Ky Fan's Inequality for Set-Valued Maps with Vector-Valued Images (Nonlinear Analysis and Convex Analysis)

Title Author(s) Citation Issue Date URL Ky Fan's Inequality for Set-Valued Maps with Vector-Valued Images (Nonlinear Analysis and Convex Analysis) Georgiev, Pando Gr.; Tanaka, Tamaki 数理解析研究所講究録 (2001), 1187: 143-154 2001-01 http://hdl.handle.net/2433/64679 Right Type Textversion Departmental Bulletin Paper publisher Kyoto University 数理解析研究所講究録 1187 巻 2001 年 143-154 143 Ky Fan’s Inequality for Set-Valued Maps with Vector-Valued Images* PANDO $\mathrm{G}\mathrm{R}$ . $\mathrm{G}\mathrm{E}\mathrm{o}\mathrm{R}\mathrm{G}\mathrm{I}\mathrm{E}\mathrm{v}\dagger_{\mathrm{a}\mathrm{n}\mathrm{d}}$ TAMAKI TANAKA (ffl $\zeta \mathrm{F}$ $\ovalbox{\tt\small REJECT}_{)}\ddagger$ Abstract: We consider four variants of Fan’s type inequality for vector-valued multifunctions in topological vector spaces with respect to a cone preorder in the target space, when the functions and the cone possess various kinds of semicontinuity and convexity properties. In order to establish these results, firstly we prove a twofunction result of Simons directly by the scalar Fan’s inequality, after that, by its help we derive a new two-function result, which is the base of our proofs. As a consequence of our Fan’s type inequalities we obtain that this new two-function result is equivalent to the scalar Fan’s inequality. Key words: pings, Fan’s inequality, vector-valued multifunctions, semicontinuous map- vex functions. $\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{o}\dot{\mathrm{n}}$ 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 and convex 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 type equivalent 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 so-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 full generality, but only for continuous functions. Our proofs are quite different from that in [1] and are based on the classical scalar Fan inequality. More precisely, in the proofs we use a new result (see Theorem 2.3) which follows from a two-function result of Simons [7, Theorem 1.2] (used in *This work is based on research 11740053 supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan. The first named author is supported by JSPS fellowship and 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 a Visiting Professor. \dagger Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1126 Sofia, -mail: pandogg@fmi. Current E-mail: -sofia.bg Bulgaria georgiev@bsp.brain.riken.go.jp \ddagger Department of Mathematical System Science, Faculty of Science and Technology, Hirosaki Universg ) sity, Hirosaki 036-8561, Japan( -mail: sltana@cc.hirosaki-u.ac.jp . ; Secondary: 2000 Mathematic Subject Classiflcation. Primary: $(7^{\grave{\backslash }})\triangleright j\mathrm{j}\mathrm{t})7\cdot\backslash y7\tau^{\nu}7\star 5\ovalbox{\tt\small REJECT}\Phi \mathrm{E}\mathrm{f}\mathrm{f}\mathrm{l}\neq\infty \mathrm{g}\beta)$ $E$ $\mathrm{u}\mathrm{n}\mathrm{i}$ $\overline{\mathrm{T}}036-8561\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{l}\hslash \mathrm{X}\mathrm{B}\mathrm{W}3$ $5.\Lambda \mathrm{f}\mathrm{f}\mathrm{i}|\star\not\in \mathrm{E}\mathrm{I}\mp\#\propto\beta$ $E$ $49\mathrm{J}53$ $49\mathrm{J}35,47\mathrm{H}04$ $\sqrt[\backslash ]{}7_{\backslash }\overline{\tau}\Delta\#.\}^{\mathrm{R}}+\#\}$ 144 Fig. 1: $f(x, y^{*})\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ . Fig.2: $f(x, y^{*})\cap$ int $C(x)=\emptyset$ . 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 a simple proof of the 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 a slightly more general form of a two-function result of Simons [7, Corollary 1.6] and as a consequence 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 the be regarded as convexity properties of the multifunctions. The four types of Fan’s nonpositivity of of the scalar the substituting by inequality Fan’s generalizations of the classical function $(f(x, y)\leq 0)$ by various types of set relations between the images of multifunction and cone; see Figures 1-4. $\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\cdot \mathrm{c}\mathrm{a}\mathrm{n}$ Fan’s inequality and a new two-function result. 