i}-c_{i}$for some $c_{i}\in C(x_{i})$ . By the locally boundedness of $F$ and from the compactness of $B(x_{0})$ , weobtain that the sequenceis precompact. Then by (3.4), passing to limits and using the factthat $C$ has aclosed graph, we obtain$\{c_{i}\}$$\lim y_{i}=y_{0}=(l+\epsilon)k_{0}-c_{0}$

,

(3.5)

60
where $c_{0}\in C(xo)$ . Since $F(xo)$ is bounded and $B(xo)$ is compact, the distance between the sets
and $[g(k_{0}, xo)-\epsilon]k_{0}-C(xo)$ is positive, so there exists at $>0$ such that
$F(x\mathrm{o})$$([g(k_{0}, x\mathrm{o})-\epsilon]k_{0}+\alpha B_{Y}-C(x\mathrm{o}))\cap F(x\mathrm{o})=\emptyset$.

By the $(-C)^{c}$ -upper semicontinuity of $C$ we obtain that for some index$y_{i}\not\in[g(k_{0}, x_{0})-\epsilon]k_{0}+\alpha BY-C(x\mathrm{o})$Hence passing to limit, by (3.3) we obtain(3.5). So we proved the lower semicontinuity ofin [2] and finish the proof.

$\forall i>i\mathit{0}$$y_{0}\not\in[l+\epsilon]k_{0}-C(x\mathrm{o})$

$g$

at

$(k_{0}, x_{0})$$i0\in \mathrm{N}$,.

, which is acontradiction with

. Now, we apply Proposition 3.1.21

1

is a Banach space and the multifunction $F$ : $Earrow 2^{Y}$ is $C(x)-$Lemma 3.4. Suppose thathas
lower semicontinuous for each $x\in E$ and locally bounded. Suppose that the multifunction
a closed graph and the cone $C(x)$ has a compact base $B(x)=(2\overline{BY}\backslash B_{Y})\cap C(x)$ for every .
is lower semicontinuous.
Then the function
$Y$

$C$

$x$

$\varphi_{1}$Proof. Firstly we shall prove that the function $g(k, x):= \sup_{y\in F(x)}h(k, x, y)$ is lower semicon. Let,
be sequences such that $x_{i}arrow x0$ andtinuous. Take $(k0, xo)$ and let
be given. There exists yo $\in F(x_{0})$ such that$\{x_{i}\}$$\epsilon>0$$k_{i}arrow k_{0}$$\{k_{i}\}$$h(k_{0}, x0,y\mathrm{o})>g(k_{0}, x\circ)-\epsilon$.

Since $F$ is -lower semicontinuous, for $\beta>0$ there exists index$C$$F(x:)\cap[y_{0}+\beta BY+C(x\mathrm{o})]\neq\emptyset$(3.6)$i\mathit{0}$such that$\forall i>i0$.

Take $y:\in F(x_{i})\cap[y_{0}+\beta BY+C(x\mathrm{o})]$ . Hence$y_{\dot{l}}=y0+\beta b+c_{i}$

,

(3.7)

md $b\in B_{Y}$ . Since $yj\in[h(k_{i}, x_{i}, y_{i})+\epsilon]k_{i}-C(x_{i})$ , we have $y_{i}\in[g(k_{i}, x_{i})+$$\epsilon]k_{i}-C(x:)$ , and hencewhere$c_{i}\in C(x\mathrm{o})$$-y_{0}-\beta b-c_{i}+[g(k_{i}, x_{i})+\epsilon]k_{i}\in C(x_{\dot{l}})$

.

(3.8)

