在本論文中，報告最近對於在無限微空間上廣義單調性算子的互補性問題的存在結果。

In this thesis, we report recent results on existence for complementarity problems in infinite-dimensional spaces under generalized monotonicity are reported.

Contents
I. Introduction
II . Notation and Preliminaries
III. Existence and Uniqueness Results
IV. Equivalence of Complementarity problem and Related problem

[1] Brézis, H., Nirenberg L. and G. Stampacchia, A remark on Ky Fan.s minimax principle, Boll. Unione Mat. Ital. (4) 6, 293-300, 1972.
[2] Barbu, V. and Precupanu, Th., Conrexity and Optimization in Banach Spaces. Sijtho¤ and Nourdho¤, Bucarest, Romania. 1986.
[3] Bensoussan, A., and Lions, J. L., Nouvelle Formulation des Problems de Controle., Impulsionnel to Applications, Comptes Rendus Hebdomadaires des Seances de I.Academie des Sciences, Series A et B, Vol. 276, pp. 1189-1192, 1973.
[4] Bensonussan, A., Gourset, M., and Lions. J. L., Controle Impulsionnel et Inequations Quasivariationnelle Stationaires, Comptes Rendus Hebdomadaires des Seances de I.Academie des Sciences, Series A et B, Vol. 276, pp. 1279-1284, 1973.
[5] Cottle, R. W., Complementarity and variational problems, Sympos. Math., Insituto Nazionable di Alta Mathematica, Volume XIX, 177-208, 1976.
[6] Cottle, R. W., and C. E. Lemke, Quadratic forms semi-de.neover convex cones, in Proceedings of the Princeton Sympoisium on Mathematical Programming, H. W. Kuhn, ed., Princeton University Press, Princeton, New Jersey, 551-565, 1970.
[7] Cottle, R. W., Notes on a fundamental theorem in quadratic programming, SIAM J. Appl. Math. 12, 663-665, 1964.
[8] Cottle, R. W., Nonlinear programs with positively bounded Jacobians, SIAM J. Appl. Math 14, 147-158, 1966.
[9] Cottle, W., Pang, J. S., and Stone, R. E., The Linear Complementarity Problem, Academic Press, New York, NY, 1992.
[10] Cottle, R.W., Numerical methods for complementarity problems in engineering and applied science, Technical Report SOL 77-24, Systems Optimization Laboratory, Department of Operations Research, Stanford University, 1977.
[11] Cottle, R.W. and J. C. Yao, Pseudomonotone complementarity problems in Hilbert space, J. Optimization Theory Appl. 75(2), 281-295, 1992.
[12] Crouzeix, J. P., Pseudomonotone Variational Inequality Problem: Existence of Solutions, Mathematical Programming, Vol.77, pp.305-314, 1997.
[13] Crouzeix, J. P., Hassouni, A., Lahlou, A., and Schaible, B., Positive Sub-definite Matrices, Generalized Monotonicity, and Linear Complementarity Problems, SIAM Journal on Matrix Analysis and Applications, Vol.22, pp. 66-85, 2000.
[14] Cubiotti, P., General nonlinear variational inequalities with (S)~operators, Appl. Math. Lett. 10 (2), 11-15, 1997.
[15] Cubiotti, P. and J. C. Yao, Multivalued (S)~operators and generalized variational inequalities, Comput. Math. Appl. 29 (12), 49-56, 1995.
[16] Cryer, C.W. and M. A. H. Dempster, Equivalence of linear complementarity problems and linear program in vector lattice Hilbert spaces, SIAM J. Control Optimization 18, 76-90, 1980.
[17] Chan, D. and J.S. Pang, The generalized quasi-variational inequality problem, Math. Open. Res. 7, 211-222, 1982.
[18] Dorn, W. S., Self-dual quadratic programs, SIAM J. Appl. Math. 9(1), 51-54, 1961.
[19] Dantzig, G.B. and R. W. Cottle, Positive semi-definite programming, in Nonlinear Programming, Abadie, ed., A course, North Holland,Amsterdam, 55-73, 1967.
[20] Dash, A. T. and S. Nanda, A complementarity problem in mathematical
programming in Banach space, J. Math. Anal. Appl. 98, 328-331, 1984.
[21] Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[22] Do, C., Bifurcation theory for elastic plates subjected to unilateral conditions, J. Math. Anal. Appl. 60, 435-448, 1977.
[23] Ekeland, I. and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
[24] Fan, K., A generalization of TychonofJ.s .xed point theorem,Math. Ann. 142, 305-310, 1961.
[25] Fang, S.C., A linearization method for generalized complementarity problems, IEEE Trans. Autom. Control 38, 363-383, 1982.
[26] Fang, S. C. and E. L. Peterson, Generalized variational inequalities, J. Optimization Theory Appl. 38, 363-383, 1982.
[27] Gownda, M. S., Pseudomonotone and Copositive Star Matrices, Linear Algebra and Applications, Vol. 113, pp. 107-118, 1989.
[28] Gownda, M. S., Complementarity Problems over Locally Compact Cones, SIAM Journal on Control and Optimization, Vol. 27, pp. 836-841, 1989.
