×

Equilibrium problems with applications to eigenvalue problems. (English) Zbl 1141.49306

Summary: We consider equilibrium problems and introduce the concept of \((S)_+\) condition for bifunctions. Existence results for equilibrium problems with the \((S)_+\) condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by [X. P. Ding and E. Tarafdar, Appl. Math. Lett. 8, No. 1, 31–36 (1995; Zbl 0824.49009)] and the generalized variational inequality studied by P. Cubiotti and J.-C. Yao [Comput. Math. Appl. 29, No. 12, 49–56 (1995; Zbl 0857.47038 )]. Finally, applications to a class of eigenvalue problems are given.

MSC:

49J40 Variational inequalities
49J35 Existence of solutions for minimax problems
90C47 Minimax problems in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DING, X. P., and TARAFDAR, E., Existence and Uniqueness of Solutions for a General Nonlinear Variational Inequality, Applied Mathematics Letters, Vol. 8, pp. 31–36, 1995. · Zbl 0824.49009 · doi:10.1016/0893-9659(94)00106-M
[2] CUBIOTTI, P., and YAO J. C., Multivalued (S) 1 C Operators and Generalized Variational Inequalities, Computers and Mathematics with Applications, Vol. 29, pp. 49–56, 1995. · Zbl 0857.47038 · doi:10.1016/0898-1221(95)00057-6
[3] BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994. · Zbl 0888.49007
[4] HADJISAVVAS, N., and SCHAIBLE, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996. · Zbl 0904.49005 · doi:10.1007/BF02192248
[5] CHADLI, O., CHBANI, Z., and RIAHI, H., Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discrete and Continuous Dynamical Systems, Vol. 5, pp. 185–195, 1999. · Zbl 0949.49008
[6] ANSARI, Q. H., WONG, N. C., and YAO, J. C., The Existence of Nonlinear Inequalities, Applied Mathematics Letters, Vol. 12, pp. 89–92, 1999. · Zbl 0940.49010 · doi:10.1016/S0893-9659(99)00062-2
[7] CHADLI, O., and YAO, J.C., Regularized Equilibrium Problems and Application to Noncoercive Hemivariational Inequalities, Preprint, 2002. · Zbl 1107.91067
[8] BIANCHI, M., and SCHAIBLE, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996. · Zbl 0903.49006 · doi:10.1007/BF02192244
[9] KONNOV, I. V., and YAO, J.C., On the Generalized Vector Variational Inequality Problem, Journal of Mathematical Analysis and Applications, Vol. 206, pp. 42–58, 1997. · Zbl 0878.49006 · doi:10.1006/jmaa.1997.5192
[10] CHADLI, O., CHBANI, Z., and RIAHI, H., Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299–323, 2000. · Zbl 0966.91049 · doi:10.1023/A:1004657817758
[11] ZHOW, J. X., and CHEN, G., Diagonal Convexity Conditions for Problems in Convex Analysis and Quasi-Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 132, pp. 213–225, 1988. · Zbl 0649.49008 · doi:10.1016/0022-247X(88)90054-6
[12] BROWDER, F.E., Nonlinear Eigenvalue Problems and Galerkin Approximations, Bulletin of American Mathematics Society, Vol. 74, pp. 651–656, 1968. · Zbl 0162.20302 · doi:10.1090/S0002-9904-1968-11979-2
[13] CUBIOTTI, P., General Nonlinear Variational Inequalities with (S) 1 C Operators, Applied Mathematics Letters, Vol. 10, pp. 11–15, 1997. · Zbl 0882.49010 · doi:10.1016/S0893-9659(97)00003-7
[14] FAN, K., A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematishe Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701 · doi:10.1007/BF01353421
[15] KÖTHE, G., Topological Vector Space I, Spinger-Verlag, Berlin, Germany, 1969.
[16] AUBIN, J. P., and EKELAND, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984. · Zbl 0641.47066
[17] BAEK, J.S., Properties of Solutions to a Class of Quasilinear Elliptic Problems, PhD Dissertation, Seoul National University, 1992.
[18] LE, V.K., Some Degree Calculations and Applications to Global Bifurcation Results of Variational Inequalities, Nonlinear Analysis, Vol. 37, pp. 473–500, 1999. · Zbl 0928.49009 · doi:10.1016/S0362-546X(98)00062-5
[19] SCHURICHT, F., Bifurcation from Minimax Solutions by Variational Inequalities in Convex Sets, Nonlinear Analysis, Vol. 26, pp. 91–112, 1996. · Zbl 0845.49004 · doi:10.1016/0362-546X(94)00181-G
[20] SZULKIN, A., Existence and Nonuniqueness of Solutions of a Noncoercive Elliptic Variational Inequality, Proceedings of Symposium in Pure Mathematics, Vol. 45, pp. 413–418, 1986. · doi:10.1090/pspum/045.2/843627
[21] YANG, J.F., Positive Solutions of Quasilinear Elliptic Obstacle Problems with Critical Exponents, Nonlinear Analysis, Vol. 25, pp. 1283–1306, 1995. · Zbl 0838.49008 · doi:10.1016/0362-546X(94)00247-F
[22] YANG, J.F., Regularity of Weak Solutions to Quasilinear Elliptic Obstacle Problems, Acta Mathematica Scientia, Vol. 17, pp. 159–166, 1997. · Zbl 0877.35023
[23] ZHOU, Y., and HUANG, Y., Existence of Solutions for a Class of Elliptic Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 250, pp. 187–195, 2000. · Zbl 0970.49009 · doi:10.1006/jmaa.2000.7069
[24] ZHOU, Y., and HUANG, Y., Several Existence Theorems for the Nonlinear Complementarity Problems, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 776–784, 1996. · Zbl 0873.90098 · doi:10.1006/jmaa.1996.0346
[25] ZEIDLER, E., Nonlinear Functional Analysis and Its Applications, Vol. 28, Springer Verlag, New York, NY, 1990. · Zbl 0684.47029
[26] KRASNOSELSKII, M.A., Topological Methods In the Theory of Nonlinear Integral Equations, Pergamon, Elmsford, NY, 1964.
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.