Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-18T12:34:14.616Z Has data issue: false hasContentIssue false
  • English
  • Français

On the implicit complementarity problem in Hilbert spaces

Published online by Cambridge University Press:  17 April 2009

G. Isac
G. Isac
Affiliation:
Department de Mathẻmatiques, Collẻge Militaire Royal, St. Jean, Quebec, Canada JOJ IRO
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider in this paper the implicit complementarity problem imposed by quasi-variational inequalities and stochastic optimal control. The principal result is an existence theorem for the implicit complementarity problem in Hilbert spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 32 , Issue 2 , October 1985 , pp. 251 - 260
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Allen, C., “Variational inequalities, complementarity problems and duality theorems”, J. Math. Anal. and 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 space”, Proc. Amer. Math. Soc. Vol. 35, no 1, (1972), 165–170.CrossRefGoogle Scholar
[3]Bensoussan, A., Gourset, M. and Lions, J. L., “Contrôle impulsionnel et inéquations quasi-variationnelles stationaires”. C. R.Acad. Sciences Paris, 276, (1973), A. 1279–1284.Google Scholar
[4]Bensoussan, A. and Lions, J.L., “Nouvelle formulation des problémes de contrôle impulsionnel et applications”, Acad. Sciences Paris. 276, (1973), 1189–1192.Google Scholar
[5]Bensoussan, A. and Lions, J.L., “Problemes de temps d'arrêt optimal et inéquations variationnelles paraboliques”. Applicable Analysis, (1973), 267–294.CrossRefGoogle Scholar
[6]Bensoussan, A. and Lions, J.L., “Nouvelles methodes en controle impulsionnel”. Applied Math. and Optimization, no 1, (1974), 289–312.Google Scholar
[7]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
[8]Berger, M.S. and Schechter, M., “On the solvability of semilinear gradient operator equations”. Advances in Math. Vol. 25, (1977), 97–132.CrossRefGoogle Scholar
[9]Borwein, J.M., “Generalized linear complementarity problems treated without fixed-point theory”. J. Opt. Theory and Appl. Vol. 43 no 3, (1984), 343–356.CrossRefGoogle Scholar
[10]Browder, F.E., “Nonlinear accretive operators”, Bull. Amer. Math. Soc. 73, (1967), 470–476.CrossRefGoogle Scholar
[11]Browder, F.E., “Nonlinear mappings of nonexpansive and accretive type in Banach spaces”, Bull. Amer. Math. Soc. 73, (1967), 875–882.CrossRefGoogle Scholar
[12]Browder, F.E., “Nonlinear monotone and accretive operators in Banach spaces”, Proc. Nat. Acad. Sci. U.S.A. 61, (1968), 388–393.CrossRefGoogle ScholarPubMed
[13]Chan, D. and Pang, J.S., “The generalized quasivariational inequality problem”, Math. of Operations Research, Vol. 7, no 2, (1982), 211–222.CrossRefGoogle Scholar
[14]Crandal, M.G. and Pazy, A., “On the range of accretive operators”. Israel J. of Math. Vol. 27, no 3–4, (1977) 235–246.CrossRefGoogle Scholar
[15]Dash, A.T. and Nanda, S., “A complementarity problem in mathematical programming in Banach Space”, J. Math. Anal. and Appl. 98, (1984), 328–331.CrossRefGoogle Scholar
[16]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
[17]Fujimoto, T., “Nonlinear complementarity problems in a function space”, Siam J. Control and Optimization Vol. 18, (1980), 621–623.CrossRefGoogle Scholar
[18]Fujimoto, T., “An extension of Tarski's fixed point theorem and its applications to isotone complementarity problems”, Math. Programming 28, (1984), 116–118.CrossRefGoogle Scholar
[19]Isac, G., “Nonlinear complementarity problem and Galerkin method”, J. Math. Anal. and Appl. 108 (1985), 563–574.CrossRefGoogle Scholar
[20]Isac, G., “Complementarity problem via nonlinear eigenvalues problems and coincidence equations on cones”, Publ. Math. Universite Limoges, (1984).Google Scholar
[21]Isac, G., “Complementarity problem and coincidence equations on convex cones”, Preprint (1984).Google Scholar
[22]Karamardian, S., “Generalized complementarity problem”, J. Optimiz. Theory and Appl. 8, (1971), 161–168.CrossRefGoogle Scholar
[23]Kato, P., “Nonlinear semigroups and evolution equations”, J. Math. Soc. Japan, (1967), 508–520.Google Scholar
[24]Kirk, W.A. and Schneberg, R., “Some results an pseudo-contractive mappings”, Pacific J. of Math. Vol. 71, no 1, (1977), 89–100.CrossRefGoogle Scholar
[25]Luna, G., “A remark on the complementarity problem”, Proc.Amer. Math. Soc. 48, no 1, (1975), 132–134.CrossRefGoogle Scholar
[26]Minty, G. H., “Monotone (nonlinear) operators in a Hilbert space”, Duke Math. J. 29, (1962), 341–346,CrossRefGoogle Scholar
[27]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 coinplementarity problems theory and Applications”, John Wiley & Sons, (1980), 271–283.Google Scholar
[28]Nanda, S. and Nanda, S., “A nonlinear complementarity problem in mathematical programming in Hilbert space”, Bull. Aust. Math. Soc. 20, (1979), 233–236.CrossRefGoogle Scholar
[29]Pang, J.S., “The implicit complementarity problem”, In: Mangasarian, O.L., Meyer, R.R. and Robinson, S.M. eds., “Nonlinear Programming 4”, Academic Press, (1981) 487–518.Google Scholar
[30]Pang, J.S., “On the convergence of a basic iterative method for the implicit complementarity problem”, J. Optimiz. Theory and Appl. 37, (1982).CrossRefGoogle Scholar
[31]Riddell, R.C., “Equivalence of nonlinear complementarity problems and least element problems in Banach lattices”, Math. of Operations Research. Vol. 6, no 3, (1981), 462–474.CrossRefGoogle Scholar
[32]Schneberg, R., “On the domain invariance theorem for accretive mappings”, J. London Math. Soc. (2) 24, (1981), 548–554.CrossRefGoogle Scholar
[33]Webb, G.F., “Nonlinear perturbation of linear operators in Banach spaces”, J. Funct. Anal. 10, (1972), 191–203.CrossRefGoogle Scholar