Fuzzy games with a countable space of actions and applications to systems of generalized quasi-variational inequalities. (English) Zbl 1345.54066
Summary: We introduce an abstract fuzzy economy (generalized fuzzy game) model with a countable space of actions, and we study the existence of fuzzy equilibrium. As application, we prove the existence of solutions for the systems of generalized quasi-variational inequalities with random fuzzy mappings, defined in this paper. Our results bring novelty to the current literature by considering random fuzzy mappings whose values are fuzzy sets over complete countable metric spaces.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 91B50 | General equilibrium theory |
| 49J40 | Variational inequalities |
| 54A40 | Fuzzy topology |
| 54E50 | Complete metric spaces |
Keywords:
abstract fuzzy economy; fuzzy equilibrium; incomplete information; random fixed point; random quasi-variational inequalities; random fuzzy mappingReferences:
| [1] | Borglin A, Keiding H: Existence of equilibrium actions and of equilibrium: a note on the ‘new’ existence theorem.J. Math. Econ. 1976, 3:313-316. · Zbl 0349.90157 · doi:10.1016/0304-4068(76)90016-1 |
| [2] | Shafer W, Sonnenschein H: Equilibrium in abstract economies without ordered preferences.J. Math. Econ. 1975, 2:345-348. · Zbl 0312.90062 · doi:10.1016/0304-4068(75)90002-6 |
| [3] | Arrow KJ, Debreu G: Existence of an equilibrium for a competitive economy.Econometrica 1954, 22:265-290. · Zbl 0055.38007 · doi:10.2307/1907353 |
| [4] | Kim WK, Lee KH: Fuzzy fixed point and existence of equilibria of fuzzy games.J. Fuzzy Math. 1998, 6:193-202. · Zbl 0903.90181 |
| [5] | Zadeh LA: Fuzzy sets.Inf. Control 1965, 8:338-353. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X |
| [6] | Kim WK, Lee KH: Generalized fuzzy games and fuzzy equilibria.Fuzzy Sets Syst. 2001, 122:293-301. · Zbl 1022.91002 · doi:10.1016/S0165-0114(00)00073-7 |
| [7] | Patriche M: Equilibrium in Games and Competitive Economies. The Publishing House of the Romanian Academy, Bucharest; 2011. |
| [8] | Wang L, Je CY, Huang N: The robustness of generalized abstract fuzzy economies in generalized convex spaces.Fuzzy Sets Syst. 2011,176(1):56-63. · Zbl 1235.91123 · doi:10.1016/j.fss.2011.03.010 |
| [9] | Patriche M: Existence of equilibrium for an abstract economy with private information and a countable space of actions.Mathem. Rep. 2013,15(65)(3):233-242. · Zbl 1389.91062 |
| [10] | Huang NJ: Random generalized nonlinear variational inclusions for random fuzzy mappings.Fuzzy Sets Syst. 1999,105(3):437-444. · Zbl 0961.49004 · doi:10.1016/S0165-0114(97)00222-4 |
| [11] | Chang S-S, Zhu Y-G: On variational inequalities for fuzzy mappings.Fuzzy Sets Syst. 1989, 32:359-367. · Zbl 0677.47037 · doi:10.1016/0165-0114(89)90268-6 |
| [12] | Noor MA: Variational inequalities for fuzzy mappings. III.Fuzzy Sets Syst. 2000, 110:101-108. · Zbl 0940.49011 · doi:10.1016/S0165-0114(98)00131-6 |
| [13] | Park JY, Lee SY, Jeong JU: Completely generalized strongly quasivariational inequalities for fuzzy mapping.Fuzzy Sets Syst. 2000, 110:91-99. · Zbl 0940.49012 · doi:10.1016/S0165-0114(98)00106-7 |
| [14] | Ding X-P, Park J-Y: A new class of generalized non linear implicit quasi-variational inclusions with fuzzy mappings.J. Comput. Appl. Math. 2002, 138:243-257. · Zbl 0996.65067 · doi:10.1016/S0377-0427(01)00379-X |
| [15] | Gwinner J, Raciti F: On a class of random variational inequalities on random sets.Numer. Funct. Anal. Optim. 2006,27(5-6):619-636. · Zbl 1097.49008 · doi:10.1080/01630560600790819 |
| [16] | Gwinner J, Raciti F: Some equilibrium problems under uncertainty and random variational inequalities.Ann. Oper. Res. 2012, 200:299-319. · Zbl 1259.60070 · doi:10.1007/s10479-012-1109-2 |
| [17] | Noor MA, Elsanousi SA: Iterative algorithms for random variational inequalities.Panam. Math. J. 1993, 3:39-50. · Zbl 0845.49007 |
| [18] | Tan K-K, Yuan GX-Z: Random equilibria of random generalized games with applications to non-compact random quasi-variational inequalities.Topol. Methods Nonlinear Anal. 1995, 5:59-82. · Zbl 0843.90142 |
| [19] | Yuan, XZ, The study of minimax inequalities and applications to economies and variational inequalities, No. 132 (1988) |
| [20] | Chang S-S: Coincidence theorems and variational inequalities for fuzzy mappings.Fuzzy Sets Syst. 1994, 61:359-368. · