Abstract
The purpose of this paper is to present a unifying approach to study various models of equilibrium theory and variational inclusions. A simple condition is established for the existence of solutions of variational relations and is applied to a number of variational problems.
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Ansari, A.H., Yao, J.C.: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 12, 53–56 (1999)
Aussel, A., Luc, D.T.: Existence conditions in general quasimonotone variational inequalities. Bull. Aust. Math. Soc. 71, 285–303 (2005)
Chowdhury, M.S.R., Tan, K.K.: Generalized variational inequalities for quasimonotone operators and applications. Bull. Pol. Acad. Sci. Math. 45, 25–54 (1997)
Daniilidis, A., Hadjisavvas, N.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 473–481 (1996)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II. Springer, New York (2003)
Flores-Bazan, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case. SIAM J. Optim. 11, 675–690 (2000)
Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)
Hadjisavvas, N., Schaible, S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 95–111 (1996)
Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)
Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Konnov, I.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 99, 165–181 (1998)
Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308 (2001)
Noor, M.A.: Some recent advances in variational inequalities, Part I, basic concepts. N.Z. J. Math. 26, 53–80 (1997)
Stampacchia, G.: Formes bilinaires coercives sur les ensembles convexes. C.R. Acad. Sci. Paris 258, 4413–4416 (1964)
Tian, G.: Generalizations of the FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, price equilibrium, and complementarity. J. Math. Anal. Appl. 170, 457–471 (1992)
Yao, J.C.: Multivalued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)
Zhou, J.X., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. 132, 213–225 (1988)
Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007)
Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic, Dordrecht (2000)
Guerraggio, A., Tan, N.X.: On general vector quasioptimization problems. Math. Methods Oper. Res. 55, 347–358 (2002)
Kalmoun, E.M., Riahi, H., Tanaka, T.: On the vector equilibrium problems: remarks on a general existence theorem and applications. Nihonkai Math. J. 12, 149–164 (2001)
Lin, L.J., Chen, H.L.: The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Glob. Optim. 32, 135–157 (2005)
Luc, D.T., Tan, N.X.: Existence conditions in variational inclusions with constraints. Optimization 53, 505–515 (2004)
Tuan, L.A., Sach, P.H.: Existence of solutions of generalized quasivariational inequalities with set-valued maps. Acta Math. Vietnam. 29, 309–316 (2004)
Tuan, L.A., Sach, P.H.: On some generalized vector equilibrium problems with set-valued maps. Acta Math. Vietnam. 32, 15–32 (2007)
Aubin, J.P., Celina, A.: Differential Inclusion. Springer, Berlin (1994)
Lin, L.J.: Systems of variational inclusions, disclusions and differential inclusions problems with applications. Preprint (2007)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Browder, F.E.: The fixed point theory of multivalued mappings in topological vector space. Math. Ann. 177, 238–301 (1968)
Lin, L.J., Chen, L.F., Ansari, Q.H.: Generalized abstract economy and systems of generalized vector quasi-equilibrium problems, J. Comput. Appl. Math. (2008, in press)
Kim, W.K., Yuan, G.X.Z.: Existence of equilibria for generalized games and generalized social systems with coordination. Nonlinear Anal. 45, 169–188 (2001)
Michael, E.: Continuous selections. I. Ann. Math. 63, 361–382 (1956)
Park, S.: Fixed point theorems for better admissible multimaps on almost convex sets. J. Math. Anal. Appl. 329, 690–702 (2007)
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Luc, D.T. An Abstract Problem in Variational Analysis. J Optim Theory Appl 138, 65–76 (2008). https://doi.org/10.1007/s10957-008-9371-9
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DOI: https://doi.org/10.1007/s10957-008-9371-9