, we obtain that
By the locally boundedness of $F$ , from (3.7) and the compactness of$\lim\infh(k:,x_{i},
y\mathrm{o})=l$(takingofgenerality
loss.
Without
Let
precompact.
isthe sequence. Then by (3.8), passing
andsubsequences) we may suppose that$C$.
obtaingraph,
weaclosedhas
thatassumption
using
the
and
to limits$>0$$ae=h(k_{0},x_{0},-\betab)$,. Since, whereHence by (3.6),
are arbitrarily small (therefore at is arbitrarily small too, by continuity of ( $k0$ , $x0$ , )), we obtain. This proves lower semicontinuity of at $(k_{0}, xo)$ . Now, we apply Proposition
the proof.
3.1.21 in [2] and
Next, we show some inherited properties from cone-quasiconvexity.
$B(x\mathrm{o})$$\{c:\}$$\mathit{1}=\lim g(k_{\dot{l}}, x_{i})$$k_{:}arrow k_{0}\in B(x\mathrm{o})$$y_{0}+\beta b\in(l+\epsilon)k_{0}-C(x\mathrm{o})$$g(k_{0}, x_{0})$$-\epsilon$$\leq h(k_{0}, x0,y\mathrm{o})\leq l+\epsilon+\alpha$$\epsilon$$h$$h(k_{0}, x0, y\mathrm{o})\leq l$$\beta$$\cdot$$g$$\mathrm{I}$$\mathrm{f}\mathrm{i}\mathrm{n}\cdot \mathrm{s}\mathrm{h}$Definition 3.3. Amultifunction $F$ : $Earrow 2^{Y}$ i called C-quasiconvex, if the set { $x\in E$ :$F(x)\cap(a-C)\neq\emptyset\}$ is convex for every $a\in Y$ .
is -quasiconvex, then is said to be-quasiconcave, which is equivalent to $(-C)$ -quasiconvex mapping.
$\mathrm{s}$$\mathrm{I}\mathrm{f}-F$

$C$

$F$

$C$

Remark 3.2. The above definition is exactly that of Ferro type (-1)-quasiconvex mapping in[6, Definition 3.5].
Definition 3.4. Amultifunction $F:Earrow 2^{Y}$ i called (in the sense of [6, Definition 3.6]$\mathrm{s}$61(a) type-(iii) $C$ -properly quasiconvex if for every two points , $x_{2}\in X$ and every
have either $F(x_{1})\subset F(\lambda x_{1}+(1-\lambda)x_{2})+C$ or $F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})+C$ .

$\lambda\in[0,$$x_{1}$1] we(b) type-(v) -properly quasiconvex if for every two points , $x_{2}\in X$ and every $\lambda\in[0,1]$ wehave either $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})-C$ or $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-C$ ;$C$$x_{1}$

is type-(iii) [resp. type-(v)] $C$-properly quasiconvex, then $F$ is said be type (iii) [resp.
type-(v) $]C$-properly quasiconcave, which is equivalent to type (iii) [resp. type (v)] $(-C)$ -properly
quasiconvex mapping.
$\mathrm{I}\mathrm{f}-F$

Remark 3.3. The convexity of (b) above is exactly that of C-quasiconvex-like multifunction in[1].
Lemma 3.5.

functionIf the multifunction$F$: $Earrow 2^{Y}$ is type-(v) $C$ -properly quasiconvex, then the$\psi_{1}(x):=\inf$$\sup h(k, y)$$k\in B_{y\in F(x)}$is quasiconvex.

Proof.

By definition, for every $\lambda\in[0,1]$ and every , $x2\in X$ we have: either $F(\lambda x_{1}+(1-$or $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-C$ . Assume that $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})-C$ .
$x_{1}$$\lambda)x_{2})\subset F(x_{1})-C$

Then$\psi_{1}(\lambda x_{1}+(1-\lambda)x_{2})$$:=$$\leq$$\inf_{k\in B}\sup\{h(k, y) :y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$$\inf_{k\in B}\sup\{h(k, y) :y\in F(x_{1})-C\}$$h(k, y-c)$$=$$\inf_{\in B}\sup_{y\in}\mathrm{c}\in CF(oe_{1}),$$\leq$(by subadditivity of $h(k$ , ))$\cdot$$k. \in B\mathrm{f}\mathrm{f}\mathrm{i}\sup_{c\in C}y\in F(x_{1})(h(k, y)+h(k, -c))$$\leq$$\psi_{1}(x_{1})$$\leq$$\max\{\psi_{1}(x_{1}), \psi_{1}(x_{2})\}$.

1

Analogously we proceed in the second case, when $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})$ -C.

Lemma 3.6.

If

$F$

is $C$ -quasiconvex, then for every $k\in B$ the function$\psi_{2}(x;k):=\inf\{h(k, y) :y\in F(x)\}$is quasiconvex.
Proof. By the definition ofsuch that$\psi_{k}$, for every$\epsilon>0$

and$x_{1}$, $x_{2}\in E$ there exist $z_{i}\in F(x_{i})$ ,$t_{i}\in \mathrm{R}$

,

(3.9)

, $i=1,2$ .