2. Theorem 2.1 (Fan). Let $X$ be a nonempty compact convex subset of a topological vector space be quasiconcave in its first variable and lower semicontinuous in its second and : variable. Then $X\cross Xarrow \mathrm{R}$ $f$ . $\min_{y\in X_{x}}\sup_{\in X}f(x, y)\leq\sup_{x\in X}f(x, x)$ Theorem 2.2 (Simons [7, Theorem 1.2]). Let be a nonempty compact convex subset lower semicontinuous in its second variable, topological vector space, : Z. Then quasiconcave in its first variable, and $f\leq g$ on $Z$ $f$ of a $g:Z\cross Zarrow \mathrm{R}$ $Z\cross Zarrow \mathrm{R}$ $Z\cross$ $\min_{y\in Z_{x}}\sup_{\in Z}f(x, y)\leq\sup_{z\in Z}g(z, z)$ Proof. Define the function co co $f$ . as a quasiconcave envelope of with respect to the first variable: $f$ $f(x, y):= \sup\{\min_{1i\in\{,\ldots,n\}}f(x_{i}, y):x=\sum_{i=1}^{n}\lambda_{i^{X_{i},X_{i}}}\in Z, \lambda_{i}\geq 0, \sum_{i=1}^{n}\lambda_{i}=1, n\in \mathrm{N}\}$ , where is the set of the natural numbers. This function satisfies the conditions of Fan’s and applying the latter, we obtain the result. Now we prove our main tool in this paper. Its proof is similar to that of [7, Corollary 1.6]. $\mathrm{N}$ $\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{l}$ Theorem 2.3. Let $X$ be a nonempty compact convex subset of a topological vector space, : $X\cross Xarrow \mathrm{R}$ quasiconvex in its lower semicontinuous in its second variable, : variable, and $x,$ $y\in X$ and $a(x, y)>0\Rightarrow b(y, x)<0$ . $a$ $b$ $X\cross Xarrow \mathrm{R}$ $se\backslash cond$ $Su\backslash$ ppose that $\inf_{x\in X}b(x, x)\geq 0$ . Then there exists $z\in X$ such that $a(x, z)\leq 0$ for all $x\in X$ . Proof. The proof is straightforward ffom 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. 1 145 3. Definitions and auxiliary results. Further let $E$ and $Y$ be topological vector spaces and $F,$ : $Earrow 2^{Y}$ two multivalued mappings and let for every $x\in E,$ $C(x)$ be a closed 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]. Denote (an open base of int $C(x)$ ), where is a neighborhood of in , and 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 $k\in$ int $C(x)$ . Moreover, we use the following notations $h(k, y)= \inf$ { : $y\in$ tk–C}, and $B=C\cap(2S\backslash \overline{S})$ , where $C$ is a convex closed cone and is a neighborhood of in $Y$ . Note again that is positively homogeneous and subadditive for every fixed . Firstly, we prove some inherited properties from cone-semicontinuity. $C$ $B(x)=(\mathrm{i}\mathrm{n}\mathrm{t}C(x))\cap(2S\backslash \overline{S})$ $0$ $S$ $\mathrm{Y}$ $t$ $S$ $h(k, \cdot)$ $0$ $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$ Definition 3.1. Let . The multifunction $F$ is -upper semicontinuous at , if for every $y\in C(\hat{x})\cup(-C(\hat{x}))$ such that , there exists an open $U\ni x_{0}$ such that for every $x\in U$ . If is a Banach space, we shall say that $F$ is $(-C)^{c}$ -upper semicontinuous at , if for any $\in>0$ and $k\in C$ such that $(k+\in B_{Y}-C)\cap F(x_{0})=\emptyset$ , there exists $\delta>0$ such that $(k+\in B_{Y}-C)\cap F(x)=\emptyset$ for every $x\in B(x_{0}; \delta)$ . $\hat{x}\in E$ $C(\hat{x})$ $x_{0}$ $F(x_{0})\subset y+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})$ $Y$ $F(x)\subset y+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})$ $x_{0}$ Definition 3.2. Let . The multifunction $F$ is -lower semicontinuous at , if for 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 is a Banach space, we shall say that $F$ is -lower $serr\iota icontinuous$ at , if for any $\in>0$ and $y_{0}\in F(x_{0})$ there exists an open $U\ni x_{0}$ such that $F(x)\cap(y_{0}+\in B_{Y}+C(\hat{x}))\neq\emptyset$ for every $x\in U$ , where denotes the open unit ball in Y. $\hat{x}\in E$ $C(\hat{x})$ $x_{0}$ $V$ $Y$ $C(\hat{x}))\neq\emptyset$ $C(\hat{x})$ $x_{0}$ $B_{Y}$ Remark 3.1. In the two definitions above, the corresponding notions for single-valued function are equivalent to the ordinary one of lower semicoIltinuity for real-valued