×

On a generalized vector equilibrium problem with bounds. (English) Zbl 1218.49010

Summary: We prove the existence of a solution to the generalized vector equilibrium problem with bounds. We show that several known theorems from the literature can be considered as particular cases of our results, and we provide examples of applications related to best approximations in normed spaces and variational inequalities.

MSC:

49J27 Existence theories for problems in abstract spaces
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Isac, G.; Sehgal, V. M.; Singh, S. P., An alternate version of a variational inequality, Indian J. Math., 41, 1, 25-31 (1999) · Zbl 1034.49005
[2] Li, Jinlu, A lower and upper bounds version of a variational inequality, Appl. Math. Lett., 13, 47-51 (2000) · Zbl 1023.49003
[3] Chadli, O.; Chiang, Y.; Yao, J. C., Equilibrium problems with lower and upper bounds, Appl. Math. Lett., 15, 327-331 (2002) · Zbl 1175.90411
[4] Ansari, Q. H.; Yao, J.-C., A fixed point theorem and its application to the system of variational inequalities, Bull. Aust. Math. Soc., 59, 433-442 (1999) · Zbl 0944.47037
[5] Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann, 142, 305-310 (1961) · Zbl 0093.36701
[6] Al-Homidan, S.; Ansari, Q. H., System of quasi-equilibrium problem with lower and upper bounds, Appl. Math. Lett., 20, 323-328 (2007) · Zbl 1114.49006
[7] Fan, Liya, Weighted quasi-equilibrium problems with lower and upper bounds, Nonlinear Anal., 70, 2280-2287 (2009) · Zbl 1155.90453
[8] Ansari, Q. H.; Konnov, I. V.; Yao, J.-C., On generalized vector equilibrium problems, Nonlinear Anal., 47, 543-554 (2001) · Zbl 1042.90642
[9] Ansari, Q. H.; Siddiqi, A. H.; Wu, S. Y., Existence and duality of generalized vector equilibrium problems, J. Math. Anal. Appl., 259, 115-126 (2001) · Zbl 1018.90041
[10] Ansari, Q. H.; Yao, J.-C., An existence result for the generalized vector equilibrium problem, Appl. Math. Lett., 12, 8, 53-56 (1999) · Zbl 1014.49008
[11] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-146 (1994) · Zbl 0888.49007
[12] Iusem, A.; Sosa, W., New existence results for the equilibrium problem, Nonlinear Anal., 52, 621-635 (2003) · Zbl 1017.49008
[13] Lin, L.-J.; Park, S., On some generalized quasi-equilibrium problems, J. Math. Anal. Appl., 224, 167-181 (1998) · Zbl 0924.49008
[14] Mitrović, Z. D., On scalar equilibrium problem in generalized convex space, J. Math. Anal. Appl., 330, 451-461 (2007) · Zbl 1117.47062
[15] Park, S., New version of the Fan-Browder fixed point theorem and existence of economic equilibria, Fixed Point Theory Appl., 37, 149-158 (2004) · Zbl 1078.54026
[16] Yuan, G. X.Z., (KKM Theory and Applications in Nonlinear Analysis. KKM Theory and Applications in Nonlinear Analysis, Pure and Applied Mathematics (1999), Marcel Dekker: Marcel Dekker New York), p. 621 · Zbl 0936.47034
[17] Fan, K., Some properties of convex sets related to fixed points theorems, Math. Ann, 266, 519-537 (1984) · Zbl 0515.47029
[18] Wan, A. H.; Fu, J. Y.; Mao, W. H., On generalized vector equilibrium problems, Acta Math. Appl. Sin. Engl. Ser., 22, 21-26 (2006) · Zbl 1114.49014
[19] Farajzadeh, A. P., On the generalized vector equilibrium problems, J. Math. Anal. Appl., 344, 999-1004 (2008) · Zbl 1147.49005
[20] Yang, H.; Yu, J., Essential component of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Lett., 15, 553-560 (2002) · Zbl 1016.91008
[21] Huang, N. J.; Li, J.; Thompson, H. B., Implicit vector equilibrium problems with applications, Math. Comput. Modelling, 37, 1343-1356 (2003) · Zbl 1080.90086
[22] Li, Jinlu, A general result proved by Fan-KKM Theorem and its applications to a variational inequality, approximation theory and fixed point theory, Far East J. of Math. Sci., 299-312 (1999), (Special Volume, Part III) · Zbl 0985.47047
[23] Fan, K., Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112, 234-240 (1969) · Zbl 0185.39503
[24] Prolla, J. B., Fixed point theorems for set-valued mappings and existence of best approximants, Numer. Funct. Anal. Optim., 5, 449-455 (1982-83) · Zbl 0513.41015
[25] Browder, F. E., A new generalization of the Schauder fixed point theorem, Math. Ann., 174, 285-290 (1967) · Zbl 0176.45203
[26] Karamardian, S., Generalized complementarity problem, J. Optim. Theory Appl., 8, 161-168 (1971) · Zbl 0218.90052
[27] Behera, A.; Panda, G. K., A generalization of Browder’s theorem, Bull. Inst. Math. Acad. Sin., 21, 183-186 (1993) · Zbl 0780.90098
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.