Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces

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2011, 1(2): 261-274. doi: 10.3934/naco.2011.1.261

Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces

Ren-You Zhong ^1, and Nan-Jing Huang ^1,

1.	Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received November 2010 Revised March 2011 Published June 2011

Abstract
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The purpose of this paper is to investigate the nonemptiness and boundedness of solution set for a generalized mixed variational inequality problem with strict feasibility in reflexive Banach spaces. We introduce a concept of strict feasibility for the generalized mixed variational inequality problem which includes the existing concepts of strict feasibility introduced for variational inequalities and complementarity problems. By using a degree theory developed in Wang and Huang [28], we prove that the monotone generalized mixed variational inequality has a nonempty bounded solution set if and only if it is strictly feasible. The results presented in this paper generalize and extend some known results in [8, 23].

Keywords: generalized $f$-projection operator, topological degree, Generalized mixed variational inequality, set-valued mapping..

Mathematics Subject Classification: Primary: 49J40; Secondary: 49J5.

Citation: Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

References:

[1]	Ya. Alber, Generalized projection operators in Banach spaces: properties and applications,, in, 1 (1994), 1.
[2]	J. P. Aubin, "Optima and Equilibria,", Springer-Verlag, (1993).
[3]	C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics,, Archive Rational Mechanics Anal., 100 (1988), 149. doi: doi:10.1007/BF00282202.
[4]	M. Bianchi, N. Hadjisavvas and S. Schaible, Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities,, J. Optim. Theory Appl., 122 (2004), 1. doi: doi:10.1023/B:JOTA.0000041728.12683.89.
[5]	I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,", Kluwer, (1990).
[6]	J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions,, Math. Program., 78 (1997), 305. doi: doi:10.1007/BF02614358.
[7]	A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities,, Math. Program., 86 (1999), 433. doi: doi:10.1007/s101070050097.
[8]	F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems,", Springer-Verlag, (2003).
[9]	J. H. Fan, X. Liu and J. L. Li, Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,, Nonlinear Anal. TMA, 70 (2009), 3997. doi: doi:10.1016/j.na.2008.08.008.
[10]	Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems,, J. Optim. Theory Appl., 129 (2006), 373. doi: doi:10.1007/s10957-006-9073-0.
[11]	S. C. Fang and E. L. Peterson, Generalized variational inequalities,, J. Optim. Theory Appl., 38 (1982), 363. doi: doi:10.1007/BF00935344.
[12]	O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems,, Math. Oper. Res., 18 (1993), 128. doi: doi:10.1287/moor.18.1.128.
[13]	P.T. Harker and J. S. Pang, Finite-dimensional Variational and Nonlinear Complementarity Problems: A survey of theory, algorithms and applications,, Math. Program., 48 (1990), 161. doi: doi:10.1007/BF01582255.
[14]	Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces,, J. Math. Anal. Appl., 330 (2007), 352. doi: doi:10.1016/j.jmaa.2006.07.063.
[15]	Y. R. He, A new projection algorithm for mixed variational inequalities,, Acta Math. Sci., 27 (2007), 215.
[16]	Y. R. He, X. Z. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces,, Acta Math. Sinica (English Series), 23 (2007), 563. doi: doi:10.1007/s10114-005-0918-5.
[17]	R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system,, J. Optim. Theory Appl., 142 (2009), 493. doi: doi:10.1007/s10957-009-9510-y.
[18]	R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces,, Optim. Lett., (): 11590. doi: doi:10.1007/s11590-010-0215-9.
[19]	V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization,, J. Math. Anal. Appl., 179 (1993), 537. doi: doi:10.1006/jmaa.1993.1368.
[20]	S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps,, J. Optim. Theory Appl., 18 (1976), 445. doi: doi:10.1007/BF00932654.
[21]	B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Degree theory for generalized variational inequalities and applications,, Eur. J. Oper. Res., 193 (2009), 12. doi: doi:10.1016/j.ejor.2007.10.028.
[22]	J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications,, J. Math. Anal. Appl., 306 (2005), 55. doi: doi:10.1016/j.jmaa.2004.11.007.
[23]	F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities,, Optimization, 60 (2011), 303. doi: doi:10.1080/02331934.2010.507985.
[24]	R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).
[25]	M. Sion, On general minimax theorems,, Pacific J. Math., 8 (1958), 171.
[26]	W. Takahashi, "Nonlinear Functional Analysis,", Yokohama Publishers, (2000).
[27]	M. M. Vainberg, "Variational Methods and Method of Monotone Operators,", Wiley, (1973).
[28]	Z. B. Wang and N.J. Huang, Degree theory for a generalized set-valued variational inequality with an application in Banach spaces,, J. Global Optim., 49 (2011), 343. doi: doi:10.1007/s10898-010-9547-3.
[29]	K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application,, Bull. Aust. Math. Soc., 73 (2006), 307. doi: doi:10.1017/S0004972700038892.
[30]	H. Y. Yin, C. X. Xu and Z. X. Zhang, The $F$-complementarity problem and its equivalence with the least element problem,, Acta Math. Sinica, 44 (2001), 679.
[31]	R. Y. Zhong and N. J. Huang, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces,, J. Optim. Theory Appl., 147 (2010), 454. doi: doi:10.1007/s10957-010-9732-z.

