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Exceptional families of elements for variational inequalities in Banach spaces. (English) Zbl 1136.49007
Summary: In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
[1]SMITH, T. E., A Solution Condition for Complementarity Problems with an Application to Spatial Price Equilibrium, Applied Mathematics and Computation, Vol. 15, pp. 61–69, 1984. · Zbl 0545.90094 · doi:10.1016/0096-3003(84)90053-5
[2]ISAC, G., and KALASHNIKOV, V. V., Exceptional Family of Elements, Leray-Schauder Alternative, Pseudomonotone Operators, and Complementarity, Journal of Optimization Theory and Applications, Vol. 109, pp. 69–83, 2001. · Zbl 0984.49006 · doi:10.1023/A:1017509704362
[3]ISAC, G., BULAVSKI, V., and KALASHNIKOV, V., Exceptional Families, Topological Degree, and Complementarity Problems, Journal of Global Optimization, Vol. 10, pp. 207–225, 1997. · Zbl 0880.90127 · doi:10.1023/A:1008284330108
[4]ZHAO, Y. B., and YUAN, J. Y., An Alternative Theorem for Generalized Variational Inequalities and Solvability of Nonlinear Quasi-P M _ Complementarity Problems, Applied Mathematics and Computation, Vol. 109, pp. 167–182, 2000. · Zbl 1021.49009 · doi:10.1016/S0096-3003(99)00019-3
[5]ZHAO, Y. B., and SUN, D., Alternative Theorems for Nonlinear Projection Equations, Nonlinear Analysis, Vol. 46, pp. 853–868, 2001. · Zbl 1047.49014 · doi:10.1016/S0362-546X(00)00154-1
[6]ZHAO, Y. B., and HAN, J., Exceptional Family of Elements for a Variational Inequality Problem and Its Applications, Journal of Global Optimization, Vol. 14, pp. 313–330, 1999. · Zbl 0932.49012 · doi:10.1023/A:1008202323884
[7]ISAC, G., and ZHAO, Y. B., Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 246, pp. 544–556, 2000. · Zbl 0966.49006 · doi:10.1006/jmaa.2000.6817
[8]ZHOU, S. Z., and BAI, M. R., A New Exceptional Family of Elements for a Variational Inequality Problem on Hilbert Space, Applied Mathematics Letters, Vol. 17, pp. 423–428, 2004. · Zbl 1076.47052 · doi:10.1016/S0893-9659(04)90084-5
[9]HAN, J., HUANG, Z. H., and FANG, S. C., Solvability of Variational Inequality Problems, Journal of Optimization Theory and Applications, Vol. 122, pp. 501–520, 2004. · Zbl 1082.49009 · doi:10.1023/B:JOTA.0000042593.74934.b7
[10]ISAC, G., and LI, J. L., Exceptional Family of Elements and the Solvability of Complementarity Problems in Uniformly Smooth and Uniformly Convex Banach Spaces, Journal of the Zhejiang University of Science, Vol. 6A, pp. 1–9, 2005.
[11]CARBONE, A., and ZABREIKO, P. P., Some Remarks on Complementarity Problems in a Hilbert Space, Zeitschrift fuer Analysis und ihre Anwendungen, Vol. 21, pp. 1005–1014, 2002.
[12]BIANCHI, M., HADJISAVVAS, N., and SCHAIBLE, S., Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 122, pp. 1–17, 2004. · Zbl 1130.49302 · doi:10.1023/B:JOTA.0000041728.12683.89
[13]KARAMARDIAN, S., Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971. · Zbl 0208.46301 · doi:10.1007/BF00932464
[14]AUBIN, J. P., and EKELAND, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984.
[15]HADJISAVVAS, N., Continuity and Maximality Properties of Pseudomonotone Operators, Journal of Convex Analysis, Vol. 20, pp. 465–475, 2003.
[16]AUSSEL, D., and HADJISAVVAS, N., On Quasimonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 121, pp. 445–450, 2004. · Zbl 1062.49006 · doi:10.1023/B:JOTA.0000037413.45495.00