(3.10)$z_{i}-t_{i}k\in-C$and$t_{i}<\psi_{k}(x_{i})+\epsilon$, by (3.9), we have $z_{i} \in t_{i}k-C\subset\max\{t_{1}, t_{2}\}k$ $C$ . Hence,
Since $s_{1}k-C\subset s2k-C$ for$F$$C$by the -quasiconvexity of , for every $\lambda\in[0,1]$ there exists $y\in F(\lambda x_{1}+(1-\lambda)x_{2})$ such that$y \in\max\{t_{1}, t_{2}\}k-C$ , which means$s_{1}\leq s_{2}$$h(k, y)$-$\leq$$\max\{t_{1}, t_{2}\}$$<$$\max\{\psi_{k}(x_{1}), \psi_{k}(x_{2})\}+\epsilon$

62(by 3.10) and since, the definition, we have$\psi_{k}(\lambda x_{1}+(1-\lambda)x_{2})=\inf\{h(k, y) :and$\epsilon>0$y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$,is arbitrarily small, we obtain $\psi_{k}(\lambda x_{1}+(1-\lambda)x_{2})\leq\max\{\psi_{k}(x_{1}), \psi_{k}(x_{2})\}$.

If the multifunction$\mathrm{I}$: $Earrow 2^{Y}$ is type-(v) $C$ -properly quasiconcave, then the$C$.
function $\psi 2(x;k)$ is quasiconcave, where

Lemma 3.7.

$F$$k\in \mathrm{i}\mathrm{n}\mathrm{t}$and every , $x_{2}\in X$ we have either
Proof. By definition, for every$F(x_{1})+C$ or $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})+C$. Assume that $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})+C$ .
Then$\lambda\in[0,1]$$\psi_{1}(\lambda x_{1}+(1-\lambda)x_{2};k)$$F(\lambda x_{1}+(1-\lambda)x_{2})\subset$$x_{1}$$=$$\inf\{h(k,y) :y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$$\geq$$\inf\{h(k,y+c) :$\geq$$\inf\{h(k,y)-h(k, -c) :$\geq$$\inf\{h(k,y) :$=$$\psi_{1}(x_{1}; k)$$\geq$$\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{n}\{\psi_{1}(x_{1} ; k), \psi_{1}(x_{2};k)\}$y\in F(x_{1}), c\in C\}$y\in F(x_{1}), c\in C\}$y\in F(x_{1})\}$.

Analogicaly we proceed in the second case, when $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})+C$ .

Lemma 3.8.

functionIf the multifunction$F$: $Earrow 2^{Y}$ is type-(iii) $C$ -properly quasiconcave, then the$\psi_{1}(x;k):=\sup\{h(k, y) :is quasiconcave, where$k\in \mathrm{i}\mathrm{n}\mathrm{t}$$C$$\mathrm{I}$y\in F(x)\}$.

, $x_{2}\in X$ we have either $F(x_{1})\subset$
By definition, for every $\lambda\in[0,1]$ and every$F(x_{2})\subset
F(\lambdax_{1}+(1-\lambda)x_{2})-C$.
or
Assume that $F(x_{1})\subset F(\lambda x_{1}+(1-\lambda)x_{2})$ $C$ . Then

Proof.

$x_{1}$$F(\lambda x_{1}+(1-\lambda)x_{2})-C$-$\psi_{2}(x_{1}; k)$$=$$\sup\{h(k, y) :y\in F(x)\}$$\leq$$\sup\{h(k, y-c) :y\in F(\lambda x_{1}+(1-\lambda)x_{2}), c\in C\}$$\leq$$\sup\{h(k,y)+h(k, -c) :$\leq$$\sup\{h(k,y) :$=$$\psi_{2}(\lambda x_{1}+(1-\lambda)x_{2;}k)$y\in F(\lambda x_{1}+(1-\lambda)x_{2}), c\in C\}$y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$,and hence $\min\{\psi_{2}(x_{1} ; k), \psi_{2}(x_{2};k)\}\leq\psi_{2}(\lambda x_{1}+(1-\lambda)x_{2};k)$.
Analogicaly we proceed in the second case, when $F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})$

4.

-

$C$

.

I

Set-valued Fan’s inequalities.

