Abstract
In this paper, a new notion of exceptional family of elements (EFE) for a pair of functions involved in the implicit complementarity problem (ICP) is introduced. Based upon this notion and the Leray–Schauder Alternative, a general alternative is obtained which gives more general existence theorems for the implicit complementarity problem. Finally, via the techniques of continuous selections, these existence theorems are extended to the multi-valued implicit complementarity problems (MIPS).
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J.-P. Aubin, in: Nonlinear Analysis and Its Economic Applications (Mir, Moscow, 1988).
E.V. Avtukhovich, Existence of solution for multi-valued complimentarity problem, Working paper, Computer Center of the Russian Academy of Sciences, Moscow (1999) (in Russian).
C. Baiocchi and A. Capelo, Variational Inequalities and Quasi-Variational Inequalities: Applications to Free Boundary Value Problems (Wiley, New York, 1984).
H. Ben-el-Mechaiekh, S. Chebbi and M. Florenzano, A Leray-Schauder type theorem for approximable maps: A simple proof, Proc. Amer. Math. Soc. 126 (1998) 2345-2349.
H. Ben-el-Mechaiekh and A. Idzik, A Leray-Schauder type theorem for approximable maps, Proc. Amer. Math. Soc. 122 (1994) 105-109.
A. Bensoussan, M. Gourset and J.-L. Lions, Contrôle impulsionnel et inéquations quasivariationnelles stationaires, C. R. Acad. Sci. Paris, Ser. A-B 276 (1973) 1279-1284.
A. Bensoussan and J.-L. Lions, Nouvelle formulation de problèmes de contrôle impulsionnel et applications, C. R. Acad. Sci. Paris, Ser. A-B 276 (1973) 1189-1192.
V.A. Bulavsky, G. Isac and V.V. Kalashnikov, Application of topological degree theory to complementarity problems, in: Multilevel Optimization: Algorithms and Applications, eds. A. Migdalas, P.M. Pardalos and P. Värbrand (Kluwer, Dordrecht, 1998).
V.A. Bulavsky, G. Isac and V.V. Kalashnikov, Application of topological degree theory to semidefinite complementarity problems, in: Proceedings of the International Conference on Operations Research OR'98 (Springer, Zurich, 1999).
J. Capuzzo-Dolcetta and U. Mosco, Implicit complementarity problems and quasi-variational inequalities, in: Variational Inequalities and Complementarity Problems: Theory and Applications, eds. R.W. Cottle, F. Giannessi and J.-L. Lions (Wiley, New York, 1980).
D. Chan and J.-S. Pang, The generalized quasi-variational problem, Math. Oper. Res. 7 (1982) 211-222.
R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem (Academic Press, New York, 1992).
J. Dugundji and A. Granas, Fixed Point Theory (PWN, Warszawa, 1982).
P.T. Harker and J.-S. Pang, Finite-dimensional variational inequalities and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming 48 (1990) 161-220.
D.H. Hyers, G. Isac and T.M. Rassias, Topics in Nonlinear Analysis and Applications (World Scientific, Singapore, 1997).
G. Isac, On the implicit complementarity problem in Hilbert spaces, Bull. Austral. Math. Soc. 32 (1985) 251-260.
G. Isac, Complementarity problem and coincidence equations on convex cones, Boll. Un. Mat. Ital. B(6) 5 (1986) 925-943.
G. Isac, Fixed point theory and complementarity problems in Hilbert spaces, Bull. Austral. Math. Soc. 36 (1987) 295-310.
G. Isac, Fixed point theory, coincidence equations on convex cones and complementarity problem, Contemp. Math. 72 (1988) 139-155.
G. Isac, A special variational inequality and the implicit complementarity problem, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 37 (1990) 109-127.
G. Isac, Complementarity Problems, Lecture Notes in Mathematics, Vol. 1528 (Springer, Berlin, 1992).
G. Isac, Tikhonov regularization and the complementarity problem in Hilbert spaces, J. Math. Anal. Appl. 174 (1993) 53-66.
G. Isac, Exceptional families of elements for k-fields in Hilbert spaces and complementarity theory, in: Proceedings of the International Conference on Optimization Techniques and Applications (ICOTA'98), Perth, Australia (1998).
G. Isac, A generalization of Karamardian's condition in complementarity theory, Nonlinear Anal. Forum 4 (1999) 49-63.
G. Isac, On the Solvability of multivalued complementarity problem: A topological method, in: Proceedings of the Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology (EFDAN'99), ed. R. Felix, Dortmund, Germany (1999).
G. Isac, Exceptional family of elements, feasibility and complementarity, J. Optim. Theory Appl. 104 (2000) 577-588.
G. Isac, Exceptional family of elements, feasibility, solvability and continuous paths of ɛ-solutions for nonlinear complementarity problems, in: Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, ed. P.M. Pardalos (Kluwer, Boston, 2000).
