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Abstract: The notion of “exceptional family of elements (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

[1] Alber, Y., 1996. Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications. In: Kartsatos, A. (Ed.), Theory and Applications of Nonlinear Operators of Monatonic and Accretive Type. Marcel Dekker, New York, p.15-50.

[2] Bulavski, V.A., Isac, G., Kalashnikov, V.V., 1998. Application of Topoligical Degree to Complementarity Problems. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (Eds.), Multilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, p.333-358.

[3] Cioranescu, I., 1990. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers.

[4] Cottle, R.W., Pang, J.S., Stone, R.E., 1992. The Linear Complementarity Problems. Academic Press, New York.

[5] Isac, G., 1992. Complementarity Problems. Lecture Notes in Math., Vol. 1528. Springer-Verlag.

[6] Isac, G., 1998. Exceptional Families of Elements for k-fields in Kilbert Spaces and Complementarity Theory. Proc. International Conf. Opt. Techniques Appl. (ICOTA’98), Perth, Australia, p.1135-1143.

[7] Isac, G., 1999a. A generalization of Karamardian’s condition in complementarity theory. Nonlinear Analysis Forum, 4:49-63.

[8] Isac, G., 1999b. On the Solvability of Multi-values Complementarity Problem: A Topological Method. Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology (EFDAN’99), Dortmund, Germany, p.51-66.

[9] Isac, G., 2000a. Topological Methods in Complementarity Theory. Kluwer Academic Publishers.

[10] Isac, G., 2000b. Exceptional family of elements, feasibility and complementarity. J. Opt. Theory Appl., 104:577-588.

[11] Isac, G., 2000c. Exceptional Family of Elements, Feasibility, Solvability and Continuous Paths of ε-solutions for Nonlinear Complementarity Problems. In: Pardalos, P. (Ed.), Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems. Kluwer Academic Publishers, p.323-337.

[12] Isac, G., 2001. Leray-Schauder type alternatives and the solvability of complementarity problems. Topol. Methods Nonlinear Analysis, 18:191-204.

[13] Isac, G., Obuchowska, V.T., 1998. Functions without exceptional families of elements and complementarity problems. J. Optim. Theory Appl., 99:147-163.

[14] Isac, G., Carbone, A., 1999. Exceptional families of elements for continuous functions: some applications to complementarity theory. J. Global Optim., 15:181-196.

[15] Isac, G., Zhao, Y.B., 2000. Exceptional family and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces. J. Math. Anal. Appl., 246:544-556.

[16] Isac, G., Li, J.L., 2001. Complementarity problems, Karamardian’s condition and a generalization of Harker-Pang condition. Nonlinear Anal. Forum, 6(2):383-390.

[17] Isac, G., Bulavski, V.A., Kalashnikov, V.V., 1997. Exceptional families, topological degree and complementarity problems. J. Global Optim., 10:207-225.

[18] Kalashnikov, V.V., 1995. Complementarity Problem and the Generalized Oligopoly Model, Habilitation Thesis. CEMI, Moscow.

[19] Kalashnikov, V.V., Isac, G., 2002. Solvability of implicit complementarity problems. Annals of Oper. Research, 116:199-221.

[20] Takahashi, W., 2000. Nonlinear Functional Analysis (Fixed Point Theory and Its Applications). Yokohama Publishers, Inc.

[21] Zhao, Y.B., 1997. Exceptional family and finite-dimensional variational inequalities over polyhedral convex sets. Applied Math. Comput., 87:111-126.

[22] Zhao, Y.B., 1998. Existence Theory and Algorithms for Finite-Dimensional Variational Inequalities and Complementarity Problems. Ph.D. Thesis, Institute of Applied Mathematics, Academia Sinica, Beijing, China (in Chinese).

[23] Zhao, Y.B., Isac, G., 2000a. Quasi-P_* and P(τ, α, β)–maps, exceptional family of elements and complementarity problems. J. Opt. Theory Appl., 105:213-231.

[24] Zhao, Y.B., Isac, G., 2000b. Properties of a multivalued mapping associated with some nonmonotone complementarity problems. SIAM J. Control Optim., 39:571-593.