Now we state the main results in this paper. The following theorem is ageneralization of that in[1]. The main difference between our result and that in [1] is the condition (iii), but it allows us
to recover the classical Fan inequality, when is the real line. The result in [1] recovers it only
for continuous functions.
$Y$

Theorem 4.1 Let $K$ be a nonempty convex subset of a topological vector space
logical vector space. Let $F:K\cross Karrow 2^{Y}$ be a multifunction. Assume that

$E$

,

$Y$

be a topO-(i) $C:Karrow 2^{Y}$ is a multifunction such that for every $x\in K$ , $C(x)$ is a closed convex cone in$Y$with int$C(x)\neq\emptyset,\cdot$

63
is a multifunction defined as(ii) W : K$K\crossYj$closed in$arrow 2^{Y}$$W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$is $C(x)$ -upper semicontinuous at(iii) for every $x,y\in K$ ,$C$ is not constant-valued, then the mapping
if the mappingof $K$ into precompact subsets of $Y$ ;$F(\cdot, y)$$x$, and the graph of W iswith closed values on $K$ andmaps the compact subsets$F(\cdot, y)$(iv) there exists a multifunction $G:K\cross Karrow 2^{Y}$ such that(a) for every $x\in K$ ,$G(x, x)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}$(b) for ever$ryx,y\in K$ ,(c) $G(x$ ,$\cdot$$)$,$C(x)$$F(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$

implies$G(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$,

is type-(v) $C(x)$ -properly quasiconcave on $K$ for every $x\in X$ ,(d) $G(x, y)$ is compact, if$G(x,y)\subset \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$ ;(v) there exists a nonempty compact convex subset $D$ of $K$ such that for every $x\in K\backslash D$ , there$C(x)$ .
exists $y\in D$ with$F(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}$Then, the solutions set$S=${ $x\in K$ :

$F$

( ,
$x$

$y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$, for all $y\in K$ }is a nonempty and compact subset of $D$ .

Proof. Put$\sup$$a(x, y):=- \inf_{k\in B(y)_{z\in-F(y,x)}}$$h(k, y, z)$ ,$b(x, y):=$ $k inf\in B(x)\sup_{z\in-G(x,y)}h(k, x, z)$.

It is easy to check that$a(x, y)>0$

if and only if$F(y, x)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(y)$, and also $b(y, x)$
by using the compactness ofcondition (d), and then $a(x, x)\leq 0$ and $b(x, x)\geq 0$ .
$\overline{F(x,y)}$Denote$<0$if$G(y, x)\subset \mathrm{i}\mathrm{n}\mathrm{t}$.

$C(y)$

by using(4.1)$S_{y}:=\{x\in D : F(x, y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)\}$i closed. Let $Y0$ be afinite
Since $a(y$ , is lower semicontinuous (by Lemma 3.1), the set
$Z$
$Z$
$K$
Obviously
is compact and convex.
.
of
hull
convexclosedsubset of . Denote by the
Lemmas 3.1, 3.5 and condition (iv) (b) show that the conditions of Theorem 2.3 are satisfied.
Now we apply Theorem 2.3 and obtain apoint $z\in Z$ such that $a(y, z)\leq 0$ for every $y\in Z$ ,which means(4.2)
for every $y\in Z$ .
$\cdot$$S_{y}$$)$$\mathrm{s}$$Y_{0}\cup D$$\mathrm{F}(\mathrm{y}, y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(z)$The conditions (v) and (4.2) imply that $z\in D$ . Relation (4.1) implies that $\cap\{S_{y} : y\in Y_{0}\}\neq$. So we proved that the family $\{S_{y} : y\in K\}$ has finite intersection property. Since $D$ 1s
compact, $\cap\{S_{y} : y\in K\}\neq\emptyset$ , which means that there exists $x0\in K$ such that $F(x0,y)\not\subset$
int $C(x_{0})$ for every $y\in K$ . So we proved that $S$ is nonempty, and since is aclosed subset of, the proof is completed.
$0$

$S$

$\mathrm{I}$$\mathrm{D}$Theorem 4.2. Let $K$ be a nonempty convex subset of a topological vector space $E$ , $Y$ a topOlogical vector space, and $F:K\cross Karrow 2^{Y}$ a multifunction. Assume that(i) $C$ : $Karrow 2^{Y}$$Y$ with int$is$a multifunction such that for every $x\in K$ , $C(x)$ is a closed convex cone in;$C(x)\neq\emptyset$

