Kim, Won Kyu

Kum, Sangho

Kyoto University Research Information Repository

TitleAn Extension of Vector Variational Inequalities With OperatorSolutions (Nonlinear Analysis and Convex Analysis)Author(s)Kum, Sangho; Kim, Won KyuCitation数理解析研究所講究録 (2004), 1365: 146-155Issue Date2004-04URL http://hdl.handle.net/2433/25347RightType Departmental Bulletin PaperTextversionpublisherKyoto University146An Extension of Vector Variational InequalitiesWith Operator SolutionsSangho Kum and Won Kyu KimDepartment of Mathematics Education, Chungbuk National University, Cheongju 361-763, Korea.Abstract. In a recent paper, Domokos and Kolumban introduced variational inequal-ities with operator solutions to provide a suitable unified approach to several kinds ofvariational inequality and vector variational inequality in Banach spaces. Inspired bytheir work, we further develop the new scheme of vector variational inequalities withoperator solutions from the single- alued case into the multi-valued one. We prove theexistence of solutions of generalized vector variational inequalities with operator solu-tions. Some applications to generalized vector variational inequalities in a normed spaceare also provided.Key Words. Vector variational inequality, $C$-pseudomonotone operator, generalizedhemicontinuity, Fan-Browder fixed point theorem1. IntroductionSince Giannessi [5] introduced the vector variational inequality, (shortly, $\mathrm{W}\mathrm{I}$) in afinite dimensional Euclidean space, many authors have intensively studied (WI) andits various extensions [1, 6, 7, 10] (see also the references therein) in abstract spaces.1365 2004 146-155147Several authors have investigated relationships between (VVI) and vector optimizationproblems, vector complementarity problem $[2, 8]$ .In a recent paper, Domokos and Kolumban [3] gave an interesting interpretation ofvariational inequalities (VI) and (VVI) in Banach space settings in terms of variationalinequalities with operator solutions (in short, OWI). They first obtained an existencetheorem of the solutions of (OWI) using Fan’s KKM Lemma [4], and then presented ageneral version of Yu and Yao [10, Theorem 3.3] in a Banach space as a main applicationand gave some other applications such as the solvability of variational inequality definedon Hausdorff topological vector space, and that of variational inequality on $L^{\infty}(\Omega)$ .However, they dealt with only the single-valued operator.Inspired by their work, in this report, we further develop the new scheme of (OWI)ffom the single- alued case into the multi-valued one, and search some applications, froma theoretical point of view, to exploit the framework of (OWI). To be more specific,we establish a multi-valued version of (OW I) called the generalized vector variationalinequality with operator solutions (in short, GOW $\mathrm{I}$). As an application of (GOW $\mathrm{I}$), weprovide a noncompact generalization of Konnov and Yao [6, Theorem 3.1] concerninga generalized (VVI) in a normed space (not necessarily Banach space). In addition,we deal with an existence theorem on (WI) concerned with upper semicontinuity ofmultifunction instead of pseudO-monotonicity. As basic tools to obtain main results, weuse a Fan-Browder type fixed point theorem due to Park [9, Theorem 5].vtheoretiII) I)mgenerali nor2. PreliminariesLet $E$ , $F$ be Hausdorff topological vector spaces, and let $X$ be a nonempty convexsubset of $E$ . A nonempty subset $P$ of $E$ is called a convex cone if$\lambda P\subset P,$ for all $A$ $>0$ and $P+P=P.