On the Connectedness of the Solution Set for the Weak Vector Variational Inequality 1

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1 Journal of Mathematcal Analyss and Alcatons 260, do: jmaa , avalable onlne at htt:.dealbrary.com on On the Connectedness of the Soluton Set for the Weak Vector Varatonal Inequalty 1 Yonghong Cheng 2 Insttute of Systems Scence, Academa Snca, Bejng , Peole s Reublc of Chna Submtted by Bors S. Mordukhoch Receved December 8, 1999; ublshed onlne June 22, 2001 The man am of ths aer s to rovde a suffcent condton hch guarantees the connected roerty of the soluton set for a knd of eak vector varatonal nequalty Academc Press 1. INTRODUCTION The study of vector varatonal nequaltes has been of great nterest snce the ath-breakng aer by Ganness n Over the ast to decades, varous vector varatonal nequaltes and ther alcatons have been ntensvely studed by Chen 2, Chen and Yang 3, Lee et al. 6, 7, and others. A large body of lterature has been devoted to the exstence of solutons for vector varatonal nequaltes. But so far, there are fe aers hch deal th the toologcal roertes of the soluton set for a vector varatonal nequalty. The urose of ths aer s to dscuss the connectedness roerty of the soluton set for a eak vector varatonal nequalty. The study of ths toc s manly motvated by the fact that t ll be helful for develong algorthms for solvng the vector varatonal nequalty. The organzaton of ths aer s as follos. In Secton 2, e ntroduce the Ž eak. vector varatonal nequalty and establsh the equvalence of the eak vector varatonal nequalty and a famly of scalar varatonal 1 Ths research as artally suorted by the Natonal Natural Scence Foundaton of Chna. 2 Current address: Deartment of Management Scence, School of Management, Fudan Unversty, Shangha 20033, Peole s Reublc of Chna X01 $35.00 Coyrght 2001 by Academc Press All rghts of reroducton n any form reserved.

2 2 YONGHONG CHENG nequaltes; some notatons are also resented n Secton 2. The man result about the connectedness of the soluton set for the eak vector varatonal nequalty s establshed n Secton WEAK VECTOR VARIATIONAL INEQUALITY Let X be a nonemty closed convex subset of R n and let f : X R n Ž 1, 2,...,. be vector-valued functons. We defne f Ž f,..., f. 1 as follos: for every x X and y R n, ž 1 / ² : ² : f x y f x, y,..., f x, y, here Ž x, y. denotes the nner roduct of the vectors x and y n the Eucldean sace. We ll denote the nonnegatve orthant and ostve orthant of R by P R and R, resectvely;.e., and R y R : y 0, 1,..., R y R : y 0, 1,...,. The vector varatonal nequalty s the follong roblem: Fnd x X, fž x.ž x x. R 0, x X. Ž VVI. The eak vector varatonal nequalty s the follong roblem: Fnd x X, fž x.ž x x. R, x X. Ž WVVI. In general, the Ž eak. vector varatonal nequalty roblem can have more than one soluton. We denote the soluton sets of Ž VVI. and Ž WVVI. by SŽ f, X. and S Ž f, X., resectvely. The study of toologcal roertes of the soluton set S Ž f, X. s of nterest n ths aer. We ll dscuss the connectedness of the soluton set S Ž f, X.. Take R 0. We consder the scalar varatonal nequalty, fnd x X, ² FŽ x,., x x: 0 x X, Ž VI. 1 here F x, Ý f x. We denote by IŽ. the soluton set of Ž VI.. Then e have the follong rooston. PROPOSITION 1. S Ž f, X. IŽ.. R0

3 SOLUTION SET CONNECTEDNESS 3 Proof. x S f, X yelds that fž x.ž x x. : x X R. Due to the convexty of X, t s clear that the set fž x.ž x x.: x X s convex. Then by the searaton theorem for convex sets there exsts R 0 such that ² : nf, fž x.ž x x. su ², y :. x X yr Ths mles that R 0. Then the rght-hand sde of the above nequalty s 0 and e have ², f Ž x.ž x x.: 0. Hence, ² F Ž x,., x x : 0 for all x X. Ths amounts to sayng that x IŽ.. Conversely, assume that x I here R 0. Then Ý ; Ý f Ž x., x x ² f Ž x., x x: ², fž x.ž x x. : mles that f x x x R. Ths comletes the roof. 3. CONNECTEDNESS OF S f, X In ths secton, e dscuss the connectedness roerty of the soluton set S Ž f, X.. We need the follong defntons. DEFINITION 1. A vector-valued functon g: R n R n s sad to be strctly seudomonotone over a closed convex subset X R n ff for any x, y X, x y, ² gž y., x y: 0 mles ² gž x., x y: 0. DEFINITION 2. A set-valued mang G from a set X to a set Y s sad to be uer semcontnuous at x X for any neghborhood W of GŽ x. there exsts a neghborhood N of x such that GŽ x. W for all x N. LEMMA 1. Assume that f Ž 1,...,. are strctly seudomonotone and contnuous and X s conex and comact. Then for eery R 0,I Ž. s nonemty and conex. Proof. Take arbtrary R 0. Frst, the asserton that IŽ. s nonemty holds snce X s comact convex and f s contnuous by the contnuty of f Ž 1,...,.. Second, e rove that IŽ. s convex. For ths, e only need to sho that ² : I L x X : F x,, x x 0, x X because t s obvous that L s convex.

