American Institute of Mathematical Sciences

Advanced
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

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 "Proceedings of the Israel Seminar Ariel," Israel. Funct. Differ. Equ., 1 (1994), 1-21.  Google Scholar

[2]

J. P. Aubin, "Optima and Equilibria," Springer-Verlag, Berlin, 1993.  Google Scholar

[3]

C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189. doi: doi:10.1007/BF00282202.  Google Scholar

[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), 1-17. doi: doi:10.1023/B:JOTA.0000041728.12683.89.  Google Scholar

[5]

I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems," Kluwer, Dordrecht, 1990.  Google Scholar

[6]

J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions, Math. Program., 78 (1997), 305-314. doi: doi:10.1007/BF02614358.  Google Scholar

[7]

A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities, Math. Program., 86 (1999), 433-438. doi: doi:10.1007/s101070050097.  Google Scholar

[8]

F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003.  Google Scholar

[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) 3997-4007. doi: doi:10.1016/j.na.2008.08.008.  Google Scholar

[10]

Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems, J. Optim. Theory Appl., 129 (2006), 373-390. doi: doi:10.1007/s10957-006-9073-0.  Google Scholar

[11]

S. C. Fang and E. L. Peterson, Generalized variational inequalities, J. Optim. Theory Appl., 38 (1982), 363-383. doi: doi:10.1007/BF00935344.  Google Scholar

[12]

O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), 128-147. doi: doi:10.1287/moor.18.1.128.  Google Scholar

[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), 161-220. doi: doi:10.1007/BF01582255.  Google Scholar

[14]

Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363. doi: doi:10.1016/j.jmaa.2006.07.063.  Google Scholar

[15]

Y. R. He, A new projection algorithm for mixed variational inequalities, Acta Math. Sci., 27 (2007), 215-220.  Google Scholar

[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), 563-570. doi: doi:10.1007/s10114-005-0918-5.  Google Scholar

[17]

R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499. doi: doi:10.1007/s10957-009-9510-y.  Google Scholar

[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.  Google Scholar

[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), 537-546. doi: doi:10.1006/jmaa.1993.1368.  Google Scholar

[20]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454. doi: doi:10.1007/BF00932654.  Google Scholar

[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), 12-22. doi: doi:10.1016/j.ejor.2007.10.028.  Google Scholar

[22]

J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55-71. doi: doi:10.1016/j.jmaa.2004.11.007.  Google Scholar

[23]

F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities, Optimization, 60 (2011), 303-310. doi: doi:10.1080/02331934.2010.507985.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[25]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  Google Scholar

[26]

W. Takahashi, "Nonlinear Functional Analysis," Yokohama Publishers, Yokohama, 2000.  Google Scholar

[27]

M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 1973.  Google Scholar

[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), 343-357 doi: doi:10.1007/s10898-010-9547-3.  Google Scholar

[29]

K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application, Bull. Aust. Math. Soc., 73 (2006), 307-317. doi: doi:10.1017/S0004972700038892.  Google Scholar

[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), 679-686.  Google Scholar

[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), 454-472. doi: doi:10.1007/s10957-010-9732-z.  Google Scholar

show all references

References:
[1]

Ya. Alber, Generalized projection operators in Banach spaces: properties and applications, in "Proceedings of the Israel Seminar Ariel," Israel. Funct. Differ. Equ., 1 (1994), 1-21.  Google Scholar

[2]

J. P. Aubin, "Optima and Equilibria," Springer-Verlag, Berlin, 1993.  Google Scholar

[3]

C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189. doi: doi:10.1007/BF00282202.  Google Scholar

[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), 1-17. doi: doi:10.1023/B:JOTA.0000041728.12683.89.  Google Scholar

[5]

I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems," Kluwer, Dordrecht, 1990.  Google Scholar

[6]

J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions, Math. Program., 78 (1997), 305-314. doi: doi:10.1007/BF02614358.  Google Scholar

[7]

A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities, Math. Program., 86 (1999), 433-438. doi: doi:10.1007/s101070050097.  Google Scholar

[8]

F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003.  Google Scholar

[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) 3997-4007. doi: doi:10.1016/j.na.2008.08.008.  Google Scholar

[10]

Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems, J. Optim. Theory Appl., 129 (2006), 373-390. doi: doi:10.1007/s10957-006-9073-0.  Google Scholar

[11]

S. C. Fang and E. L. Peterson, Generalized variational inequalities, J. Optim. Theory Appl., 38 (1982), 363-383. doi: doi:10.1007/BF00935344.  Google Scholar

[12]

O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), 128-147. doi: doi:10.1287/moor.18.1.128.  Google Scholar

[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), 161-220. doi: doi:10.1007/BF01582255.  Google Scholar

[14]

Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363. doi: doi:10.1016/j.jmaa.2006.07.063.  Google Scholar

[15]

Y. R. He, A new projection algorithm for mixed variational inequalities, Acta Math. Sci., 27 (2007), 215-220.  Google Scholar

[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), 563-570. doi: doi:10.1007/s10114-005-0918-5.  Google Scholar

[17]

R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499. doi: doi:10.1007/s10957-009-9510-y.  Google Scholar

[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.  Google Scholar

[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), 537-546. doi: doi:10.1006/jmaa.1993.1368.  Google Scholar

[20]

S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454. doi: doi:10.1007/BF00932654.  Google Scholar

[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), 12-22. doi: doi:10.1016/j.ejor.2007.10.028.  Google Scholar

[22]

J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55-71. doi: doi:10.1016/j.jmaa.2004.11.007.  Google Scholar

[23]

F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities, Optimization, 60 (2011), 303-310. doi: doi:10.1080/02331934.2010.507985.  Google Scholar

[24]

R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[25]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.  Google Scholar

[26]

W. Takahashi, "Nonlinear Functional Analysis," Yokohama Publishers, Yokohama, 2000.  Google Scholar

[27]

M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 1973.  Google Scholar

[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), 343-357 doi: doi:10.1007/s10898-010-9547-3.  Google Scholar

[29]

K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application, Bull. Aust. Math. Soc., 73 (2006), 307-317. doi: doi:10.1017/S0004972700038892.  Google Scholar

[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), 679-686.  Google Scholar

[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), 454-472. doi: doi:10.1007/s10957-010-9732-z.  Google Scholar

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