function whenever and $C=[0, \infty)$ . When the cone consists 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]. $Y=\mathrm{R}$ $C(\hat{x})$ $V+\mathrm{i}\mathrm{n}\mathrm{t}C(\hat{x})=$ Proposition 3.1 If for some $x_{0}\in E,$ is a compact subset and multivalued mapping has a closed graph, then there exists an open set $U\ni x_{0}$ such that every $x\in U$ . In particular is lower semicontinuous. $A\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x_{0})$ $W(\cdot):=Y\backslash \{\mathrm{i}\mathrm{n}\mathrm{t}C(\cdot)\}$ $C$ $A\subset C(x)fo\mathit{7}^{\cdot}$ Proof. Assume the coIltrary. Then there exists a net such that for every converging to there exists . Since is compact, we may assume that $a_{i}arrow a\in A$ . Since $W$ has a closed , it follows that $a\in W(x_{0})$ , which is a contradiction. $\{x_{i}\}$ $i$ $a_{i}\in A\backslash C(x_{i})$ $x_{0}$ $A$ 1 $\mathrm{g}\mathrm{T}\mathrm{a}\mathrm{p}\mathrm{h}$ Lemma 3.1. Suppose that rnultifunction $W$ : $Earrow 2^{Y}$ defined as closed graph. If the rnultifunction $F$ is $(-C(x))$ -upper semicontinuous at (the restriction of the function $W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $x$ for each $x\in E$ has a , then $\varphi_{1}|x$ $\varphi_{1}(x).--\inf_{k\in B(x)_{y\in F^{1}(x)}}\mathrm{s}^{\backslash }\mathrm{u}\mathrm{p}h(k, x, y)$ to the set $(P)$ ) is upper for every $Y$ If the $X$ $x\in X$ , that is, rnapping $sernicontir\iota uous$ , if $(F, X)$ satisfies the properiy there exisis an open is compact. $U\ni x$ such that the set is constant-valued, then $\varphi_{1}$ ; $F(U\cap X)$ $\overline{F(U\cap X)}$ $C$ $(P)$ is upper semicontinuous. is precompact in 146 Proof. Assume that $(F, X)$ has property of there exists $k_{0}\in B(x_{0})$ such that $(P)$ . Let $\in>0$ and $x_{0}\in X$ be given. By the definition $\varphi_{1}$ $h(k_{0}, x_{0}, y)<\varphi_{1}(x_{0})+\in$ $\sup$ . $y\in F^{\gamma}(x_{0})$ Since $\sup_{y\in F^{\urcorner}(x_{0})}h(k_{0}, x_{0}, y)=\inf\{t:F(x_{0})\subset tk_{0}-C(x_{0})\}$ , we can take . $\inf\{t:F(x_{0})\subseteq tk_{0}-C(x_{0})\}<t_{0}<\varphi_{1}(x_{0})+\in$ Since $F$ is $(-C(x_{0}))$ -upper semicontinuous at $F(x)\subset t_{0}k_{0}$ By Proposition 3.1 and property that $(P)$ , there exists an open –int $C(x_{0})$ for every , for $t_{0}<t’<\varphi_{1}(x_{0})+\in$ and $F(x)\subset t’k_{0}-intC(x)$ $x_{0}$ $x\in U$ such that . , there exists an open for every $k_{0}\in B(x)$ $U\ni x_{0}$ $x\in U_{1}\cap X$ $U_{1}\subset U$ such . Then inf $=$ $\varphi_{1}.(x)$ $\sup h(k, x, y)$ $k\in B(x)_{y\in F^{7}(x)}$ $h(k_{0}, x, y)$ $\sup$ $\leq$ $y\in t’k0-C(x)$ $=$ $t’h(k_{0}, x, k_{0})+$ $h(k_{0}, x, y)$ $\sup$ $y\in-C(x)$ $\leq$ $t’$ $\leq$ $\varphi_{1}(x_{0})+\in$ . 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 $(P)$ . $C$ 1 Lemma 3.2. Suppose that the multifunction and the multifunction $W$ : $Earrow 2^{Y}$ defined by (the restriction of function $F$ is $-C(x)$ -lower semicontinuous for each $x\in E$ has a closed graph. Then the $W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $\varphi_{2}|x$ $\varphi_{2}(x):=\inf_{k\in B(x)}\inf_{y\in F(x)}h(k, x, y)$ to the set $X$ ) is upper semicontinuous, if $(F, X)$ satisfies the property constant-valued, then is upper semicontinuous. $(P)$ . If the mapping $C$ is $\varphi_{2}$ Proof. Let $\in>0$ and $x_{0}\in E$ be given. By the definition of , for there exists $k_{0}\in B(x_{0}),$ int $C(x_{0})$ , and $z_{0}\in F(x_{0})$ such that $z_{0}-t_{0}k_{0}\in$ -int $C(x_{0})$ . By Proposition 3.1, there exists an open set such that $\varphi_{2}$ $t_{0}\in(\varphi_{2}(x_{0}), \varphi_{2}(x_{0})+\in)$ $k_{0}\in$ $U_{1}\ni x_{0}$ $z_{0}-t_{0}k_{0}\in$ -int $C(x)$ and $k_{0}\in \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ for every $x\in U_{1}$ . Therefore $h(k_{0}, x, z_{0})\leq t_{0}$ $U_{2}$ Let $\gamma<\in/2$ . By such that $(-C(x_{0}))$ for every -lower semicontinuity of $G(x).