64
is a multifunction defined as(ii) W : K
the graph of W is closed in $K\cross Y$ ;$arrow 2^{Y}$$W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}$$C(x)$, for every x $\in K$ such that

is $C(x)$ -louter semicontinuous with closed values on $K$ and if(iii) for every , $y\in K$ ,$C$
the mapping is not constant-valued, then the mapping $F(\cdot,y)$ , for every $y\in K$ , maps thecompact subsets of $K$ into precompact subsets of $Y$ ;$F(\cdot, y)$$x$(iv) there exists a multifunction $G:K\cross Karrow 2^{Y}$ such that(a) for every $x\in K$ ,$G(x, x)\cap \mathrm{i}\mathrm{n}\mathrm{t}C(x)=\emptyset$(b) for every , $y\in K$ ,$x$(c) $G(x$ ,$\cdot$$)$$F(x, y)\cap \mathrm{i}\mathrm{n}\mathrm{t}$,$C(x)\neq\emptyset$

implies$G(x, y)\cap \mathrm{i}\mathrm{n}\mathrm{t}C(x)\neq\emptyset$,

is $C(x)$ -quasiconcave on $K$ for every $x\in K$ ;(v) there exists a nonempty compact convex subset $D$ of $K$ such that for every.
eists $y\in D$ with

$x\in K\backslash D$, there$F(x, y)\cap \mathrm{i}\mathrm{n}\mathrm{t}$ $C(x)\neq\emptyset$

Then, the solutions set$S=${ $x\in K$ :

$F$

( ,
$x$

$C(x))=\emptyset$$y)\cap(\mathrm{i}\mathrm{n}\mathrm{t}$, for all $y\in K$ }is a nonempty and compact subset of $D$ .
Proof. Put$a(x,y):=- \inf_{k\in B}\inf_{(y)z\in-F(y,x)}h(k,y, z)$,$b(x, y):=\underline{\mathrm{i}}\mathrm{n}\mathrm{k}h(k(x), x, z)z\in x,y)$ ’where the function is any fixed selection of the multivalued mapping
int $C(x)$ for every $x\in K$ . It is easy to check that$k$$a(x, y)>0$

if and only if$b(y, x)<0$ if and only if$a(x, x)\leq 0$,$F(y, x)\cap(\mathrm{i}\mathrm{n}\mathrm{t}C(y))$$x-+\mathrm{i}\mathrm{n}\mathrm{t}C(x)$$\mathrm{i}.\mathrm{e}.$, $k(x)\in$I ,$\emptyset$$G(y, x)\cap(\mathrm{i}\mathrm{n}\mathrm{t}C(y))\neq\emptyset$$b(x, x)\geq 0$

,

,

.

Lemmas 3.2, 3.6 and condition (iv) (6) show that the conditions of Theorem 2.3 are satisfied.
Further the proof is the same as that of Theorem 4.1, but in this case $S_{y}:=\{x\in D$ :.
(int $C(x)$ )$F(x, y)\cap \mathrm{I}$$=\emptyset\}$Theorem 4.3. Let $K$ be a nonempty convex subset of a topological vector space
space, and $F:K\mathrm{x}Karrow 2^{Y}$ a multifunction. Assume that

$E$

,

$Y$

a Banach(i) $C:Karrow 2^{Y}$ is a multifunction with a closed graph and $C(x)$ is a closed convex cone witha compact base $B(x)=(2\overline{BY}\backslash B_{Y})\cap C(x)$ for every $Xj$(ii) for every $y\in K$ ,$F(\cdot, y)$is $(-C)^{c}$ -upper semicontinuous and locally bounded;(iii) there exists a multifunction $G:K\mathrm{x}Karrow 2^{Y}$ such that(a) for every $x\in K$ , $G(x, x)\cap(-C(x))\neq\emptyset$ ,(b) for every $x,y\in K$ , $F(x,y)\cap(-C(x))=\emptyset$ implies $G(x, y)\cap(-C(x))=\emptyset$ ,(c) $G(x$ , is type-(v) $C(x)$ -properly quasiconcave on $K$ for every $x\in K$ ;$\cdot$$)$(iv) there exists a nonempty compact convex subset $D$ of $K$ such that for every
exists $y\in D$ with $F(x,y)\cap(-C(x))=\emptyset$ .

$x\in K\backslash D$, there65
Then, the solutions set{ $x\in K$ :

$S=$

$F$

( , $y)\cap(-C(x))\neq\emptyset$ , for$x$$y\in K$$\mathrm{a}1$}

is a nonempty and compact subset of $D$ .
Proof. Putinf$a(x, y):=$$$h(k, y, z)$ ,$b(x, y):=-$$k\in B(y)z\in F(y,x)$

inf

inf

$h(k, x, z)$$k\in B(x)z\in G(x,y)$.

