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2013, 3(3): 567-581. doi: 10.3934/naco.2013.3.567

## Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072

Received  September 2011 Revised  April 2013 Published  July 2013

The notions of $\alpha$-well-posedness and generalized $\alpha$-well-posedness for a system of constrained variational inequalities involving set-valued mappings (for short, (SCVI)) are introduced in Hilbert spaces. Existence theorems of solutions for (SCVI) are established by using penalty techniques. Metric characterizations of $\alpha$-well-posedness and generalized $\alpha$-well-posedness, in terms of the approximate solutions sets, are presented. Finally, the equivalences between (generalized) $\alpha$-well-posedness for (SCVI) and existence and uniqueness of its solutions are also derived under quite mild assumptions.
Citation: Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567
##### References:
 [1] R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings,, J. Inequal. Appl., 7 (2002), 807. Google Scholar [2] H. Attouch, "E.D.P.associées à de sous-différentiels,", Thèse de Doctorat d'état ES Sciences Mathématiques, (1976). Google Scholar [3] L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities,, J. Optim. Theory Appl., 139 (2008), 109. doi: 10.1007/s10957-008-9428-9. Google Scholar [4] J. W. Chen, Z. Wan and Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems,, Math. Meth. Oper. Res., 77 (2013), 33. doi: 10.1007/s00186-012-0414-5. Google Scholar [5] J. W. Chen and Z. Wan, Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces,, J. Inequal. Appl., 49 (2011). doi: 10.1186/1029-242X-2011-49. Google Scholar [6] Y. J. Cho, Y. P. Fang, N. J. Huang and N. J. Hwang, Algorithms for systems of nonlinear variational inequalities,, J. Korean Math. Soc., 41 (2004), 203. Google Scholar [7] Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints,, Comput. Math. Appl., 55 (2008), 89. doi: 10.1016/j.camwa.2007.03.019. Google Scholar [8] M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces,, J. Optim. Theory Appl., 5 (1970), 225. doi: 10.1007/BF00927717. Google Scholar [9] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequalities problems with functional constraints,, J. Ind. Manag. Optim., 3 (2007), 671. doi: 10.3934/jimo.2007.3.671. Google Scholar [10] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of vector variational inequality problems with functional constraints,, Numer. Funct. Anal. Optim., 31 (2010), 671. doi: 10.1080/01630563.2010.485296. Google Scholar [11] R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Well-posedness of systems of equilibrium problems,, Taiwanese J. Math., 14 (2010), 2435. Google Scholar [12] R. Hu, Y. P. Fang and N. J. Huang, Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities,, J. Ind. Manag. Optim., 6 (2010), 465. doi: 10.3934/jimo.2010.6.465. Google Scholar [13] G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points,, Publ. Math. Debrecen, 54 (1999), 267. Google Scholar [14] J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces,, J. Convex Anal., 11 (2004), 235. Google Scholar [15] K. Kuratowski, "Topology,", (Vols. 1 and 2), (1968). Google Scholar [16] C. S. Lalitha and G. Bhatia, Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints,, Optim., 59 (2010), 997. doi: 10.1080/02331930902878358. Google Scholar [17] M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,, J. Glob. Optim., 16 (2000), 57. doi: 10.1023/A:1008370910807. Google Scholar [18] M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in:, (2001), 367. Google Scholar [19] M. B. Lignola and J. Morgan, α-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints,, J. Glob. Optim., 36 (2006), 439. doi: 10.1007/s10898-006-9020-5. Google Scholar [20] P. L. Lions, Two remarks on the convergence of convex functions and monotone operator,, Nonlinear Anal., 2 (1978), 553. Google Scholar [21] R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimimum problems with applications to variational inequalities,, Numer. Funct. Anal. Optim., 3 (1981), 461. Google Scholar [22] P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems,, J. Optim. Theory Appl., 138 (2008), 459. doi: 10.1007/s10957-008-9433-z. Google Scholar [23] A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities,, Optim. Lett., 6 (2012), 451. doi: 10.1007/s11590-010-0271-1. Google Scholar [24] M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities,, Nonlinear Anal., 70 (2009), 2700. doi: 10.1016/j.na.2008.03.057. Google Scholar [25] D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type,", Martinus Nijhoff, (1978). Google Scholar [26] J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems,, Optim. Lett., 4 (2010), 501. doi: 10.1007/s11590-010-0179-9. Google Scholar [27] J. W. Peng and J. Tang, α-well-posedness for mixed quasi-variational-like inequality problems,, Abstr. Appl. Anal., 2011 (2011), 1. Google Scholar [28] G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes,, CR Acad. Sci. Paris, 258 (1964), 4413. Google Scholar [29] Y. Tang and L. W. Liu, The penalty method for a new system of generalized variational inequalities,, Int. J. Math. Math. Sci., 2010 (2010), 1. doi: 10.1155/2010/614276. Google Scholar [30] A. N. Tykhonov, On the stability of the functional optimization problem,, USSR J. Comput. Math. Math. Phys., 6 (1966), 631. Google Scholar [31] Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares,, Numer. Algebra Control Optim., 1 (2011), 15. doi: 10.3934/naco.2011.1.15. Google Scholar [32] R. Y. Zhong and N. J. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces,, Numer. Algebra Control Optim., 1 (2011), 261. doi: 10.3934/naco.2011.1.261. Google Scholar

