×

Existence results for densely pseudomonotone variational inequalities. (English) Zbl 0974.49006

Let \(K\) be a nonempty convex subset of a Hausdorff topological vector space \(X\) and let \(f:K\rightarrow X^*\) be a nonlinear operator. The paper deals with the study of the following class of variational inequalities: find \(x_0\in K\) such that \(\langle f(x_0),x-x_0\rangle\geq 0\), for all \(x\in K\). Problems of this type were originally studied in the 60’s by Stampacchia. The author of this paper proves several existence results for the above class of densely pseudomonotone variational inequalities. There are also given some particular cases in reflexive Banach spaces which include previously known results. In another section of the paper, using recession directions, there are derived existence criteria for monotone and densely psudomonotone variational inequalities. Some of these results are applied in the last part of the paper to find generalized versions of the Browder-Minty theorem on the surjectivity of monotone operators.

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baiocchi, C.; Capelo, A., Variational and Quasivariational Inequalities: Applications to Free Boundary Problems (1984), Wiley: Wiley New York · Zbl 0551.49007
[2] Baiocchi, C.; Buttazzo, G.; Gastaldi, F.; Tomarelli, F., General existence results for unilateral problems in continuum mechanics, Arch. Rational Mech. Anal., 100, 149-189 (1988) · Zbl 0646.73011
[3] Brezis, H., Opérateurs Maximaux Monotones et Semi-groupes de Contraction dans les Espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam · Zbl 0252.47055
[4] Cottle, R. W.; Yao, J. C., Pseudomonotone complementary problems in Hilbert spaces, J. Optim. Theory Appl., 78, 281-295 (1992) · Zbl 0795.90071
[5] Crouzeix, J. P.; Ferland, J. A., Criteria for differentiable generalized monotone maps, Math. Programming, 75, 399-406 (1996) · Zbl 0892.49008
[6] Dedieu, J. P., Cone asymptote d’un ensemble non convexe, Application à l’optimisation, C.R. Acad. Sci. Paris, 287, 501-503 (1977) · Zbl 0363.46003
[7] Fan, K., A generalizaton of Tychonoff’s fixed point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701
[8] Giannessi, F.; Maugeri, A., Variational Inequalities and Network Equilibrium Problems (1995), Plenum: Plenum New York · Zbl 0834.00044
[9] Glowinski, R.; Lions, J. L.; Tremolieres, R., Analyses Numériques des Inéquations Variationelles: Méthodes Mathématiques de l’Informatique (1976), Dunod: Dunod Paris
[10] Granas, A.; Lassonde, M., Sur un principe géométrique en analyse convexe, Studia Math., 101, 1-18 (1991) · Zbl 0826.49003
[11] Hadjisavvas, N.; Schaible, S., Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90, 95-111 (1996) · Zbl 0904.49005
[12] Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problem: A Survey of theory, algorithms and applications, Math. Programming, 48, 161-220 (1990) · Zbl 0734.90098
[13] Hartman, G. J.; Stampacchia, G., On some nonlinear elliptic differential functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102
[14] Holmes, R. B., Geometric Functional Analysis and Its Applications (1975), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0336.46001
[15] Isac, G., Complementarity Problems. Complementarity Problems, Lecture Notes in Math., 1528 (1992), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0795.90072
[16] Karamardian, S., Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18, 445-454 (1976) · Zbl 0304.49026
[17] Karamardian, S.; Schaible, S., Seven kinds of monotone maps, J. Optim. Theory Appl., 66, 37-46 (1990) · Zbl 0679.90055
[18] Karamardian, S.; Schaible, S.; Crouzeix, J. P., Characterizations of generalized monotone maps, J. Optim. Theory Appl., 76, 399-413 (1993) · Zbl 0792.90070
[19] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[20] Luc, D. T., Theory of Vector Optimization. Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319 (1989), Springer-Verlag: Springer-Verlag New York/Berlin
[21] Luc, D. T., Recession maps and applications, Optimization, 27, 1-15 (1993) · Zbl 0823.46037
[22] Luc, D. T., Characterizations of quasiconvex functions, Bull. Austral. Math. Soc., 48, 393-405 (1993)
[23] Luc, D. T., On generalized convex nonsmooth functions, Bull. Austral. Math. Soc., 49, 139-149 (1994) · Zbl 0811.90096
[24] D. T. Luc, and, J. P. Penot, Convergence of asymptotic directions, preprint, University of Pau, 1999.; D. T. Luc, and, J. P. Penot, Convergence of asymptotic directions, preprint, University of Pau, 1999.
[25] Luc, D. T.; Schaible, S., Generalized monotone nonsmooth maps, Convex Anal., 3, 195-205 (1996) · Zbl 0870.90091
[26] Noor, M. A., Some recent advances in variational inequalities. Part I. Basic concepts, New Zealand J. Math., 26, 53-80 (1997) · Zbl 0886.49004
[27] Noor, M. A., Some recent advances in variational inequalities. Part II. Other concepts, New Zealand J. Math., 26, 229-255 (1997) · Zbl 0889.49006
[28] Noor, M. A.; Noor, K. I.; Rassias, Th. M., Some aspects of variational inequalities, J. Comput. Appl. Math., 47, 285-312 (1993) · Zbl 0788.65074
[29] Shih, M. H.; Tan, K. K., Browder-Stampacchia variational inequalities for multivalued monotone operators, J. Math. Anal. Appl., 134, 431-440 (1988) · Zbl 0671.47043
[30] Stampacchia, G., Formes bilinéaires coercives sur les ensembles convexes, C.R. Acad. Sci. Paris, 258, 4413-4416 (1964) · Zbl 0124.06401
[31] Thera, M., A note on the Hartman-Stampacchia theorem, (Lakshmikantham, V., Nonlinear Analysis and Applications (1987), Dekker: Dekker New York), 573-577 · Zbl 0644.49006
[32] Yao, J. C., Variational inequalities with generalized monotone operators, Math. Oper. Res., 19, 691-705 (1994) · Zbl 0813.49010
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.