Applications of order-theoretic fixed point theorems to discontinuous quasi-equilibrium problems
Tóm tắt
In this paper, we apply order-theoretic fixed point theorems and isotone selection theorems to study quasi-equilibrium problems. Some existence theorems of solutions to quasi-equilibrium problems are obtained on Hilbert lattices, chain-complete lattices and chain-complete posets, respectively. In contrast to many papers on equilibrium problems, our approach is order-theoretic and all results obtained in this paper do not involve any topological continuity with respect to the considered mappings.
Từ khóa
Tài liệu tham khảo
Noor, MA, Oettli, W: On general nonlinear complementarity problems and quasi-equilibria. Matematiche 49, 313-331 (1994)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994)
Harker, PT: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81-94 (1991)
Harker, PT: A note on the existence of traffic equilibria. Appl. Math. Comput. 18, 277-283 (1986)
Lin, LJ, Park, S: On some generalized quasi-equilibrium problems. J. Math. Anal. Appl. 224, 167-181 (1998)
Park, S: Fixed points and quasi-equilibrium problems. Math. Comput. Model. 32, 1297-1304 (2000)
Fu, JY: Symmetric vector quasi-equilibrium problems. J. Math. Anal. Appl. 285, 708-713 (2003)
Khaliq, A: Implicit vector quasi-equilibrium problems with applications to variational inequalities. Nonlinear Anal. 63, 1823-1831 (2005)
Ding, XP: Quasi-equilibrium problems with applications to infinite optimization and constrained games in general topological spaces. Appl. Math. Lett. 13, 21-26 (2000)
Cubiotti, P: Existence of solutions for lower semicontinuous quasi-equilibrium problems. Comput. Math. Appl. 30, 11-22 (1995)
Noor, MA: Auxiliary principle for generalized mixed variational-like inequalities. J. Math. Anal. Appl. 215, 75-85 (1997)
Lin, LJ, Huang, YJ: Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal. 66, 1275-1289 (2007)
Bianchia, M, Kassayb, G, Pinic, R: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 1454-1464 (2007)
Ding, XP, Ding, TM: KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces. J. Math. Anal. Appl. 331, 1230-1245 (2007)
Al-Homidan, S, Ansari, QH: Fixed point theorems on product topological semilattice spaces, generalized abstract economies and systems of generalized vector quasi-equilibrium problems. Taiwan. J. Math. 15(1), 307-330 (2011)
Al-Homidan, S, Ansari, QH, Yao, J-C: Collectively fixed point and maximal element theorems in topological semilattice spaces. Appl. Anal. 96(6), 865-888 (2011)
Lin, L-J, Yu, Z-T, Ansari, QH, Lai, L-P: Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities. J. Math. Anal. Appl. 284, 656-671 (2003)
Lin, L-J, Ansari, QH: Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296, 455-472 (2004)
Fujimoto, T: An extension of Tarski’s fixed point theorem and its application to isotone complementarity problems. Math. Program. 28, 116-118 (1984)
Chitra, A, Subrahmanyam, P: Remarks on nonlinear complementarity problem. J. Optim. Theory Appl. 53, 297-302 (1987)
Borwein, J, Dempster, M: The linear order complementarity problem. Math. Oper. Res. 14, 534-558 (1989)
Nishimura, H, Ok, EA: Solvability of variational inequalities on Hilbert lattices. Math. Oper. Res. 37(4), 608-625 (2012)
Li, J, Ok, EA: Optimal solutions to variational inequalities on Banach lattices. J. Math. Anal. Appl. 388, 1157-1165 (2012)
Li, J, Yao, JC: The existence of maximum and minimum solutions to general variational inequalities in Hilbert lattices. Fixed Point Theory Appl. (2011). doi:10.1155/2011/904320
Ok, EA: Order theory (2011). https://files.nyu.edu/eo1/public/books.html
Meyer-Nieberg, P: Banach Lattices. Universitext. Springer, Berlin (1991)
Smithson, RE: Fixed points of order preserving multifunctions. Proc. Am. Math. Soc. 28, 304-310 (1971)
Tarski, A: The lattice theoretical fixed point theorem and its applications. Pac. J. Math. 5, 285-309 (1955)
Li, J: Several extensions of the Abian-Brown fixed point theorem and their applications to extended and generalized Nash equilibria on chain-complete posets. J. Math. Anal. Appl. 409, 1084-1092 (2014)