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References:

[1]	Ya. Alber, Generalized projection operators in Banach spaces: properties and applications,, in, 1 (1994), 1.
[2]	J. P. Aubin, "Optima and Equilibria,", Springer-Verlag, (1993).
[3]	C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics,, Archive Rational Mechanics Anal., 100 (1988), 149. doi: doi:10.1007/BF00282202.
[4]	M. Bianchi, N. Hadjisavvas and S. Schaible, Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities,, J. Optim. Theory Appl., 122 (2004), 1. doi: doi:10.1023/B:JOTA.0000041728.12683.89.
[5]	I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,", Kluwer, (1990).
[6]	J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions,, Math. Program., 78 (1997), 305. doi: doi:10.1007/BF02614358.
[7]	A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities,, Math. Program., 86 (1999), 433. doi: doi:10.1007/s101070050097.
[8]	F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems,", Springer-Verlag, (2003).
[9]	J. H. Fan, X. Liu and J. L. Li, Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,, Nonlinear Anal. TMA, 70 (2009), 3997. doi: doi:10.1016/j.na.2008.08.008.
[10]	Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems,, J. Optim. Theory Appl., 129 (2006), 373. doi: doi:10.1007/s10957-006-9073-0.
[11]	S. C. Fang and E. L. Peterson, Generalized variational inequalities,, J. Optim. Theory Appl., 38 (1982), 363. doi: doi:10.1007/BF00935344.
[12]	O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems,, Math. Oper. Res., 18 (1993), 128. doi: doi:10.1287/moor.18.1.128.
[13]	P.T. Harker and J. S. Pang, Finite-dimensional Variational and Nonlinear Complementarity Problems: A survey of theory, algorithms and applications,, Math. Program., 48 (1990), 161. doi: doi:10.1007/BF01582255.
[14]	Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces,, J. Math. Anal. Appl., 330 (2007), 352. doi: doi:10.1016/j.jmaa.2006.07.063.
[15]	Y. R. He, A new projection algorithm for mixed variational inequalities,, Acta Math. Sci., 27 (2007), 215.
[16]	Y. R. He, X. Z. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces,, Acta Math. Sinica (English Series), 23 (2007), 563. doi: doi:10.1007/s10114-005-0918-5.
[17]	R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system,, J. Optim. Theory Appl., 142 (2009), 493. doi: doi:10.1007/s10957-009-9510-y.
[18]	R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces,, Optim. Lett., (): 11590. doi: doi:10.1007/s11590-010-0215-9.
[19]	V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization,, J. Math. Anal. Appl., 179 (1993), 537. doi: doi:10.1006/jmaa.1993.1368.
[20]	S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps,, J. Optim. Theory Appl., 18 (1976), 445. doi: doi:10.1007/BF00932654.
[21]	B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Degree theory for generalized variational inequalities and applications,, Eur. J. Oper. Res., 193 (2009), 12. doi: doi:10.1016/j.ejor.2007.10.028.
[22]	J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications,, J. Math. Anal. Appl., 306 (2005), 55. doi: doi:10.1016/j.jmaa.2004.11.007.
[23]	F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities,, Optimization, 60 (2011), 303. doi: doi:10.1080/02331934.2010.507985.
[24]	R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).
[25]	M. Sion, On general minimax theorems,, Pacific J. Math., 8 (1958), 171.
[26]	W. Takahashi, "Nonlinear Functional Analysis,", Yokohama Publishers, (2000).
[27]	M. M. Vainberg, "Variational Methods and Method of Monotone Operators,", Wiley, (1973).
[28]	Z. B. Wang and N.J. Huang, Degree theory for a generalized set-valued variational inequality with an application in Banach spaces,, J. Global Optim., 49 (2011), 343. doi: doi:10.1007/s10898-010-9547-3.
[29]	K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application,, Bull. Aust. Math. Soc., 73 (2006), 307. doi: doi:10.1017/S0004972700038892.
[30]	H. Y. Yin, C. X. Xu and Z. X. Zhang, The $F$-complementarity problem and its equivalence with the least element problem,, Acta Math. Sinica, 44 (2001), 679.
[31]	R. Y. Zhong and N. J. Huang, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces,, J. Optim. Theory Appl., 147 (2010), 454. doi: doi:10.1007/s10957-010-9732-z.

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