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Variational inclusions problems with applications to Ekeland’s variational principle, fixed point and optimization problems. (English) Zbl 1190.90212
Summary: We prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.

90C30Nonlinear programming
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
65K10Optimization techniques (numerical methods)
Full Text: DOI
[1] Aubin J.P. and Cellina A. (1994). Differential Inclusion. Springer, Berlin, Germany
[2] Birbil S., Bouza G., Frenk J.B.G. and Still G. (2006). Equilibrium constrained optimization problems. Eur. J. Oper. Res. 169: 1108--1127 · Zbl 1079.90152 · doi:10.1016/j.ejor.2004.07.075
[3] Deguire P., Tan K.K. and Yuan G.X.Z. (1999). The study of maximal elements, fixed point for L s ajorized mappings and quasi-variational inequalities in product spaces. Nonlinear Anal. 37: 933--951 · Zbl 0930.47024 · doi:10.1016/S0362-546X(98)00084-4
[4] Ekeland I. (1972). Remarques sur les problems variationals. I. C. R.. Acad. Sci. Paris Ser. A-B 275: 1057--1059 · Zbl 0249.49004
[5] Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324--353 · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0
[6] Fan K. (1952). Fixed point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38: 121--126 · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121
[7] Gopfer A., Tammer Chr. and Zálinescu C. (2000). On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39: 909--922 · Zbl 0997.49019 · doi:10.1016/S0362-546X(98)00255-7
[8] Hamel A.H. (2003). Phelp’s lemma, Dane’s drop theorem and Ekeland’s principle in locally convex spaces. Proc. Am. Math. Soc. 131(10): 3025--3038 · Zbl 1033.49010 · doi:10.1090/S0002-9939-03-07066-7
[9] Himmelberg C.J. (1972). Fixed point of compact multifunctions. J. Math. Anal. Appl. 38: 205--207 · Zbl 0225.54049 · doi:10.1016/0022-247X(72)90128-X
[10] Huang N.J. (2001). A new class of generalized set-valued implicit variational inclusions in Banach spaces with applications. Comput. Math. Appl. 41(718): 937--943 · Zbl 0998.47044 · doi:10.1016/S0898-1221(00)00331-X
[11] Isac G. (2004). Nuclear cones in product spaces, Pareto efficiency and Ekeland’s-type variational principle in locally convex spaces. Optimization 53(3): 253--268 · Zbl 1059.49024 · doi:10.1080/02331930410001720923
[12] Lin L.J. (2007). Mathematical program with systems of equilibrium constraints. J. Glob. Optim. 37: 275--286 · Zbl 1140.90038 · doi:10.1007/s10898-006-9049-5
[13] Lin L.J. and 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 · Zbl 1101.49022 · doi:10.1016/j.jmaa.2005.10.005
[14] Lin L.J. and Ansari Q.H. (2004). Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296: 455--472 · Zbl 1051.54028 · doi:10.1016/j.jmaa.2004.03.067
[15] Lin L.J. and Still G. (2006). Mathematical programs with equilibrium constraints: The existence of feasible points. Optimization 55: 205--216 · Zbl 1124.90033 · doi:10.1080/02331930600703635
[16] Lin, L.J.: Systems of generalized quasi-variational inclusions problems with applications to variational analysis and optimization problems. J. Glob. Optim. 38, (2007) · Zbl 1124.49006
[17] Lin L.J. and Yu Z.T. (2001). On some equilibrium problems for multimaps. J. Comput. Appl. Math. 129: 171--183 · Zbl 0990.49003 · doi:10.1016/S0377-0427(00)00548-3
[18] Lin L.J. and Hsu H.W. (2007). Existence theorems of vector quasi-equilibrium problems with equilibrium constraints. J. Glob. Optim. 37: 195--213 · Zbl 1149.90136 · doi:10.1007/s10898-006-9044-x
[19] Lin L.J. and Huang Y.J. (2007). Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal. Theory Methods Appl. 66: 1275--1289 · Zbl 1192.90192 · doi:10.1016/j.na.2006.01.025
[20] Lin, L.J., Du, W.S.: Systems of Equilibrium problems with applications to generalized Ekeland’s variational principles and systems of semi-infinite problems. J. Glob. Optim. (2007)
[21] Luc D.T. (1989). Theory of Vector Optimization, Lectures Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin, Germany · Zbl 0688.90051
[22] Luo Z.Q., Pang J.S. and Ralph D. (1997). Mathematical Program with Equilibrium Constraint. Cambridge University Press, Cambridge · Zbl 0898.90006
[23] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I,II. Springer, Berlin, Heidelberg, New York (2006)
[24] Robinson S.M. (1979). Generalized equation and their solutions, part I: basic theory. Math. Program. Study 10: 128--141 · Zbl 0404.90093
[25] Tan N.X. (1985). Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Math. Nachr. 122: 231--245 · doi:10.1002/mana.19851220123
[26] Wong C-W. (2007). A drop theorem without vector topology. J. Math. Anal. Appl. 329: 452--471 · Zbl 1118.46008 · doi:10.1016/j.jmaa.2006.06.086