–F(x)\cap$ [ $z_{0}+\gamma k_{0}$ –int $C(x_{0})$ ] $F$ $x\in U_{1}$ , there exists an open set $\neq\emptyset$ for every Hence $G(U_{2}\cap X)\subset z_{0}+\gamma k_{0}$ (3.1) . –int $C(x_{0})$ $x\in U_{2}$ . $U_{2}\subset U_{1},$ $x_{0}\in$ (3.2) 147 and $\overline{G(U_{2}\cap X)}\subset z_{0}+2\gamma k_{0}$ By Proposition 3.1 there exists an open $U_{3}\subset U_{2},$ $\overline{G(U_{2}\cap X)}\subset z_{0}+2\gamma k_{0}$ –int $C(x_{0})$ . $U_{3}\ni x_{0}$ such that –int $C(x)$ for every $x\in U_{3}$ . This implies $F(x)\cap$ Take $c_{x}\in$ ( $z_{0}+2\gamma k_{0}$ and $y_{x}\in F(x)\cap$ ( -int $C(x)$ . We obtain $x\in U_{3}\cap X$ $\varphi_{2}(x_{0})+\in$ –int $C(x)$ ) $z_{0}+2\gamma k_{0}$ for every $\neq\emptyset$ $x\in U_{3}\cap X$ –int $C(x)$ ). Therefore $\geq$ $t_{0}$ $\geq$ $h(k_{0}, x, z_{0})$ $=$ $h(k_{0}, x, y-2\gamma k_{0}-c_{x})$ $\geq$ $h(k_{0}, x, y)-h(k_{0}, x, 2\gamma k_{0})-h(k_{0}, x, c_{x})$ $\geq$ $h(k_{0}, x, y)-2\gamma$ $\geq$ $\varphi_{2}(x)-\in$ . $y_{x}=z_{0}+2\gamma k_{0}+c_{x}$ , where (by (3.1)) (by subadditivity of $h(k_{0},$ $x,$ $\cdot)$ ) . Hence for every $\varphi_{2}(x_{0})+2\in\geq\varphi_{2}(x)$ $x\in U_{3}\cap X$ . 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 $(P)$ . 1 Lemma 3.3. Suppose that 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 $x_{0}\in E$ there exisis an open set $U\ni x_{0}$ and $p>0$ such thai $F(x)\subset pB_{Y}$ for every $x\in U$ , where denotes the open unit ball in ). Suppose that the multifunction 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. $Y$ $B_{Y}$ $Y$ $C$ $x$ Proof. Firstly we shall prove that the function tinuous. It is easy to see that $\varphi_{2}$ $g(k, x):= \inf_{y\in F(x)}h(k, x, y)$ is lower semicon- $g(k, x)= \inf\{t:(tk-C(x))\cap F(x)\neq\emptyset\}$ for every , we put $g(k,$ $x)=+\infty$ ). Take $(k_{0}, x_{0})\in Y\cross E$ and let be sequences such that . Let $\lim$ inf $h(k_{i}, x_{i})=l$ . There exists a and subsequence of such that $k_{i_{n}}arrow k_{0}\in B(x_{0})$ and $l= \lim g(k_{i_{n}}, x_{i_{n}})$ . Assume that $l<g(k_{0}, x_{0})$ . Then there exists $\in>0$ such that (if $(tk-C(x))\cap F(x)=\emptyset$ $t$ $\{x_{i}\},$ $\{k_{i}\}$ $x_{i}arrow x_{0}$ $\{(k_{i_{n}}, x_{i_{n}})\}$ $k_{i}arrow k_{0}$ $\{(k_{i}, x_{i})\}$ $l+\in<g(k_{0}, x_{0})-\in$ . (3.3) By the definition of , there exists $g$ $y_{i}\in F(x_{i})\cap[(g(k_{i}, x_{i})+\in)k_{i}-C(x_{i})]$ $\forall i\in \mathrm{N}$ . Hence (3.4) $y_{i}=[g(k_{i}, x_{i})+\in]k_{i}-c_{i}$ for some $c_{i}\in C(x_{i})$ . By the locally boundedness of $F$ and from the compactness of $B(x_{0})$ , we obtain that the sequence is precompact. Then by (3.4), passing to limits and using the fact that has a closed graph, we obtain $\{c_{i}\}$ $C$ $\lim y_{i}=y_{0}=(l+\in)k_{0}-c_{0}$ , (3.5) 148 is bounded and $B(x_{0})$ is compact, the distance between the sets and $[g(k_{0}, x_{0})-\in]k_{0}-C(x_{0})$ is positive, so there exists $\alpha>0$ such that where $c_{0}\in C(x_{0})$ $F(x_{0})$ . Since $F(x_{0})$ $([g(k_{0}, x_{0}) - \in]k_{0}+\alpha B_{Y}-C(x_{0}))\cap F(x_{0})=\emptyset$ By the $(-C)^{c}$ -upper semicontinuity of $C$ . we obtain that for some index $y_{i}\not\in[g(k_{0}, x_{0})-\in]k_{0}+\alpha B_{Y}-C(x_{0})$ $\forall i>i_{0}$ $i_{0}\in \mathrm{N}$ , . Hence passing to limit, by (3.3) we obtain $y_{0}\not\in[l+\in]k_{0}-C(x_{0})$ , which is a contradiction with . Now, we apply Proposition 3.1.21 (3.5). So we proved the lower semicontinuity of at in [2] and finish the proof. I $g$ $(k_{0}, x_{0})$ is a Banach space and the multifunction $F$ : $Earrow 2^{Y}$ is $C(x)-$ Lemma 3.4. Suppose that lower semicontinuous for each $x\in E$ and locally bounded. Suppose that the multifunction 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. $Y$ $C$ $x$ $\varphi_{1}$ Proof. Firstly we shall prove that the function $g(k, x):= \sup_{y\in be sequences such that tinuous. Take and let be given. There exists $y_{0}\in F(x_{0})$ such that $(k_{0}, x_{0})$ $\{x_{i}\},$ $\{k_{i}\}$ $x_{i}arrow x_{0}$ $h(k_{0}, x_{0}, y_{0})>g(k_{0}, x_{0})-\in$ Since $F$ is $C$ -lower semicontinuous, for $\beta>0$ $y_{i}\in F(x_{i})\cap[y_{0}+\beta B_{Y}+C(x_{0})]$ and $b\in B_{Y}$ . Since , and hence $c_{i}\in C(x_{0})$ $\in]k_{i}-C(x_{i})$ (3.6) $i_{0}$ such that $\forall i>i_{0}$ . . Hence $y_{i}=y_{0}+\beta b+c_{i}$ where and is lower