$148Let $C_{1}$ : $X\supset F$ be a multifunction such that for each $x\in X$ , $C_{1}(x)$ is a convex cone in$F$ with int $C_{1}(x)\neq\emptyset$ and $C_{1}(x)\neq F$ . Let $L$(E, $F$) be the space of all continuous linearoperators ffom $E$ to $F$ and $T_{1}$ : $X$ = $L(E, F)$ a multifunction.Then $T_{1}$ is said to be(1) $C_{1}$ -psettdomonotone if for any $x,y\in X$ and for any $s\in$ Ti (x), we have$\langle s, y-x\rangle$ $\not\in-\mathrm{i}\mathrm{n}\mathrm{t}C_{1}$ $(x)$ implies $(t, t -x\rangle\not\in-\mathrm{i}\mathrm{n}\mathrm{t}C_{1}(x)$ for all $t\in T_{1}(y)$ ; and(2) generalized hemicontinuous if for any $x$ , $y\in X$ and $\alpha\in[0,1]$ , the multifunction$\alpha\mapsto\langle T_{1}(x+\alpha(y-x)), y-x\rangle$is upper semicontinuous at $0^{+}$ , where$\langle 7_{1}(x+\alpha(y-x))\mathrm{t}y-x\rangle=\{\{s, y-x\rangle|s\in Tz(x +\alpha(y- x))\}$ .Now we pay our attention to generalized variational inequalities with operator solu-tions (in short, GOW $\mathrm{I}$). Prom now on, unless otherwise specified, we work under thefollowing settings:Let $X’$ be $\backslash \mathrm{a}$ nonempty convex subset of $L(E, F)$ and $T:X’\supset E$ be a multifunction.Let $C$ : $X’\supset F$ be a multifunction such that for each $f\in X’$ , $C(f)$ is a convex conein $F$ with 0 ( $\mathrm{C}(\mathrm{f})$ . Then (GOW I) is defined as follows:Find $f_{0}\in X’$ such that $\forall f\in X’,$ $\exists x\in$ T(/0) with $\langle f-f_{0},x\rangle\not\in$ C(f). (GOWI)Then $T$ is single valued, (GO$\mathrm{V}\mathrm{V}\mathrm{I}$) reduces to (OWI) due to Domokos and Kolumb&n[3]. As pointed out in [3], the notation (GOW I) is motivated by the fact that thesolutions are sought in the space of continuous linear operators.149In regard to monotonicity and continuity of $T$, two analogous definitions to those of $T_{1}$in the above are necessary; $T$ : $X’3$ $E$ is said to be(1)’ $C$-pseudomonotone if for any $f,g\in X’$ and for any $s\in T(f)$ , we have$\langle g-f, s\rangle\not\in C(f)$ implies $\langle g-f, t\rangle\not\in C(f)$ for all $t\in T(g)$ ; and(2)’ generalized hemicontinuous if for any $f$, $g\in X’$ and $\alpha\in[0,1]$ , the multifunction$\alpha\mapsto\langle g-f, T(f+\alpha(g-f))\rangle$is upper semicontinuous at $0^{+}$ , where$\langle g-f, T(f+\alpha(g-f))\rangle$ $=$ $\{\langle g-f, s\rangle|s\in \mathrm{j} (f+\alpha(g-f))\}$ .In order to prove our main result, we need the following fixed point theorem whichis a particular form of Park [9, Theorem 5].Lemma 2.1. Let $X$ be a nonempty convex subset of a real (not necessarily) Hausdorfftopological vector space $E$ , $K$ a nonempty compact subset of $X$ . Let $A$ , $B$ : $X$ = $X$be two multifunction. Suppose that(i) for each $x\in X$, $Ax\subset Bx;$(ii) for each $x\in X$ , $Bx$ is convex;(iii) for each $x\in K$, $Ax$ is nonempty ;(iv) for each $y\in X,$ $A^{-1}y=$ {$x\in X\cdot|y\in$ Ax} is open in $X$;and(v) for each finite subset $N$ of $X$ , there exists a nonempty compact convex subset $L_{N}$of $X$ containing $N$ such that for each $x\in L_{N}\backslash K$, $Ax\cap L_{N}\neq\emptyset$.Then $B$ has a fixed point $\mathrm{x}\mathrm{o}$ ; that is, $x_{0}\in$ Bxq.1503. Generalized vector variational inequality with operator soultionsWe begin with the following lemma to get the main result.Lemma 3.1. Let $T:X’=3$ $E$ be a $C$-pseudomonotone and generalized hemicontinuousmultifunction with $T(f)\neq\emptyset$ for all $f\in X’$ . Let $W$ : $X’3$ $F$ be defined by $W(f)=$$F\backslash C(f)$ such that the graph $Gr(W)$ of $W$ is closed in $X’ \mathrm{x}F$ where $L(E, F)$ isendowed with the topology of pointwise convergence. Then the following two problemsare equivalent:(i) Find $f\in X’$ such that $lg$ $\in X’$ , $\exists x\in T(f)$ with $\langle g-f, x\rangle\not\in$ C(f).(ii) Find $f\in X’$ such that $\forall g\in X’$ , $\forall x\in T(g)$ , $\langle g-f,x\rangle\not\in C(f)$ .Using Lemma 3.1, we prove the following which is a multi-valued version of (OWI) in[3]Theorem 3.1. Let $T$ : $X’\supset E$ be a $C$-pseudomonotone and generalized hemicon-tinuous multifunction with $T(f)\neq\emptyset$ for all $f\in X’$ . Let $W$ : $X’\supset F$ be defined by$W(f)=F\backslash C(f)$ such that the graph $Gr(W)$ of $W$ is closed in $X’\mathrm{x}$ $F$ where $L(E, F)$is endowed with the topology of pointwise convergence. Let $K’$ be a nonempty compactsubset of $X’$ . Assume that for each finite subset $N’$ of $X’$ , there exists a nonemptycompact convex subset $L_{N’}$ of $X’$ containing $N’$ such that for each $f\in L_{N’}\backslash K’$ , thereexists $g\in L_{N’}$ satisfying$\langle g-f, x\rangle\in C(f)$ for some $x\in T(g)$ .Then (GOVVI) is solvable.151Proof. First note that $L(E, F)$ equipped with the topology of pointwise convergenceis a Hausdorff t.v.s. We define two multifunctions $A$ , $B$ : $X’\supset X’$ to be$A(f)$ $:=$ {$g\in X’$ $|\exists x\in T(g)$ such that $\langle g-f$, $x\rangle\in C(f)$ },$B(f)$ $:=$ $\{g\in X’|\forall x\in T(f), \langle g-f, x\rangle\in C(f)\}$ .The proof is organized in the following parts.(i) Since $T$ is $C$-pseudomonotone, we have $\mathrm{B}(\mathrm{f})\subset B(f)$ for all $f\in X’$ .(ii) For each $f\in X’$ , $B(f)$ is convex. Indeed, let $g_{1}$ and $g_{2}$ be in $B(f)$ . For all $t\in[0,1]$and $x\in T(f)$ : we have$\langle tg_{1}+ (1-t)02 -f, x\rangle$ $=t\langle g_{1}-f, x\rangle+$ $($1- $t)02-f$, $x\rangle$ $\in C(f)$ ,which implies that $tg_{1}+(1-t)g_{2}\in B(f)$ . Hence $B(f)$ is convex.(iii) Clearly $B$ has no fixed point because 0 ( $C(f)$ for all $f\in X’$ .(iv) For each $g\in X’$ , $A^{-1}(g)$ is open in $X’$ . In fact, let $\{f_{\lambda}\}$ be a net in $(A^{-1}(g))\mathrm{c}$convergent to $f\in X’$ . Then $g\not\in A(f_{\lambda})$ and hence for each $c$ $\in T(g)$ ,$\langle g-f_{\lambda}, x\rangle\not\in C(f_{\lambda})$ .Thus $\langle g-f_{\lambda}, x\rangle$ $\in W(f_{\lambda})$ . Since ( $f_{\lambda}$ , $(g-f_{\lambda},x\rangle)\in Gr(W)$ and $L(E, F)$ is endowed withthe topology of pointwise convergence, by virtue of the closedness of $Gr(W)$ , we have$(f, (g-f,x))\in$ Gr(W), that is, $\langle g .-f, x\rangle$ $\not\in$ C(/) for every $x\in T(g)$ . Hence $g\not\in A(f)$ ,so $f\in(A^{-1}(g))^{\mathrm{c}}$ . This shows that $(A^{-1}(g))^{\mathrm{c}}$ is closed, therefore $A^{-1}(g)$ is open in $X’$ .