Abstract
In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.
Similar content being viewed by others
References
Aliprantis C.D., Border K.C. (1999). Infinite Dimensional Analysis. Springer Verlag, Berlin, Germany
Ansari Q.H., Lin L.J., Su L.B. (2005). Systems of simultaneous generalized vector quasiequilibrium problems and applications. J. Optim. Theory Appl. 127: 27–44
Aubin J.P., Cellina A. (1994). Differential Inclusion. Springer Verlag, Berlin, Germany
Blum E., Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Students 63: 123–145
Caristi J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215: 241–251
Chen M.P., Lin L.J., Park S. (2003). Remarks on generalized quasi-equilibrium problems. Nonlinear Anal. 52: 433–444
Chen G.Y., Huang X.X., Yang X.Q. (2005). Vector Optimization. Springer-Verlag Berlin Heidelberg,, Germany
Dancš S., Hegedüs M., Medvegyev P. (1983). A general ordering and fixed point principle in complete metric spaces. Acta Sci. Math. 46: 381–388
Deguire P., Tan K.K., Yuan G.X.Z. (1999). The study of maximal elements, fixed point for Ls-majorized mappings and the quasi-variational inequalities in product spaces. Nonlinear Anal. 37: 933–951
Ekeland I. (1972). Remarques sur les problémes variationnels. I, C. R. Acad. Sci. Paris Sér. A-B. 275: 1057–1059
Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353
Ekeland I. (1979). Nonconvex minimization problems. Bull. Amer. Math. Soc. 1: 443–474
Fu J.Y., Wan A.H. (2002). Generalized vector equilibria problems with set-valued mappings. Math. Meth. Oper. Res. 56: 259–268
Göpfert A., Tammer Chr., Zălinescu C. (2000). On the vectorial Ekeland’s variational principle and minimal points in product Spaces. Nonlinear Anal. 39: 909–922
Hamel A. (1994). Remarks to an equivalent formulation of Ekeland’s variational principle. Optimization 31(3): 233–238
Hamel A. (2003). Phelps’ lemma, Dancš’ drop theorem and Ekeland’s principle in locally convex spaces. Pro. Amer. Math. Soc. 131: 3025–3038
Hamel A., Löhne A.: A minimal point theorem in uniform spaces, In Agarwal, R.P., O’Regan, D. (eds) Nonlinear Analysis and Applications to V. Lakshmikantham on his 80th Birthday, vol. 1, pp. 577–593. Kluwer Academic Publisher (2003)
Hyers D.H., Isac G., Rassias T.M. (1997). Topics in Nonlinear Analysis and Applications. World Scientific, Singapore
Kada O., Suzuki T., Takahashi W. (1996). Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jan. 44: 381–391
Lin L.J., Ansari Q.H. (2004). Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296: 455–472
Lin L.J., Du W.S. (2006). Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 323: 360–370
Lin L.J., Still G. (2006). Mathematical programs with equilibrium constraints: The existence of feasible points. Optimization 55(3): 205–219
Lin, L.J.: Existence theorems for bilevel problems with applications to mathematical programs with equilibrium constraints and semi-infinite problems J. Optim. Theory and Appl. (to appear)
Lin L.J., Yu Z.T., Kassay G. (2002). Existence of equilibria for multivalued mappings and its application to vectorial equilibria. J. Optim. Theory Appl. 114: 189–208
Lin L.J., Yu Z.T. (2001). On some equilibrium problems for multimaps. J. Computional Appl. Math. 129: 171–183
Luc D.T. (1989). Theory of Vector Optimization, Vol. 319. Lecture notes in economics and mathematical systems. Springer Verlag, Berlin, Germany
McLinden L. (1982). An application of Ekeland’s theorem to minimax problems. Nonlinear Anal. 6(2): 189–196
Oettli W., Théra M. (1993). Equivalents of Ekeland’s principle. Bull. Austral. Math. Soc. 48: 385–392
Park S. (2000). On generalizations of the Ekeland-type variational principles. Nonlinear Anal. 39: 881–889
Penot J.-P. (1986). The drop theorem, the petal theorem and Ekeland’s variational principle. Nonlinear Anal. 10(9): 813–822
Stein O., Still G. (2002). On generalized semi-infinite optimization and bilevel optimization. Europ. J. Operat Res 142(3): 442–462
Suzuki T. (2001). Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253: 440–458
Takahashi W. (2000). Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan
Tammer Chr. (1992). A generalization of Ekeland’s variational principle. Optimization 25: 129–141
Tan N.X. (1985). Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Mathematicsche Nachrichten 122: 231–245
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, LJ., Du, WS. Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems. J Glob Optim 40, 663–677 (2008). https://doi.org/10.1007/s10898-007-9146-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9146-0