The numerical range theory and boundedness of solutions of the complementarity problem. (English) Zbl 0722.47008

Isac, G.

J. Math. Anal. Appl. 143, No. 1, 235-251 (1989).

Let H be a Hilbert space, let \(K\subset H\) be a closed convex cone. Let \(K^*\) be the dual of K defined by \(K^*=\{y\in H:\) \(<x,y>\geq 0\), \(\forall x\in K\}\). Let f: \(K\to H\) be a mapping, the complementary problem associated to f and K denoted by c.p.(f,K) is: find \(x_*\in K:\) \(f(x_*)\in K^*\), \(<x_*,f(x_*)>=0.\)
This problem is an interesting one and is very important from the standpoints of both practical and theoretical interest, it has physical applications.
In this article the author studied the problem in a Hilbert space and used a method based on the theory of numerical range of an operator. Using this method the author finds as a particular case the results proved in [Pordalos and Ronsen, Bounds for the solution set of linear complementarity problems, Discrete Appl. Math.]. Also the author supplied solutions to the linear and nonlinear complementary problems.

Reviewer: M.Kutkut (Jeddah)

MSC:

47A12	Numerical range, numerical radius
47J25	Iterative procedures involving nonlinear operators
47L07	Convex sets and cones of operators
49J27	Existence theories for problems in abstract spaces

Keywords:

adjoint; positive definite; selfadjoint; Hilbert space; closed convex cone; complementary problem; numerical range

References:

[1]	Benci, V., Positive solutions of some eigenvalue problems in the theory of variational inequalities, J. Math. Anal. Appl., 61, 165-187 (1977) · Zbl 0432.35059
[2]	Berman, A., Complementarity problem and duality over convex cones, Canad. Math. Bull., 17, No. 1, 19-25 (1974) · Zbl 0337.90055
[3]	Bershchanskii, Y. M.; Meerov, M. V., The complementarity problem: Theory and method of solution, Automat. Remote Control, 44, No. 6, 687-710 (1983), Part 1 · Zbl 0527.90095
[4]	Bonsall, F. F.; Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (1971), Cambridge University Press: Cambridge University Press London/New York · Zbl 0207.44802
[5]	Cottle, R. W., Complementarity and variational problems, (Symposia Math., 19 (1976)), 177-208 · Zbl 0349.90083
[6]	Do, C., Problèmes de valeurs propres pour une inéquation variationnelle sur un cône et application au flambement unilatéral d’une plaque mince, C.R. Acad. Sci. Paris Ser. A, 280, 45-48 (1975) · Zbl 0298.73077
[7]	Giles, J. R., Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129, 436-446 (1967) · Zbl 0157.20103
[8]	Isac, G., Problèmes de complémentarité [En dimension infinie] (Mini-cours) (1985), Dépt. Math. Univ. Limoges
[9]	Isac, G., Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl., 108, No. 2, 563-574 (1985)
[10]	Isac, G., Complementarity problem and coincidence equations on convex cone, Boll. Un. Mat. Ital. B, 6/5, 925-943 (1986)
[11]	Isac, G.; Théra, M., Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate, J. Optim. Theory Appl., 58, No. 2, 241-257 (1988) · Zbl 0631.49005
[12]	Larsen, R., Functional Analysis (1973), Dekker: Dekker New York · Zbl 0261.46001
[13]	Lemke, C. E., A survey of complementarity theory, (Cottle, R. W.; Giannessi, F.; Lions, J. I., Variational Inequalities and Complementarity Problems: Theory and Applications (1980), Wiley: Wiley New York), 213-239 · Zbl 0439.90089
[14]	Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc., 100, 29-43 (1961) · Zbl 0102.32701
[15]	Mangasarian, O. L., Characterisations of bounded solutions of linear complementarity problem, Math. Programming Stud., 19, 153-166 (1982) · Zbl 0487.90088
[16]	Mangasarian, O. L., Simple computable bounds for solutions of linear complementarity problems and linear programs, Math. Programming Stud., 25, 1-12 (1985) · Zbl 0582.90098
[17]	Mangasarian, O. L.; McLinden, L., Simple bounds for solutions of monotone complementarity problems and convex programs, Math. Programming, 32, 32-40 (1985) · Zbl 0567.90093
[18]	Martin, D. H., The spectra of bounded linear self-adjoint operators relative to a cone in a Hilbert space, Nonlinear Anal. Theory, Methods Appl., 5, No. 3, 293-301 (1981) · Zbl 0452.47023
[19]	P. M. Pardalos and J. B. RosenDiscrete Appl. Math.; P. M. Pardalos and J. B. RosenDiscrete Appl. Math. · Zbl 0611.90094
[20]	Riddel, R. C., Eigenvalue problems for nonlinear elliptic variational inequalities on a cone, J. Funct. Anal., 26, 333-355 (1977) · Zbl 0369.47039