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Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. (English) Zbl 1127.49015
Summary: This paper introduces a vectorial form of equilibrium version of Ekeland-type variational principle. Some equivalent results to our variational principle are given. As applications, we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi-metric spaces with a $$W$$-distance. Caristi-Kirk fixed point theorem for multivalued maps is also established in a more general setting.

##### MSC:
 49J53 Set-valued and variational analysis 49J27 Existence theories for problems in abstract spaces 47H10 Fixed-point theorems
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##### References:
 [1] Ansari, Q.H., Vector equilibrium problems and vector variational inequalities, (), 1-16 · Zbl 1198.90343 [2] Ansari, Q.H.; Konnov, I.V.; Yao, J.C., Existence of a solution and variational principles for vector equilibrium problems, J. optim. theory appl., 110, 3, 481-492, (2001) · Zbl 0988.49004 [3] Ansari, Q.H.; Konnov, I.V.; Yao, J.C., Characterizations of solutions for vector equilibrium problems, J. optim. theory appl., 113, 3, 435-447, (2002) · Zbl 1012.90055 [4] Aubin, J.-P., Mathematical methods of game and economic theory, (1979), North-Holland Amsterdam · Zbl 0452.90093 [5] Aubin, J.-P.; Ekeland, I., Applied nonlinear analysis, (1984), Wiley New York [6] Aubin, J.-P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser Boston [7] Bianchi, M.; Hadjisavvas, N.; Schaible, S., Vector equilibrium problems with generalized monotone bifunctions, J. optim. theory appl., 92, 527-542, (1997) · Zbl 0878.49007 [8] Bianchi, M.; Kassay, G.; Pini, R., Existence of equilibria via Ekeland’s principle, J. math. anal. appl., 305, 502-512, (2005) · Zbl 1061.49005 [9] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 1-4, 123-145, (1994) · Zbl 0888.49007 [10] Caristi, J.; Kirk, W.A., Geometric fixed point theory and inwardness conditions, (), 74-83 [11] Chen, G.Y.; Huang, X.X., Stability results for Ekeland’s &z.epsiv; variational principle for vector valued functions, Math. methods oper. res., 48, 97-103, (1998) · Zbl 0941.49011 [12] Chen, G.Y.; Huang, X.X., Ekeland’s &z.epsiv;-variational principle for set-valued mappings, Math. methods oper. res., 48, 181-186, (1998) · Zbl 0936.49012 [13] Chen, G.Y.; Yang, X.Q.; Yu, H., A nonlinear scalarization function and generalized quasi-vector equilibrium problems, J. global optim., 32, 451-466, (2005) · Zbl 1130.90413 [14] Ekeland, I., Sur LES probléms variationnels, C. R. acad. sci. Paris, 275, 1057-1059, (1972) · Zbl 0249.49004 [15] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015 [16] Ekeland, I., On convex minimization problems, Bull. amer. math. soc. (N.S.), 1, 3, 445-474, (1979) [17] Facchinei, F.; Pang, J.-S., Finite dimensional variational inequalities and complementarity problems, I, (2003), Springer-Verlag New York · Zbl 1062.90001 [18] Feng, Y.Q.; Liu, S.Y., Fixed point theorems for multivalued contractive mappings and multi-valued Caristi type mappings, J. math. anal. appl., 317, 103-112, (2006) · Zbl 1094.47049 [19] Flores-Bazán, F., Existence theory for finite-dimensional pseudomonotone equilibrium problems, Acta appl. math., 77, 249-297, (2003) · Zbl 1053.90110 [20] Gerth (Tammer), Chr.; Weidner, P., Nonconvex separation theorems and some applications in vector optimization, J. optim. theory appl., 67, 297-320, (1990) · Zbl 0692.90063 [21] Giannessi, F., Theorems of the alternative, quadratic programs and complementarity problems, (), 151-186 · Zbl 0484.90081 [22] () [23] Göpfert, A.; Tammer, Chr.; Riahi, H.; Zălinescu, C., Variational methods in partially ordered spaces, (2003), Springer-Verlag New York [24] Göpfert, A.; Tammer, Chr.; Zălinescu, C., On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear anal., 39, 909-922, (2000) · Zbl 0997.49019 [25] Kada, O.; Suzuki, T.; Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. japonica, 44, 2, 381-391, (1996) · Zbl 0897.54029 [26] Lin, L.J., Existence results for primal and dual generalized vector equilibrium problems with applications to generalized semi-infinite programming, J. global optim., 33, 579-595, (2005) · Zbl 1097.90069 [27] Luc, T.D., Theory of vector optimization, Lecture notes in econom. and math. systems, vol. 319, (1989), Springer-Verlag Berlin [28] Oettli, W.; Théra, M., Equivalents of Ekeland’s principle, Bull. austral. math. soc., 48, 385-392, (1993) · Zbl 0793.54025 [29] Park, S., On generalizations of the Ekeland-type variational principles, Nonlinear anal., 39, 881-889, (2000) · Zbl 1044.49500 [30] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Yokohama, Japan [31] Tammer, Chr., A generalization of Ekeland’s variational principle, Optimization, 25, 129-141, (1992) · Zbl 0817.90098
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