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A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi$f_i$expansive mappings in convex metric spaces
Existence of solutions and $\alpha$wellposedness for a system of constrained setvalued variational inequalities
1.  School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 
2.  School of Mathematics and Statistics, Wuhan University, Wuhan, 430072 
References:
[1] 
R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasivariational inclusions with setvalued mappings, J. Inequal. Appl., 7 (2002), 807828. 
[2] 
H. Attouch, "E.D.P.associées à de sousdifférentiels," Thèse de Doctorat d'état ES Sciences Mathématiques, Université Paris 6, 1976. 
[3] 
L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Wellposedness for mixed quasivariationallike inequalities, J. Optim. Theory Appl., 139 (2008), 109125. doi: 10.1007/s1095700894289. 
[4] 
J. W. Chen, Z. Wan and Y. J. Cho, LevitinPolyak wellposedness by perturbations for systems of setvalued vector quasiequilibrium problems, Math. Meth. Oper. Res., 77 (2013), 3364. doi: 10.1007/s0018601204145. 
[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: 10.1186/1029242X201149. 
[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), 203210. 
[7] 
Y. P. Fang, R. Hu and N. J. Huang, Wellposedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89100. doi: 10.1016/j.camwa.2007.03.019. 
[8] 
M. Furi and A. Vignoli, About wellposed optimization problems for functions in metric spaces, J. Optim. Theory Appl., 5 (1970), 225229. doi: 10.1007/BF00927717. 
[9] 
X. X. Huang and X. Q. Yang, LevitinPolyak wellposedness in generalized variational inequalities problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671684. doi: 10.3934/jimo.2007.3.671. 
[10] 
X. X. Huang and X. Q. Yang, LevitinPolyak wellposedness of vector variational inequality problems with functional constraints, Numer. Funct. Anal. Optim., 31 (2010), 671684. doi: 10.1080/01630563.2010.485296. 
[11] 
R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Wellposedness of systems of equilibrium problems, Taiwanese J. Math., 14 (2010), 24352446. 
[12] 
R. Hu, Y. P. Fang and N. J. Huang, LevitinPolyak wellposedness for variational inequalities and for optimization problems with variational inequalities, J. Ind. Manag. Optim., 6 (2010), 465481. doi: 10.3934/jimo.2010.6.465. 
[13] 
G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points, Publ. Math. Debrecen, 54 (1999), 267279. 
[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), 235243. 
[15] 
K. Kuratowski, "Topology," (Vols. 1 and 2), Academic Press, New York, 1968. 
[16] 
C. S. Lalitha and G. Bhatia, Wellposedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints, Optim., 59 (2010), 9971011. doi: 10.1080/02331930902878358. 
[17] 
M. B. Lignola and J. Morgan, Wellposedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 5767. doi: 10.1023/A:1008370910807. 
[18] 
M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$wellposedness for variational inequalities and Nash equilibria, in: "Decision and Control in Management Science," Kluwer Academic Publishers, (2001), 367378. 
[19] 
M. B. Lignola and J. Morgan, αwellposedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, J. Glob. Optim., 36 (2006), 439459. doi: 10.1007/s1089800690205. 
[20] 
P. L. Lions, Two remarks on the convergence of convex functions and monotone operator, Nonlinear Anal., 2 (1978), 553562. 
[21] 
R. Lucchetti and F. Patrone, A characterization of Tykhonov wellposedness for minimimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461476. 
[22] 
P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems, J. Optim. Theory Appl., 138 (2008), 459477. doi: 10.1007/s109570089433z. 
[23] 
A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities, Optim. Lett., 6 (2012), 451458. doi: 10.1007/s1159001002711. 
[24] 
M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities, Nonlinear Anal., 70 (2009), 27002706. doi: 10.1016/j.na.2008.03.057. 
[25] 
D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Martinus Nijhoff, The Hague, 1978. 
[26] 
J. W. Peng and S. Y. Wu, The generalized Tykhonov wellposedness for system of vector quasiequilibrium problems, Optim. Lett., 4 (2010), 501512. doi: 10.1007/s1159001001799. 
[27] 
J. W. Peng and J. Tang, αwellposedness for mixed quasivariationallike inequality problems, Abstr. Appl. Anal., 2011 (2011), 117. 
[28] 
G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes, CR Acad. Sci. Paris, 258 (1964), 44134416. 
[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), 18. doi: 10.1155/2010/614276. 
[30] 
A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Math. Phys., 6 (1966), 631634. 
[31] 
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numer. Algebra Control Optim., 1 (2011), 1534. doi: 10.3934/naco.2011.1.15. 