(v) By the given hypothesis, we know that for each finite subset $N’$ of $X’$ , there existsa nonempty compact convex subset $L_{N’}$ of $X’$ containing $N’$ such that for each $f\in$$L_{N’}\mathrm{S}$ $K’$ , there exists $g\in L_{N’}$ satisfying $g\in A(f)$ , hence $L_{N’}$ rl $A(f)$ $\neq\emptyset$ .152(vi) Prom $(\mathrm{i})-(\mathrm{v})$ , we see, by Lemma 2.1, there must be an $f_{0}\in K’$ such that $A(f_{0})=\emptyset$ ,namely,$\langle g-\mathrm{j}\mathrm{o}, x\rangle$ $\not\in$ C(fo) for any $7\in X’$ , $x\in T(g)$ .It follows from Lemma 3.1 that $f_{0}$ is a solution of (GOVVI). This completes the proof.As an application ofTheorem 3.1 in multi-valued settings, we can obtain the existenceof a solution of a generalized (WI) in a normed space :Theorem 3.2, Let $\mathrm{Y}$ and $Z$ be two normed spaces. Let $X$ be a nonempty convexsubset of $\mathrm{Y}$ and $C_{1}$ : $X\supset Z$ be a multifunction such that for each $x\in X$, $C_{1}(x)$ is aconvex cone in $Z$ with $\mathrm{i}\mathrm{n}\mathrm{t}C_{1}(x)\neq\emptyset$ and $C_{1}(x)\neq Z.$ Let $T_{1}$ : $X\supset$ L(y, $Z$) be a $C_{1^{-}}$pseudomonotone and generalized hemicontinuous multifunction with nonempty values.Let $1_{1}$ : $X\supset Z$ be defined by $W_{1}(x)=Z\backslash -\mathrm{i}\mathrm{n}\mathrm{t}C_{1}(\mathrm{x})$ such that the graph Gr(Wi) of$W_{1}$ is weakly closed in $X\cross Z.$ Assume that $K$ is a nonempty weakly compact subset of$X$ and for each finite subset $N$ of $X$ , there exists a nonempty weakly compact convexsubset $L_{N}$ of $X$ containing $N$ such that for each $x\in L_{N}\backslash K,$ there exists $y\in L_{N}$satisfying$\langle s, y-x\rangle$ $\in-\mathrm{i}\mathrm{n}\mathrm{t}C_{1}(x)$ for some $s\in T_{1}(y)$ .Then there exists $x_{0}^{!}\in X$ such that$\mathrm{I}x$ $\in X$, $\exists t\in T_{1}(x_{0})$ with $(t, x-x_{0}\rangle\not\in-\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{C}17_{1} (x_{0})$ .Remark 3,1. Theorem 3.2 is a noncompact generalization of Konnov and Yap [6,Theorem 3.1] in normed spaces (not necessarily $\mathrm{B}\mathrm{a}\mathrm{n}.\underline{\mathrm{f}\mathrm{l}}\mathrm{A}$ spaces) without assuming theconvex cone $C_{1}(x)$ being closed.153Now we are interested in (GOVVI) concerned with the upper semicontinuity of $T$instead of pseudomonotonicity and hemicontinuity. To this end, we replace the topologyof pointwise convergence by that of bounded convergence on $L(E, F)$ .Theorem 3.3. Suppose that $L(E, F)$ is endowed with the topology of bounded con-vergence. Let $T:X’\supset E$ be an upper semicontinuous multifunction such that $T(f)$ isa nonempty compact subset of $E$ for all $f\in X’$ , and the range $\mathrm{T}(\mathrm{X}’)$ is contained in acompact subset of $E$ . Let $W$ : $X’\supset$ $F$ be defined by $W(f)=F\backslash C(f)$ such that thegraph $Gr(W)$ of $W$ is closed in $X’\mathrm{x}F$ . Let $K’$ be a nonempty compact subset of $X’$ .Assume that for each finite subset $N’$ of $X’$ , there exists a nonempty compact convexsubset $L_{N’}$ of $X’$ containing $N’$ such that for each $f\in L_{N’}\mathrm{s}$ $K’$ , there exists $g\in L_{N’}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\phi \mathrm{i}\mathrm{n}\mathrm{g}$$\langle g-f, x\rangle\in C(f)$ for all $x\in T(f)$ .Then (GOVVI) is solvable.As a direct consequence of Theorem 3.3, we obtain the following.Theorem 3.4. Let $\mathrm{Y}$ and $Z$ be two normed spaces. Let $X$ be a nonempty convexsubset of $\mathrm{Y}$ and $C_{1}$ : $X\supset$ $Z$ be a multifunction such that for each $x\in X$, $C_{1}(x)$ is aconvex cone in $Z$ with intCi $(\mathrm{x})\neq\emptyset$ and $C_{1}(x)\neq Z.