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Fixed point theory and complementarity problems in Hilbert space

Published online by Cambridge University Press:  17 April 2009

G. Isac
G. Isac
Affiliation:
Département de mathématiques, Collège militaire royal de Saint-Jean, Saint-Jean, Quebec, Canada, JOJ IRO
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Abstract

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In this paper we study both the implicit and the explicit complementarity problem using some special and interesting connections between the complementarity problem and fixed point theory in Hilbert space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 2 , October 1987 , pp. 295 - 310
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Allen, G., “Variational inequalities, complementarity problems and duality theorems”, J. Math. Anal. Appl. 58 (1977), 1–10.CrossRefGoogle Scholar
[2]Bazaraa, M.S., Goode, J.J. and Nashed, M.Z., “A nonlinear complementarity problem in mathematical programming in Banach spaces”, Proc. Amer. Math. Soc. 35 (1972), 165–170.CrossRefGoogle Scholar
[3]Bensoussan, A., “Variational inequalities and optimal stopping time problems”, In: Russel, D.L. ed: Calculus of variations and control theory. (Academic Press 1976), 219–244.Google Scholar
[4]Bensoussan, A. and Lions, J.L., “Nouvelle formulation des problèmes de contrôle impulsionnel et applications”, C.R. Acad. Sci. Ser. I. Math. Paris 276 (1973), 1189–1192.Google Scholar
[5]Bensoussan, A. and Lions, J.L., “Problèmes de temps d'arrêt optimal et inequations variationnelles paraboliques”, Applicable Anal. (1973), 267–294.Google Scholar
[6]Bensoussan, A. and Lions, J.L., “Nouvelles méthodes en contrôle impulsionnel”, Appl. Math. Optim. 1 (1975), 289–312.Google Scholar
[7]Bensoussan, A., Gourset, M. and Lions, J.L., “Contrôle impulsionnel et inéquation quasi-variationnelles stationnaires”, C.R. Acad. Sci. Ser. I. Math. Paris 276 (1973), 1279–1284.Google Scholar
[8]Borwein, J.M., “Generalized linear complementarity problems treated without fixed-point theory”, J. Opt. Theory Appl. 43 (1984), 343–356.Google Scholar
[9]Boyd, D.W. and Wang, J.S.W., “On nonlinear contractions”, Proc. Amer. Math. Soc. 20 (1969), 458–464.Google Scholar
[10]Browder, F.E., “Non-expansive non-linear operators in Banach spaces”, Proc. Nat. Acad. Sci. U.S.A. (1965), 1041–1044.CrossRefGoogle Scholar
[11]Darbo, G., “Punti uniti in transformazioni a codomenio non compatto”, Rend. Sem. Math. Univ. Padova 24 (1955), 84–92.Google Scholar
[12]Dash, A.T. and Nanda, S., “A complementarity problem in mathematical programming in Banach space”, J. Math. Anal. Appl. 98 (1984), 328–331.Google Scholar
[13]Dolcetta, I.C. and Mosco, V., “Implicit complementarity problems and quasi-variational inequalities”, In: Cottle, R.W., Giannessi, F. and Lions, J.L. eds: Variational inequalities and complementarity problems. Theory and Appl. (John Wiley & Sons 1980) 75–87.Google Scholar
[14]Fujimoto, T., “Nonlinear complementarity problems in a function space”, SIAM J. Control Optim. 18 (1980), 621–623.CrossRefGoogle Scholar
[15]Fujimoto, T., “An extension of Tarski's fixed point theorem and its applications to isotone complementarity problems”, Math. Programming, 18 (1984), 116–118.CrossRefGoogle Scholar
[16]Isac, G., “On the implicit complementarity problem in Hilbert spaces”, Bull. Austral. Math. Soc. 32 (1985), 251–260.Google Scholar
[17]Isac, G., “Sur l'existence par comparaison des valeurs propres positives pour des opérateurs non-linéaires”, Ann. Sci. Math. Québec VII, (1983), 159–184.Google Scholar
[18]Isac, G., “Nonlinear complementarity problem and Galerkin method”, J. Math. Anal. Appl. 108 (1985), 563–574.Google Scholar
[19]Isac, G., “Complementarity problem and coincidence equations on convex cones”, Bull. Un. Mat. Ital. (6) 5-B (1986), 925–943.Google Scholar
[20]Isac, G., “Problèmes de complémentarité (En dimension infinie)”, Minicours. Publ. Dept. de Math. et Inf. Univ. de Limoges (1985).Google Scholar
[21]Isac, G. and Nemeth, A.B., “Monotonicity of metric projections onto positive cones of ordered euclidean spaces. Arch. Math. (Basel) 46, (1986), 568–576.Google Scholar
[22]Isac, G. and Nemeth, A.B., “Ordered Hilbert spaces”, Preprint, 1986.Google Scholar
[23]Isac, G. and Téra, M., “Complementarity problem and post-critical equilibrium state of thin elastic plates”, J. Opt. Theory Appl. (to appear).Google Scholar
[24]Isac, G. and Téra, M., “A variational principle. Application to the nonlinear complementarity problems”, Preprint, 1986.Google Scholar
[25]Karamardian, S., “The nonlinear complementarity problem with applications”, J. Opt. Theory Appl. 4 (1969), 87–98.Google Scholar
[26]Karamardian, S., “Generalized complementarity problem”, J. Optim. Theory Appl. 8 (1971), 161–168.Google Scholar
[27]Luna, G., “A remark on the complementarity problem”, Proc. Amer. Math. Soc. 48 (1975), 132–134.CrossRefGoogle Scholar
[28]Moreau, J., “Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires”, C.R. Acad. Sci. Paris Ser. I. Math. 225 (1962), 238–240.Google Scholar
[29]Mosco, V., “On some non-linear quasi-variational inequalities and implicit complementarity problems in stochastic control theory”, In: Cottle, R.W., Giannessi, F. and Lions, J.L. eds: Variational inequalities and complementarity problems. Theory and Appl. (John Wiley & Sons 1980), 271–283.Google Scholar
[30]Nanda, S. and Nanda, S., “A nonlinear complementarity problem in mathematical programming in Hilbert space”, Bull. Austral. Math. Soc. 20 (1979), 233–236.Google Scholar
[31]Peressini, A.L., Ordered Topological Vector Spaces, (Harper & Row, New York 1967).Google Scholar
[32]Riddell, R.C., “Equivalence of nonlinear complementarity problems and least element in problems in Banach lattices”, Math. Oper. Res. 6 (1981), 462–474.CrossRefGoogle Scholar
[33]Sadovski, B.N., “A fixed point principle”, Functional Anal. Appl. 1 (1967), 151–153.Google Scholar
[34]Zarantonello, E.H., “Projections on convex sets in Hilbert space and spectral theory”, In: Zarantonello, E.H. ed.: Contribution to Nonlinear Functional Analysis. (Academic Press. New York 1971), 237–424.Google Scholar