It is easy to check that$a(x, y)\leq 0$

if and only if $F(y, x)\cap(-C(y))\neq\emptyset$ ,$b(y, x)\geq 0$

if and only if $G(y, x)\cap(-C(y))\neq\emptyset$ ,$a(x, x)\leq 0$and$b(x, x)\geq 0$.

Lemmas 3.3, 3.7 and condition (iii) (b) show that the conditions of Theorem 2.3 are satisfied.
Denote $S_{y}:=\{x\in D : F(x, y)\cap(-C(x))\neq\emptyset\}$ . Since $a(y$ , is lower semicontinuous (byLemma 3.3), the set
is closed. Let $Y$ be afinite subset of $K$ . Denote by $Z$ the intersection of$K$ and the linear hull of $Y\cup D$ . Obviously $Z$ is compact and convex. Now we apply
Theorem 2.3
and obtain apoint $z\in Z$ such that$\cdot$$)$$S_{y}$$a(y, z)\leq 0$

for every $y\in Z$(4.3)

which means$F(z, y)\cap(-C(x))\neq\emptyset$

for every$y\in Z$.

(4.4)

Assumption (iv) and condition (4.4) imply that $z\in D$ , and condition (4.4) implies also:
$D$
. So thehasintersection property. Since
is compact,, which completes the proof.
$\cap\{S_{y}$$y\in Y\}\neq\emptyset$$\mathrm{f}\mathrm{a}\mathrm{m}\mathrm{i}1\mathrm{y}\{S_{y}:y\in 01\mathrm{e}\mathrm{t}\mathrm{s}K\}$$\cap\{S_{y} :$.\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}$y\in K\}\neq\emptyset$

Theorem 4.4. Let $K$ be a nonempty convex subset of a topological vector space $E$ , $Y$ a Banach
space, and $F:K\cross Karrow 2^{Y}$ a multifunction. Assume that(i) $C:Karrow 2^{Y}$a multifunction with a closed graph such that $C(x)$ is a closed convex conewith a compact base $B(x)=(2\overline{B}_{Y}\backslash B_{Y})\cap C(x)$ for every ;
$is$

$x$

(ii) for every$x_{\dot{\mathit{1}}}y\in K$,$F(\cdot, y)$is $C(x)$ -lower semicontinuous and locally bounded;(iii) there exists a multifunction $G:K\cross Karrow 2^{Y}$ such that(a) for every $x\in K.,$ $G(x, x)\subset-C(x)$ ,,is type- $(iii)$ $C(x)$ -properly quasiconcave on $K$ for every $x\in K$ ;(b) for every(c)$G(x_{\dot{J}}\cdot$ $)$$x_{\backslash }y\in K$, $F(x, y)\not\subset-C(x)$ implies$G(x_{\grave{l}}y)\not\subset-C(x)$(iv) there exists a nonempty compact convex subset $D$ of $K$ such that for every $x\in K\backslash D$ , there
exists $y\in D$ with $F(x, y)\not\subset-C(x)$ .

Then, the solutions set$S=${ $x\in K$ :$F$

is a nonempty and compact subset of $D$ .

( , $y)\subset-C(x)$ , for all $y\in K$ }$x$

66
Proof. Putinf$a(x, y):=$$\sup$$h(k, y, z)$ ,$b(x, y):=-$$k\in B(y)_{z\in F(y,x)}$i#$\sup$$h(k, x, z)$.

$k\in B(x)_{z\in G(x,y)}$It is easy to check that$a(x, y)\leq 0$

if and only if $F(y, x)\subset-C(y)$ ,$b(y, x)\geq 0$

if and only if $G(y, x)\subset-C(y)$ ,$a(x, x)\leq 0$and$b(x, x)\geq 0$.

Lemmas 3.4, 3.8 and condition (iii) (b) show that the conditions of Theorem 2.3 are satisfied.
Further the proof is the same as that of Theorem 4.3, but in this case $S_{y}:=\{x\in D$ : $F(x, y)\subset$$-C(x)\}$

5.

.

I

Conclusions.

We have presented four type generalizations of the scalar Fan’s inequality in the following setting:(i) set-valued maps with vector-valued images instead of scalar functions;(ii) tw0-function type instead of single function type;(iii) parametric ordering structure instead of fixed ordering structure;(iv) complete extensions including the result of [1],As acorollary from any of Theorems 4.1-4.4, we obtain that Theorem 2.3 implies the scalar Fan
inequality.

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[7] S. Simons (1986). TwO-function minimax theorems and variational inequalities for functions
on compact and noncompact sets, with some comments on fied-point theorems, Proc. Symp.
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