G. Isac, Topological Methods in Complementarity Theory (Kluwer, Dordrecht, 2000).
G. Isac, V.A. Bulavsky and V.V. Kalashnikov, Exceptional families, topological degree and complementarity problems, J. Global Optim. 10 (1997) 207-225.
G. Isac and A. Carbone, Exceptional families of elements for continuous functions, some applications to complementarity theory, J. Global Optim. 15 (1999) 181-196.
G. Isac and D. Goeleven, Existence theorems for the implicit complementarity problems, Internat. J. Math. Math. Sci. 16 (1993) 67-74.
G. Isac and D. Goeleven, The implicit general order complementarity problem: Models and iterative methods, Ann. Oper. Res. 44 (1993) 63-92.
G. Isac and V.V. Kalashnikov, Exceptional families of elements, Leray-Schauder alternatives, pseudomonotone operators and complementarity, J. Optim. Theory Appl. 109 (2001) 69-83.
G. Isac and W.T. Obuchowska, Functions without exceptional families of elements and complementarity problems, J. Optim. Theory Appl. 99, (1998) 147-163.
G. Isac and Y.B. Zhao, Exceptional families of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces, Preprint, University of Beijing (1999).
G. Isac and Y.B. Zhao, Exceptional families of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces, J. Math. Anal. Appl. 246 (2000) 544-556.
V.V. Kalashnikov, Complementarity problem and generalized oligopoly models, Habilitation thesis, Central Economics and Mathematics Institute, Moscow (1995) (in Russian).
V.V. Kalashnikov, Fixed point existence theorems based upon topological degree theory, Working paper, Central Economics and Mathematics Institute, Moscow (1995) (in Russian).
S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971) 161-168.
D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, (Academic Press, New York, 1980).
E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956) 233-238.
U. Mosco, Implicit Variational Problems and Quasi-Variational Inequalities, Lecture Notes in Mathematics, Vol. 543 (Springer, New York, 1976).
U. Mosco, On some nonlinear quasi-variational inequalities and implicit complementarity problems in stochastic control theory, in: Variational Inequalities and Complementarity Problems, Theory and Applications, eds. R.W. Cottle, F. Giannessi and J.-L. Lions (Wiley, New York, 1980).
H. Nikaido, Convex Structures and Mathematical Economics (Academic Press, New York, 1968).
J.-S. Pang, The implicit complementarity problem, in: Nonlinear Programming IV, eds. O.L. Mangasarian, R.R. Meyer and S.M. Robinson (Academic Press, New York, 1981).
J.-S. Pang, On the convergence of a basic iterative method for the implicit complementarity problem, J. Optim. Theory Appl. 37 (1982) 149-162.
D.B. Silin, Some properties of upper semicontinuous multi-valued mappings, Proc. of Math. Inst. of the Russian Academy of Science (Steklov Institute) 185 (1998) 222-235 (in Russian).
M.J. Smith, A descent algorithm for solving monotone variational inequalities and monotone complementarity problems, J. Optim. Theory Appl. 44 (1984) 485-496.
Y.B. Zhao, Exceptional family and finite-dimensional variational inequality over polyhedral convex set, Appl. Math. Comput. 87 (1997) 111-126.
Y.B. Zhao, Existence theory and applications for finite-dimensional variational inequality and complementarity problems, Ph.D. thesis, Institute of Applied Mathematics, Academia Sinica, Beijing, China (1998).
Y.B. Zhao and J.Y. Han, Exceptional family of elements for a variational inequality problem and its applications, J. Global Optim. 14 (1999) 313-330.
Y.B. Zhao, J.Y. Han and H.D. Qi, Exceptional family and existence theorems for variational inequality problems, J. Optim. Theory Appl. 101 (1999) 475-495.
Y.B. Zhao and G. Isac, Properties of a multivalued mapping and existence of central path for some nonmonotone complementarity problems, Preprint, University of Beijing (1999).
Y.B. Zhao and G. Isac, Quasi-P * and P(τ,α,α)-maps, exceptional family of elements and complementarity problems, J. Optim. Theory Appl. 105 (2000) 213-231.
Y.B. Zhao and G. Isac, Properties of a multivalued mapping associated with some non-monotone complementarity problems, SIAM J. Control Optim. 39 (2000) 571-593.
Y.B. Zhao and D.F. Sun, Alternative theorems for nonlinear projection equations and applications to generalized complementarity problem, Nonlinear Anal. 46 (2001) 853-868.
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Kalashnikov, V.V., Isac, G. Solvability of Implicit Complementarity Problems. Annals of Operations Research 116, 199–221 (2002). https://doi.org/10.1023/A:1021388515849
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DOI: https://doi.org/10.1023/A:1021388515849