$ Let $T_{1}$ : $X\supset$ $\mathrm{L}(\mathrm{E}, Z)$ be an uppersemicontinuous multifunction with nonempty compact values and the range $T_{1}(X)$ becontained in a compact subset of $L(\mathrm{Y}, Z)$ where $L(\mathrm{Y}, Z)$ is the normed space of thecontinuous linear operators between $\mathrm{Y}$ and $Z$ with the usual norm. Let $\mathrm{S}_{1}$ : $X\supset$ $Z$be defined by $\mathrm{W}(\mathrm{f})=Z\backslash -\mathrm{i}\mathrm{n}\mathrm{t}C_{1}$ $(x)$ such that the graph Gr(W) of $W_{1}$ is closed in$X\mathrm{x}Z$ . Assume that $K$ is a nonempty compact subset of $X$ and for each finite subset154$N$ of $X$ , there exists a nonempty compact convex subset $L_{N}$ of $X$ containing $N$ suchthat for each $x\in L_{N}\backslash K,$ there exists $y\in L_{N}$ satisfying$\langle s, y-x\rangle\in-\mathrm{i}\mathrm{n}\mathrm{t}C_{1}$ $(x)$ for all $s\in \mathrm{r}1$ $(\mathrm{x})$Then there exists $x_{0}\in X$ such that$lx$ $\in X$, $\exists t\in T_{1}(x_{0})$ with $\langle t, x-x_{0}\rangle\not\in-\mathrm{i}\mathrm{n}\mathrm{t}C_{1}$ $(x_{0})$ .AcknowledgementThis work was supported by Korea Research Foundation Grant KRF-2002-041-C00037.References1. Chen, G. Y. (1992), Existence of solutions for a vector variational inequality: anextension of Hartman-Stampacchia theorem, Journal of Optimization Theory andApplications 74, 445-456.2. Chen, G. Y. and Yang, X. Q. (1990), The vector complementarity problem and itsequivalences with weak minimal elements in ordered spaces, Journal ofMathematicalAnalysis and Applications 153, 136-158.3. Domokos, A. and Kolumb&n,$\sim$J. (2002), Variational inequalities with operator solu-Lions, Journal of Global Optimization 23, 99-110.4. Fan, K. (1961), A generalization of Tychonoff’s fixed-point theorem, MathematischeAnnalen 142, 305310.5. Giannessi, F. (1980), Therorems of alternative, quadratic programs and complemen-tarity problems, in Cottle, R. W., Giannessi F. and Lions, J. L. $(\mathrm{e}\mathrm{d}\mathrm{s}.)$ , Vaniational$\sim \mathrm{J}$mm generali-3ri155Inequalities and Complementarity Problems, John Wiley and Sons, Chichester, Eng-land, 151-186.6. Konnov, I. V. and Yao, J. C. (1997), On the generalized vector variational inequalityproblem, Journal of Mathematical Analysis and Applications 206, 42-58.7. Lai, T. C. and Yao, J. C. (1996), Existence results for VVIP, Applied MathematicsLetters 9, 17-19.8. Lee, G. M., Kim, D. S., Lee, B. S. and Yen, N. D. (1998), Vector variational inequalityas a tool for studying vector optimization problems, Nonlinear Analysis. 34, 745-765.9. Park, S. (1994), Foundations of the KKM theory via coincidences of composites ofupper semicontinuous maps, Jour$mal$ of the Korean Mathematical Society 31, 493519.10. Yu, S. J. and Yao, J. C. (1996), On vector variational inequalities, Journal of Opti-mization Theory and Applications 89, 749-769.mtrn l -5

An Extension of Vector Variational Inequalities With Operator Solutions (Nonlinear Analysis and Convex Analysis)

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An Extension of Vector Variational Inequalities With Operator Solutions (Nonlinear Analysis and Convex Analysis)

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