semicon. Let $\in>0$ $k_{i}arrow k_{0}$ . there exists index $F(x_{i})\cap[y_{0}+\beta B_{Y}+C(x_{0})]\neq\emptyset$ Take F(x)}h(k, x, y)$ , (3.7) $y_{i}\in[h(k_{i}, x_{i}, y_{i})+\in]k_{i}-C(x_{i})$ $-y_{0}-\beta b-c_{i}+[g(k_{i}, x_{i})+\epsilon]k_{i}\in C(x_{i})$ , we have $y_{i}\in[g(k_{i}, x_{i})+$ . (3.8) By the locally boundedness of $F$ , from (3.7) and the compactness of $B(x_{0})$ , we obtain that the sequence is precompact. Let $\lim$ inf $h(k_{i}, x_{i}, y_{0})=l$ . Without loss of generality (taking subsequences) we may suppose that $k_{i}arrow k_{0}\in B(x_{0})$ and $l= \lim g(k_{i}, x_{i})$ . Then by (3.8), passing to limits and using the assumption that has a closed graph, we obtain $y_{0}+\beta b\in(l+\in)k_{0}-C(x_{0})$ . Hence by (3.6), $g(k_{0}, x_{0})-\in\leq h(k_{0}, x_{0}, y_{0})\leq l+\in+\alpha$ , where $\alpha=h(k_{0}, x_{0}, -\beta b)$ . ), we obtain are arbitrarily small (therefore is arbitrarily small too, by continuity of $h(k_{0}, x_{0}, y_{0})\leq l$ . This proves lower semicontinuity of at . Now, we apply Proposition 3.1.21 in [2] and finish the proof. Next, we show some inherited properties from cone-quasiconvexity. $\{c_{i}\}$ $C$ $\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\in>0,$ $h(k_{0},$ $\alpha$ $g$ $x_{0},$ $\beta$ $\cdot)$ $(k_{0}, x_{0})$ 1 A multifunction : $Earrow 2^{Y}$ is called -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 $(-C)$ -quasiconcave, which is equivalent to -quasiconvex mapping. Definition 3.3. $F$ $C$ $\mathrm{I}\mathrm{f}-F$ $C$ $F$ $C$ Remark 3.2. The above definition is exactly that of Ferro type [6, Definition 3.5]. Definition 3.4. A multifunction $F:Earrow 2^{Y}$ $(-1)$ -quasiconvex mapping in is called (in the sense of [6, Definition 3.6]) 149 (a) $type-(\mathrm{i}\mathrm{i}\mathrm{i})C$ -properly quasiconvex if for every two points have either (b) $F(x_{1})\subset F(\lambda x_{1}+(1-\lambda)x_{2})+C$ or $x_{2}\in X$ $x_{1},$ and every $\lambda\in[0,1]$ $F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})+C$ we . $x_{2}\in X$ and every $\lambda\in[0,1]$ we -properly quasiconvex if for every two points have either $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})-C$ or $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-C$; $type-(\mathrm{v})C$ $x_{1},$ If $-F$ is type-(iii) [resp. type-(v)] $C$-properly quasiconvex, then $F$ is said be type-(iii) [resp. type-(v) -properly quasiconcave, which is equivalent to type-(iii) [resp. type-(v)] $(-C)$ -properly quasiconvex mapping. $]$ $C$ Remark 3.3. The convexity of (b) above is exactly that of C-quasiconvex-like multifunction in [1]. Lemma 3.5. function If the multifunction $F$ : is $Earrow 2^{Y}$ $\psi_{1}(x):=$ inf $type-(\mathrm{v})C$ -properly quasiconvex, ihen the $\sup h(k, y)$ $k\in B_{y\in F(x)}$ is quasiconvex. Proof. By definition, for every $\lambda)x_{2})\subset F(x_{1})-C$ or $\lambda\in[0,1]$ and every $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-C$ $x_{1},$ $x_{2}\in X$ . Assume that we have: either $F(\lambda x_{1}+(1-$ $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})-C$ . Then $\psi_{1}(\lambda x_{1}+(1-\lambda)x_{2})$ $:=$ $\inf_{k\in B}\sup\{h(k, y) : $\leq$ y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$ $\inf_{k\in B}\sup\{h(k, y) : inf $=$ y\in F(x_{1})-C\}$ $h(k, y-c)$ $\sup$ $k\in B_{y\in F()}$ inf $\leq$ $\sup(h(k, y)+h(k, -c))$ (by subadditivity of $h(k,$ $\cdot)$ ) $k\in B_{y\in F^{\backslash }(x)}$ $\leq$ $\psi_{1}(x_{1})$ $\leq$ $\max\{\psi_{1}(x_{1}), \psi_{1}(x_{2})\}$ Analogously we proceed in the second case, when Lemma 3.6. If $F$ is $C$ -quasiconvex, then . . $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})-C$ for every $k\in B$ $\psi_{2}(x;k):=\inf\{h(k, y) : the 1 function y\in F(x)\}$ is quasiconvex. Proof. By the definition of such that $\psi_{k}$ , for every $\in>0$ and $x_{1},$ $z_{i}-t_{i}k\in-C$ $x_{2}\in E$ there exist , $z_{i}\in F(x_{i}),$ $t_{i}\in \mathrm{R}$ (3.9) and $t_{i}<\psi_{k}(x_{i})+\in,$ $i=1,2$ . (3.10) Since $s_{1}k-C\subset s_{2}k-C$ for , by (3.9), we have $z_{i} \in t_{i}k-C\subset\max\{t_{1}, t_{2}\}k-C$ . Hence, by