4 YONGHONG CHENG x I yelds that, for each x X, 0 ² Ž 0. 0: Ý ² Ž 0. 0: F x,, x x f x, x x 0. Ths mles that, for each x X, 1 ² 0: Ý ² 0: F x,, x x f x, x x 0. 1 If not, there ould exst x X such that Ý ² f Ž x., x x : Then, n ve of R 0, e have ² : f x, x x 0, 1,..., and x x. Thus, by the strct seudomonotonty of f, e deduce that ² : f x, x x 0. Hence, there exsts x X such that Ý ² f Ž x., x x : 1 0, a contra- dcton. So, t follos that x LŽ. 0. On the other sde, e assume that ² FŽ y,., y x0: 0, y X. The assumton that X s convex guarantees that ty Ž 1 t. x0 X for every t Ž 0, 1.. So, or ² Ž. : F ty 1 t x,, ty 1 t x x 0 0 ² Ž. : F ty 1 t x,, y x 0. Snce f Ž 1,...,. are contnuous, F s contnuous. Lettng t 0, e deduce that ² : F x,, y x 0. Hence, x IŽ.. Thus, e have checked that IŽ. LŽ.. Ths comletes the roof of Lemma 1. LEMMA 2. Let X be comact and conex, and let f Ž 1,...,. be contnuous. Then the soluton mang I s nonemty, comact-alued, and uer semcontnuous for eery R 0.

5 SOLUTION SET CONNECTEDNESS 5 Proof. Take arbtrary R 0. It holds that IŽ. s nonemty from Lemma 1. Next, e rove that the soluton mang I s closed. Then takng nto account the comactness of X, e obtan that I s comact valued and uer semcontnuous. For ths am, e take x IŽ. th x x and. x IŽ. n n n n n n mles that xn X and ² : F x,, y x 0, y X. n n n Then x X, and from the contnuty of F e get ² FŽ x,., y x : ² FŽ x,., y x :. n n n Hence, ² FŽ x,., y x: 0, y X. Ths amounts to sayng that x IŽ.. The roof s comleted. No, e ll resent the man result about the connectedness of the soluton set S Ž f, X. for Ž WVVI.. THEOREM 1. Assume that X s comact and conex n Rn and f Ž 1,...,. are contnuous and strctly seudomonotone. Then the soluton set S Ž f, X. for Ž WVVI. s connected. Proof. From Prooston 1, e kno that S Ž f, X. IŽ.. R 0 Then the asserton holds mmedately from Lemma 1 and Lemma 2. REFERENCES 1. J. P. Aubn and I. Ekeland, Aled Nonlnear Analyss, Wley, Ne York, G. Y. Chen, Exstence of solutons for a vector varatonal nequalty: An extenson of the HartmanStamaccha theorem, J. Otm. Theory Al. 7 Ž 1992., G. Y. Chen and X. Q. Yang, The comlementarty roblems and ther equvalence th the eak mnmal element n ordered saces, J. Math. Anal. Al. 153 Ž 1990., F. Ganness, Theorems of alternatve, quadratc rograms, and comlementarty roblem, n Varatonal Inequalty and Comlementarty Problems ŽR. W. Cottle, F. Ganness, and J.-L. Lons, Eds.., Wley, Ne York, P. T. Harker and J. S. Pang, Exstence of otmal solutons to mathematcal rograms th equlbrum constrants, Oer. Res. Lett. 7 Ž 1988., G. M. Lee, D. S. Km, and B. S. Lee, Generalzed vector varatonal nequalty, Al. Math. Lett. 9 Ž 1996., G. M. Lee, D. S. Km, B. S. Lee, and N. D. Yen, Vector varatonal nequalty as a tool for studyng vector otmzaton roblems, Nonlnear Anal. 3 Ž 1998., D. T. Luc, Theory of Vector Otmzaton, Lecture Notes n Economcs and Mathematcal Systems, Vol. 319, Srnger-Verlag, Berln, 1989.