[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), 261274. doi: 10.3934/naco.2011.1.261. 
show all references
References:
[1] 
R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasivariational inclusions with setvalued mappings, J. Inequal. Appl., 7 (2002), 807828. 
[2] 
H. Attouch, "E.D.P.associées à de sousdifférentiels," Thèse de Doctorat d'état ES Sciences Mathématiques, Université Paris 6, 1976. 
[3] 
L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Wellposedness for mixed quasivariationallike inequalities, J. Optim. Theory Appl., 139 (2008), 109125. doi: 10.1007/s1095700894289. 
[4] 
J. W. Chen, Z. Wan and Y. J. Cho, LevitinPolyak wellposedness by perturbations for systems of setvalued vector quasiequilibrium problems, Math. Meth. Oper. Res., 77 (2013), 3364. doi: 10.1007/s0018601204145. 
[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: 10.1186/1029242X201149. 
[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), 203210. 
[7] 
Y. P. Fang, R. Hu and N. J. Huang, Wellposedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89100. doi: 10.1016/j.camwa.2007.03.019. 
[8] 
M. Furi and A. Vignoli, About wellposed optimization problems for functions in metric spaces, J. Optim. Theory Appl., 5 (1970), 225229. doi: 10.1007/BF00927717. 
[9] 
X. X. Huang and X. Q. Yang, LevitinPolyak wellposedness in generalized variational inequalities problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671684. doi: 10.3934/jimo.2007.3.671. 
[10] 
X. X. Huang and X. Q. Yang, LevitinPolyak wellposedness of vector variational inequality problems with functional constraints, Numer. Funct. Anal. Optim., 31 (2010), 671684. doi: 10.1080/01630563.2010.485296. 
[11] 
R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Wellposedness of systems of equilibrium problems, Taiwanese J. Math., 14 (2010), 24352446. 
[12] 
R. Hu, Y. P. Fang and N. J. Huang, LevitinPolyak wellposedness for variational inequalities and for optimization problems with variational inequalities, J. Ind. Manag. Optim., 6 (2010), 465481. doi: 10.3934/jimo.2010.6.465. 
[13] 
G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points, Publ. Math. Debrecen, 54 (1999), 267279. 
[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), 235243. 
[15] 
K. Kuratowski, "Topology," (Vols. 1 and 2), Academic Press, New York, 1968. 
[16] 
C. S. Lalitha and G. Bhatia, Wellposedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints, Optim., 59 (2010), 9971011. doi: 10.1080/02331930902878358. 
[17] 
M. B. Lignola and J. Morgan, Wellposedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 5767. doi: 10.1023/A:1008370910807. 
[18] 
M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$wellposedness for variational inequalities and Nash equilibria, in: "Decision and Control in Management Science," Kluwer Academic Publishers, (2001), 367378. 
[19] 
M. B. Lignola and J. Morgan, αwellposedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, J. Glob. Optim., 36 (2006), 439459. doi: 10.1007/s1089800690205. 
[20] 
P. L. Lions, Two remarks on the convergence of convex functions and monotone operator, Nonlinear Anal., 2 (1978), 553562. 
[21] 
R. Lucchetti and F. Patrone, A characterization of Tykhonov wellposedness for minimimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461476. 
[22] 
P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems, J. Optim. Theory Appl., 138 (2008), 459477. doi: 10.1007/s109570089433z. 
[23] 
A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities, Optim. Lett., 6 (2012), 451458. doi: 10.1007/s1159001002711. 
[24] 
M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities, Nonlinear Anal., 70 (2009), 27002706. doi: 10.1016/j.na.2008.03.057. 
[25] 
D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Martinus Nijhoff, The Hague, 1978. 
[26] 
J. W. Peng and S. Y. Wu, The generalized Tykhonov wellposedness for system of vector quasiequilibrium problems, Optim. Lett., 4 (2010), 501512. doi: 10.1007/s1159001001799. 
[27] 
J. W. Peng and J. Tang, αwellposedness for mixed quasivariationallike inequality problems, Abstr. Appl. Anal., 2011 (2011), 117. 
[28] 
G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes, CR Acad. Sci. Paris, 258 (1964), 44134416. 
[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), 18. doi: 10.1155/2010/614276. 
[30] 
A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Math. Phys., 6 (1966), 631634. 
[31] 
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numer. Algebra Control Optim., 1 (2011), 1534. doi: 10.3934/naco.2011.1.15. 
[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), 261274. doi: 10.3934/naco.2011.1.261. 
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