the -quasiconvexity of , for every 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 $s_{1}\leq s_{2}$ $C$ $F$ $\lambda\in[0,1]$ $h(k, y)$ $\leq$ $\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{x}\{t_{1}, t_{2}\}$ $<$ $\max\{\psi_{k}(x_{1}), \psi_{k}(x_{2})\}+\in$ 150 (by 3.10) and since, the definition, we have $\psi_{k}(\lambda x_{1}+(1-\lambda)x_{2})=\inf\{h(k, y) : $\mathrm{a}\mathrm{n}\mathrm{d}\in>0$ is arbitrarily small, we obtain Lemma 3.7. function If the multifunction $\psi_{2}(x;k)$ or : is quasiconcave, where Proof. By definition, for every $F(x_{1})+C$ $F$ $\lambda\in[0,1]$ y\in F(\lambda x_{1}+(1-\lambda)x_{2})\})$ I . $\psi_{k}(\lambda x_{1}+(1-\lambda)x_{2})\leq\max\{\psi_{k}(x_{1}), \psi_{k}(x_{2})\}$ $Earrow 2^{Y}$ . $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$ is $type-(\mathrm{v})C$ -properly quasiconcave, then the $x_{2}\in X$ we have either and every . Assume that $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{1})+C$ . $F(\lambda x_{1}+(1-\lambda)x_{2})\subset$ $x_{1},$ $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})+C$ Then $\psi_{1}(\lambda x_{1}+(1-\lambda)x_{2}; k)$ $=$ $\inf\{h(k, y) : $\geq$ $\inf\{h(k, y+c) : $\geq$ $\inf\{h(k, y)-h(k, -c) : $\geq$ $\inf\{h(k, y) : $=$ $\psi_{1}(x_{1};k)$ $\geq$ $\min\{\psi_{1}(x_{1};k), \psi_{1}(x_{2)}k)\}$ Analogicaly we proceed in the second case, when Lemma 3.8. functio n If the rnultifunction $F$ : y\in F(\lambda x_{1}+(1-\lambda)x_{2})\}$ $k\in \mathrm{i}\mathrm{n}\mathrm{t}C$ y\in F(x_{1}), c\in C\}$ y\in F(x_{1})\}$ . I . $F(\lambda x_{1}+(1-\lambda)x_{2})\subset F(x_{2})+C$ $Earrow 2^{Y}$ is $type-(\mathrm{i}\mathrm{i}\mathrm{i})C$ $\psi_{1}(x;k):=\sup\{h(k, y) : is quasiconcave, where y\in F(x_{1}), c\in C\}$ -properly quasiconcave, then the y\in F(x)\}$ . $x_{2}\in X$ we have either By definition, for every $\lambda\in[0,1]$ and every $F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})-C$ or . Assume that $F(x_{1})\subset F(\lambda x_{1}+(1-\lambda)x_{2})-C$ . Then Proof. $x_{1},$ $F(x_{1})\subset$ $F(\lambda x_{1}+(1-\lambda)x_{2})-C$ $\psi_{2}(x_{1}; k)$ $=$ $\sup\{h(k, y) : y\in F(x)\}$ $\leq$ $\sup\{h(k, y-c) : y\in F(\lambda x_{1}+(1-\lambda)x_{2}), c\in C\}$ $\leq$ $\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 $F(x_{2})\subset F(\lambda x_{1}+(1-\lambda)x_{2})-C$. 4. I Set-valued Fan’s inequalities. state the main results in this paper. The following theorem is a generalization 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 $Y$ is the real line. The result in [1] recovers it only for c\={o}ntinuous functions. $\mathrm{N}\mathrm{o}\mathrm{w}|\mathrm{w}\mathrm{e}$ 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 (i) $C$ : $Y$ with int $Karrow 2^{Y}$ is a muliifunction $C(x)\neq\emptyset$ ; such that for every $x\in K,$ $C(x)$ $E,$ $Y$ be a topo- is a closed convex cone in 151 : is a closed in $K\cross Y$ ; (ii) $W$ $Karrow 2^{Y}$ multifunction defined as $W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ is $C(x)$ -upper semicontinuous at for every $y\in K,$ $noi$ constant-valued, then the mapping the is mapping if of into precompact subsets of ; (iii) $F(\cdot, y)$ $x,$ $C$ $W$ is with closed values on $K$ and maps the compact subsets $Y$ multifunc tion (iv) there exists a (b) of $F(\cdot, y)$ $K$ (a) $x$ , and the graph for for every $x\in K,$ every $x,$ (c) $G(x, \cdot)$ (d) $G(x, y)$ is $G(x, x)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $y\in K,$ , $F(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $type-(\mathrm{v})C(x)$ is compact, if such that $G:K\cross Karrow 2^{Y}$ implies -properly quasiconcave on $y\in D$ with $K$ for every , $x\in X$ , $G(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)_{i}$ (v) there exists a nonempty compact convex subset exists $G(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $F(x, y)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ . $D$ of $K$ such that for every $x\in K\backslash D$ , there Then, the solutions set $S=$ { $x\in K$ : $F(x,$ is a nonempty and compact subset $y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ , for all $y\in K$ } of . $D$ Proof. Put $h(k, y, z)$ , $\sup$ $a(x, y):=- \inf_{k\in By)_{z\in-F(y,x)}}$ $b(x, y):=$ inf $\sup$ $k\in Bx)_{z\in-G(x,y)}\mathrm{f}$ $h(k, x, z)$ . It is easy to check that if and only if $a(x, y)>0$ by using the compactness of condition (d), and then $a(x, x)\leq Denote , and also and $b(x, x)\geq $\overline{F(x,y)}$ 0$ $F(y, x)\subset \mathrm{i}\mathrm{n}\mathrm{t}C(y)$ $<0$ $b(y, x)$ 0$ if $G(y, x)\subset$ int $C(y)$ by using . $S_{y}:=\{x\in D : F(x, y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(x)\}$ (4.1) . be a finite is closed. Let is lower semicontinuous (by Lemma 3.1), the set . Obviously $Z$ is compact and convex. subset of $K$ . Denote by $Z$ the closed convex hull of 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 a point $z\in Z$ such that $a(y, z)\leq 0$ for every $y\in Z$ , which means (4.2) for every $y\in Z$ . Since $a(y, \cdot)$ $Y_{0}$ $S_{y}$ $Y_{0}\cup D$ $F(z, y)\not\subset \mathrm{i}\mathrm{n}\mathrm{t}C(z)$ 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 $D$ is compact, $\cap\{S_{y} : y\in K\}\neq\emptyset$ , which means that there exists $x_{0}\in K$ such that int $C(x_{0})$ for every $y\in K$ . So we proved that is nonempty, and since is a closed subset of $D$ , the proof is completed. $\emptyset$ $F(x_{0}, y)\not\subset$ $S$ $S$ I Theorem 4.2. Let $K$ be a nonempty convex subset of a topological vector space logical vector space, and $F:K\cross Karrow 2^{Y}$ a multifunction. Assume that (i) $C$ $Y$ : $Karrow 2^{Y}$ is a with int multifunction $C(x)\neq\emptyset$ ; such that for every $x\in K,$ $C(x)$ $E,$ $Y$ a topo- is a closed convex cone in 152 is a multifunction defined as the graph of $W$ is closed in $K\cross Y$ ; (ii) $W$ (iii) for : $Karrow 2^{Y}$ $W(x)=Y\backslash \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ , for every $x\in K$ such that every $y\in K,$ is $C(x)$ -lower semicontinuous with closed values on $K$ and if , for every $y\in K$ , maps the the mapping is not constant-valued, then the mapping $K$ compact subsets of into precompact subsets of ; $F(\cdot, y)$ $x,$ $C$ $F(\cdot, y)$ $Y$ multifunction $G:K\cross Karrow 2^{Y}$ (iv) there exists a (a) (b) (c) for every $x\in K,$ $G(x, x)\cap$ int for every $y\in K,$ $F(x, y)\cap$ int $C(x)=\emptyset$ $x,$ $G(x, \cdot)$ is on $C(x)$ -quasiconcave such that , implies $C(x)\neq\emptyset$ $K$ $G(x, y)\cap$ int $C(x)\neq\emptyset$ , for every $x\in K$ ; (v) there exists a nonempty compact convex subset . exists $y\in D$ with $D$ of such ihat for every $K$ , there $x\in K\backslash D$ $F(x, y)\cap \mathrm{i}\mathrm{n}\mathrm{t}C(x)\neq\emptyset$ Then, the solutions set $S=$ { $x\in K$ : is a nonempty and compact subset $F(x,$ $y)\cap(\mathrm{i}\mathrm{n}\mathrm{t}C(x))=\emptyset$ , for all $y\in K$ } of . $D$ Proof. Put $a(x, y):=- \inf_{k\in By)}\inf_{z\in-Fy,x)}h(k, y, z)$ , $b(x, y):= \inf_{z\in-Gx,y)}h(k(x), x, z)$ 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 $a(x, x)\leq 0$ , $xrightarrow \mathrm{i}\mathrm{n}\mathrm{t}C(x)$ $F(y, x)\cap(\mathrm{i}\mathrm{n}\mathrm{t}C(y))\neq\emptyset$ $G(y, x)\cap(\mathrm{i}\mathrm{n}\mathrm{t}C(y))\neq\emptyset$ $b(x, x)\geq 0$ , , i.e., $k(x)\in$ , , . Lemmas 3.2, 3.6 and condition (iv) $(b)$ 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 $F:K\cross Karrow 2^{Y}$ a multifunction. Assume that (i) is a multifunction with a closed graph and a compact base $B(x)=(2\overline{B_{Y}}\backslash B_{Y})\cap C(x)$ for every ; $C:Karrow 2^{Y}$ ’ $C(x)$ $E,$ $Y$ a Banach is a closed convex cone with $x$ (ii) for every $y\in K,$ (iii) there xists a $\mathrm{e}$ (a) (b) (c) for for $F(\cdot, y)$ is $(-C)^{c}$ -upper semicontinuous and locally bounded; multifunction $G:K\cross Karrow 2^{Y}$ every $x\in K,$ $G(x, x)\cap(-C(x))\neq\emptyset$ every $x,$ $G(x, \cdot)$ is such that , $y\in K,$ $F(x, y)\cap(-C(x))=\emptyset$ $type-(\mathrm{v})C(x)$ implies -properly quasiconcave on (iv) there exists a nonempty compact convex subset exists $y\in D$ with $F(x, y)\cap(-C(x))=\emptyset$ . $D$ of $K$ $G(x, y)\cap(-C(x))=\emptyset$ $K$ for , every $x\in K_{f}$. such that for every $x\in