show all references

##### References:
 [1] R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings,, J. Inequal. Appl., 7 (2002), 807. Google Scholar [2] H. Attouch, "E.D.P.associées à de sous-différentiels,", Thèse de Doctorat d'état ES Sciences Mathématiques, (1976). Google Scholar [3] L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities,, J. Optim. Theory Appl., 139 (2008), 109. doi: 10.1007/s10957-008-9428-9. Google Scholar [4] J. W. Chen, Z. Wan and Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems,, Math. Meth. Oper. Res., 77 (2013), 33. doi: 10.1007/s00186-012-0414-5. Google Scholar [5] J. W. Chen and Z. Wan, Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces,, J. Inequal. Appl., 49 (2011). doi: 10.1186/1029-242X-2011-49. Google Scholar [6] Y. J. Cho, Y. P. Fang, N. J. Huang and N. J. Hwang, Algorithms for systems of nonlinear variational inequalities,, J. Korean Math. Soc., 41 (2004), 203. Google Scholar [7] Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints,, Comput. Math. Appl., 55 (2008), 89. doi: 10.1016/j.camwa.2007.03.019. Google Scholar [8] M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces,, J. Optim. Theory Appl., 5 (1970), 225. doi: 10.1007/BF00927717. Google Scholar [9] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequalities problems with functional constraints,, J. Ind. Manag. Optim., 3 (2007), 671. doi: 10.3934/jimo.2007.3.671. Google Scholar [10] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of vector variational inequality problems with functional constraints,, Numer. Funct. Anal. Optim., 31 (2010), 671. doi: 10.1080/01630563.2010.485296. Google Scholar [11] R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Well-posedness of systems of equilibrium problems,, Taiwanese J. Math., 14 (2010), 2435. Google Scholar [12] R. Hu, Y. P. Fang and N. J. Huang, Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities,, J. Ind. Manag. Optim., 6 (2010), 465. doi: 10.3934/jimo.2010.6.465. Google Scholar [13] G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points,, Publ. Math. Debrecen, 54 (1999), 267. Google Scholar [14] J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces,, J. Convex Anal., 11 (2004), 235. Google Scholar [15] K. Kuratowski, "Topology,", (Vols. 1 and 2), (1968). Google Scholar [16] C. S. Lalitha and G. Bhatia, Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints,, Optim., 59 (2010), 997. doi: 10.1080/02331930902878358. Google Scholar [17] M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,, J. Glob. Optim., 16 (2000), 57. doi: 10.1023/A:1008370910807. Google Scholar [18] M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in:, (2001), 367. Google Scholar [19] M. B. Lignola and J. Morgan, α-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints,, J. Glob. Optim., 36 (2006), 439. doi: 10.1007/s10898-006-9020-5. Google Scholar [20] P. L. Lions, Two remarks on the convergence of convex functions and monotone operator,, Nonlinear Anal., 2 (1978), 553. Google Scholar [21] R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimimum problems with applications to variational inequalities,, Numer. Funct. Anal. Optim., 3 (1981), 461. Google Scholar [22] P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems,, J. Optim. Theory Appl., 138 (2008), 459. doi: 10.1007/s10957-008-9433-z. Google Scholar [23] A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities,, Optim. Lett., 6 (2012), 451. doi: 10.1007/s11590-010-0271-1. Google Scholar [24] M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities,, Nonlinear Anal., 70 (2009), 2700. doi: 10.1016/j.na.2008.03.057. Google Scholar [25] D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type,", Martinus Nijhoff, (1978). Google Scholar [26] J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems,, Optim. Lett., 4 (2010), 501. doi: 10.1007/s11590-010-0179-9. Google Scholar [27] J. W. Peng and J. Tang, α-well-posedness for mixed quasi-variational-like inequality problems,, Abstr. Appl. Anal., 2011 (2011), 1. Google Scholar [28] G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes,, CR Acad. Sci. Paris, 258 (1964), 4413. Google Scholar [29] Y. Tang and L. W. Liu, The penalty method for a new system of generalized variational inequalities,, Int. J. Math. Math. Sci., 2010 (2010), 1. doi: 10.1155/2010/614276. Google Scholar [30] A. N. Tykhonov, On the stability of the functional optimization problem,, USSR J. Comput. Math. Math. Phys., 6 (1966), 631. Google Scholar [31] Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares,, Numer. Algebra Control Optim., 1 (2011), 15. doi: 10.3934/naco.2011.1.15. Google Scholar [32] R. Y. Zhong and N. J. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces,, Numer. Algebra Control Optim., 1 (2011), 261. doi: 10.3934/naco.2011.1.261. Google Scholar
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