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 and 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.

[2]

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

[3]

C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189. 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), 1-17. doi: doi:10.1023/B:JOTA.0000041728.12683.89.

[5]

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

[6]

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

[7]

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

[8]

F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 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) 3997-4007. 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), 373-390. 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), 363-383. 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), 128-147. 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), 161-220. doi: doi:10.1007/BF01582255.

[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.

[15]

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

[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.

[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.

[18]

R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces, Optim. Lett., DOI 10.1007/s11590-010-0215-9. 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), 537-546. 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), 445-454. 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), 12-22. 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), 55-71. 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), 303-310. doi: doi:10.1080/02331934.2010.507985.

[24]

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

[25]

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

[26]

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

[27]

M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 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), 343-357 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), 307-317. 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), 679-686.

[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.

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.

[2]

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

[3]

C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189. 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), 1-17. doi: doi:10.1023/B:JOTA.0000041728.12683.89.

[5]

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

[6]

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

[7]

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

[8]

F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 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) 3997-4007. 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), 373-390. 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), 363-383. 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), 128-147. 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), 161-220. doi: doi:10.1007/BF01582255.

[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.

[15]

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

[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.

[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.

[18]

R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces, Optim. Lett., DOI 10.1007/s11590-010-0215-9. 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), 537-546. 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), 445-454. 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), 12-22. 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), 55-71. 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), 303-310. doi: doi:10.1080/02331934.2010.507985.

[24]

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

[25]

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

[26]

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

[27]

M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 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), 343-357 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), 307-317. 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), 679-686.

[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.

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