K\backslash D$ , there 153 Then, the solutions set { $x\in K$ : $F(x,$ $S=$ is a nonempty and compact subset of $D$ $y)\cap(-C(x))\neq\emptyset$ , for all $y\in K$ } . Proof. Put inf $a(x, y):=$ inf $h(k, y, z)$ , inf $b(x, y):=-$ $k\in B(y)z\in F(y,x)$ $h(k, x, z)$ . inf $k\in B(x)z\in G(x,y)$ It is easy to check that if and only if $F(y, x)\cap(-C(y))\neq\emptyset$ , $a(x, y)\leq 0$ $b(y, x)\geq 0$ if and only if $a(x, x)\leq 0$ $G(y, x)\cap(-C(y))\neq\emptyset$ 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 is lower semicontinuous (by Lemma 3.3), the set is closed. Let be a finite subset of $K$ . Denote by $Z$ the intersection of $K$ and the linear hull of $Y\cup D$ . Obviously is compact and convex. Now we apply Theorem 2.3 $z\in Z$ and obtain a point such that $a(y, \cdot)$ $Y$ $S_{y}$ $Z$ for every $a(y, z)\leq 0$ (4.3) $y\in Z$ which means for every $F(z, y)\cap(-C(x))\neq\emptyset$ $y\in Z$ . (4.4) Assumption (iv) and condition (4.4) imply that $z\in D$ , and condition (4.4) implies also . So the family $\{S_{y} : y\in K\}$ has finite intersection property. Since $D$ is , which completes the proof. $\cap\{S_{y}$ : $y\in Y\}\neq\emptyset$ $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}$ $\cap\{S_{y} : y\in K\}\neq\emptyset$ Theorem 4.4. Let $K$ be a nonempty convex subset of a topological vector space space, and $F:K\cross Karrow 2^{Y}$ a multifunction. Assume that (i) is a multifunction with a closed graph such that with a cornpact base $B(x)=(2\overline{B}_{Y}\backslash B_{Y})\cap C(x)$ for every $C:Karrow 2^{Y}$ $E,$ $Y$ a Banach is a closed convex cone $C(x)$ $x_{i}$ (ii) for every $x,$ $y\in K,$ (iii) there exists a (a) (b) (c) $F(\cdot, y)$ is $C(x)$ -lower semicontinuous and locally bounded; rnultifunction $G:K\cross Karrow 2^{Y}$ for every $x\in K,$ $G(x, x)\subset-C(x)$ , for every $y\in K,$ $F(x, y)\not\subset-C(x)$ $x,$ $G(x, \cdot)$ is $type-(\mathrm{i}\mathrm{i}\mathrm{i})C(x)$ such that irnplies $G(x, y)\not\subset-C(x)$ -properly quasiconcave on (iv) there exists a nonernpty compact convex subset exists $y\in D$ with $F(x, y)\not\subset-C(x)$ . $D$ of $K$ $K$ for every { $x\in is a nonernpty and cornpact subset K:F(x,$ of . $D$ $y)\subset-C(x)$ $x\in K$ ; such that for every Then, the solutions set $S=$ , , for all $y\in K$ } $x\in K\backslash D$ , ihere 154 Proof. Put $a(x, y):= \inf_{k\in B(y)}\sup_{z\in F(y,x)}h(k, y, z)$ It is $\mathrm{e}a\mathrm{s}\mathrm{y}$ , $b(x, y)$ $:=- \mathrm{i}\mathrm{r}\mathrm{l}\mathrm{f}\sup_{z\in G(x,y)}h(k, x, z)k\in B(x)$ . to check that $a(x, y)\leq 0$ if and only if $b(y, x)\geq 0$ if and only if $G(y, x)\subset-C(y)$ , $a(x, x)\leq 0$ and $F(y, x)\subset-C(y)$ , $b(x, x)\geq 0$ . Lemrnas 3.4, 3.8 and condition (iii) (b) show that the conditions of Theorem 2.3 are satisfied. Further the proof is the sarne as that of Theorern 4.3, but in this case $S_{y}:=\{x\in D$ : $F(x, y)\subset$ 1 $-C(x)\}$ . 5. 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) two-function type instead of single function type; (iii) parametric ordering structure instead of fixed ordering structure; (iv) cornplete extensions including the result of [1]. As a corollary from any of Theorems 4.1-4.4, we obtain that Theoreni 2.3 irnplies the scalar Fan inequality. References $,[1]$ Q. H. Ansari and J.-C. Yao (1999). An $exister\iota ce$ result for the generalized vector equilibrium problem, Appl. Math. Lett., 12, 53 56. [2] J.-P. Aubin and I. Ekeland (1984). Applied Nonlinear Analysis, Wiley Interscience, New York. [3] J.-P.Aubin and H. Frankowska (1990). Set- Valued Analysis, Birkh\"auser, Boston. [4] P. G. Georgiev and T. Tanaka (2000). Vector-valued set-valued variants ity, J. Nonlinear and Convex Analysis, 1 (3), 245-254. of Ky Fan’s inequal- [5] Y. Kimura, K. Tanaka, and T. Tanaka (1999). On semicontinuity of set-valued maps and marginal functions, 181 188 in Nonlinear Anlysis and Convex Analysis –Proceedings of the International Conference (W. Takahashi and T. Tanaka, eds.), World Scientific, Singapore. [6] D. Kuroiwa, T. Tanaka, and T.X.D. Ha (1997). 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