×

A class of operator equilibrium problems. (English) Zbl 1072.47061

The authors study a class of operator equilibrium problems defined as follows: find \(f\in K\) such that \[ F(f,g)\notin -C(f),\qquad \forall g\in K, \] where \(K\subset L(X,Y),\) \(X,Y\) are Hausdorff topological vector spaces, \(F:K\times K\to Y,\) and \(C:K\to 2^Y\) has solid open convex cone images, with \(0\notin C(f)\), for every \(f\in K.\) Under the assumptions on the bi-operator \(F\) of \(C(f)\)-pseudo monotonicity and hemicontinuity in the first argument, and natural quasi \(P\)-convexity in the second one, a Minty-type lemma is proved, providing the equivalence between the set of the Stampacchia and Minty solutions. An existence theorem is then established, using quite standard techniques. Other existence results are proved under \(B\)-\(C(f)\)-pseudo monotonicity assumptions, and in topological (not necessarily Hausdorff) vector spaces.
Reviewer: Rita Pini (Milano)

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
90C47 Minimax problems in mathematical programming
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
91B50 General equilibrium theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brézis, H., Equations et inequalities nolineares dans les espace vectorials on dualite, Ann. Inst. Fourier (Grenoble), 18, 115-175 (1968)
[2] Brézis, H.; Nierenberg, L.; Stampaccia, G., A remark on Ky Fan’s minimax principle, Boll. Un. Mat. Ital., 6, 293-300 (1972) · Zbl 0264.49013
[3] Chen, G.-Y., Existence of solutions for a vector variational inequalities: An extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl., 74, 445-456 (1992) · Zbl 0795.49010
[4] Chowdhury, M. S.R.; Tan, K.-K., Generalization of Ky Fan’s minimax inequality with applications to generalized variational inequalities for pseudomonotone operators and fixed point theorem, J. Math. Anal. Appl., 204, 910-929 (1996) · Zbl 0879.49011
[5] Domokos, A.; Kolumbán, J., Variational inequalities with operator solutions, J. Global Optim., 23, 99-110 (2002) · Zbl 1009.47064
[6] Fan, K., A generalization of Tychnoff’s fixed-point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701
[7] Hadjisavas, N.; Schaible, S., From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl., 96, 297-309 (1998) · Zbl 0903.90141
[8] Hu, S. H.; Papageorigiu, N. S., Handbook of Multivalued Analysis, vol. 1 (1997), Kluwer Academic: Kluwer Academic Dordrecht
[9] Kazmi, K. R., A variational principle for vector equilibrium problems, Proc. Indian Acad. Sci. (Math. Sci.), 111, 465-470 (2001) · Zbl 1019.49017
[10] Kazmi, K. R., On vector equilibrium problem, Proc. Indian Acad. Sci. (Math. Sci.), 110, 213-223 (2000) · Zbl 1032.90039
[11] Kazmi, K. R., Some remarks on vector optimization problems, J. Optim. Theory Appl., 96, 133-138 (1998) · Zbl 0897.90166
[12] Lee, G. M.; Kim, D. S.; Lee, B. S., On noncooperative vector equilibrium, Indian J. Pure Appl. Math., 27, 735-739 (1996) · Zbl 0858.90141
[13] Showalter, R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, (Math. Surveys Monogr., vol. 49 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0870.35004
[14] Yu, S. J.; Yao, J. C., On vector variational inequalities, J. Optim. Theory Appl., 89, 749-769 (1996) · Zbl 0848.49012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.