[29] Gownda, M. S., A¢ ne Pseudomonotone Mappings and the Linear Complementarity Problem, SIAM Journal on Matrix Analysis and Applications, Vol. 11, pp. 373-380, 1990.
[30] Gownda, M. S., and Seidman, T. I., Generalized Linear Complementarity Problems, Mathematical Programming, Vol. 46, pp. 329-340, 1990.
[31] Guo, J. S. and J. C. Yao, Extension of strongly nonlinear quasivariational inequalities, Appl. Math. Lett. 5(3), 35-38, 1992.
[32] Guo, J. S. and J. C. Yao, Variational inequalities with non-monotone operators, J. Optimization Theory Appl. 80(1), 63-74, 1994.
[33] Habetler, G. T., and Price, A. L., Existence Theory for Generalized Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 7, pp. 223-239, 1971.
[34] Hadjisavvas, N. and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optimization Theory Appl. 90, 95-111, 1996.
[35] Hadjisavvas, N. and S. Schaible, On strong pseudomonotonicity and (semi)strict quasimonotonicity, J. Optimization Theory Appl. 79, 139-155, 1993.
[36] Harker, P. T. and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems : A survey of theory, algorithms and applications, Math. Program., Series B, 48, 161-220, 1990.
[37] Hartmann, P. and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115, 153-188, 1966.
[38] Holmes, R. B., Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, 1975.
[39] Ingleton, A. W., A problem in linear inequalities, Proc. London Math. Soc. 16, 519-536, 1966.
[40] Isac, G., Complementarity Problems, Lecture Notes in Mathematics, 1528, Springer-Verlag, Berlin Heidelberg, 1992.
[41] Saigal,R.,Extension of the generalized complementarrity problem, Math. Oper. Res. 1, 260-266, 1976.
[42] Karamardian, S., Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 8, pp. 161-168, 1971.
[43] Karamardian, S., Complementarity Problems over Cones with Monotone and Pseudomontone Maps, Journal of Optimization Theory and Applications, 18, pp. 445-454, 1976.
[44] Karamardian, S., Duality in Mathematical Programming, P.h.D. Thesis, University of California, Berkley, 1966.
[45] Karamardian, S. and S. Schaible, Seven kinds of monotone maps, J. Optimization Theory Appl. 66, 37-46, 1990.
[46] Karamardian, S., An existence theorem for the complementarity problem, J. Optimization Theory Appl. 19 (2), 227-232, 1976.
[47] Kelley, J. L. and 1. Namioka, Linear Topolical Spaces, Springer, New York, 1976.
[48] Lemke, C. E., Bimatrix equilibrium points and mathematical programming, Management Sciences 11(7), 681-689, 1965.
[49] Lions, J. L. and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20, 493-519, 1967.
[50] Luna, G., A remark on the nonlinear complementarity problem, Proc. Amer. Math. Soc. 48, 132-134, 1975.
[51] Mancino, O. and G. Stampacchia, Convex programming and variational in equalities, J. Optimization Theory Appl. 9, 3-23, 1972.
[52] More, J. J. and W. C. Rheinboldt, On P .and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra Appl. 6, 45-68, 1973.
[53] Megiddo, N. and M. Kojima, On the existence and uniquencess of solutions in non linear complementarity theory, Math. Program. 12, 110-130, 977.
41
[54] Mosco, U., Implicit Variational Problems and Quasi-Variational Inequalities, Lecture Notes in Mathematics, 543, Springer, Berlin, 1976.
[55] Nanda, S., Nonlinear Complementarity Problem of Mathematical Programming in Banach Space. Indian J. Pure Appl. Math. 18, 215-218, 987.
[56] Nanda, S., A Nonlinear Complementarity Problem in Banach Space. Bull Austral. .Math. Soc. 21, 351-356, 1980.
[57] Rheinboldt, W. C., On M-functions and their application to nonlinear
Gauss-Seidel iterations and network .ows, J. Math. Anal. Appl. 32, 274-307, 1970.
[58] Riddel, R. C., Equivalence of nonlinear complementarity problems and least
element problems in Banach spaces, Math. Oper. Res. 6, 191-200, 981.
[59] Stampacchia G., Variational inequalities, in Theory and Applications of
Monotone Operator, A. Ghizzetti ed., Proceedings of the NATO Advanced Study Institute, Venice Italy (Edizioni Oderisi, Gubbio, Italy), 02-192, 1968.
[60] Stampacchia, G., Variational Inequalities. In Theory and Applications of
Monotone Operators (A. Ghizzetti, Ed.), Edizioni Oderisi, Gubbio, Italy, 1969.
[61] Shisheng Zhang, Jian Li, Complementarity problems in Banach spaces,
Appl. Math. J. Chinese Univ. 9, 75-83, 1994.
[62] Schaible, S. and J. C. Yao, On the equivalence of nonlinear complementarity problems and least-element problems, Math.Program. 70, 191-200, 1995.
[63] Weiping Guo, Lixue Qu, Complementarity problems in re.exive Banach
spaces, J. Math. Study 31, 1998.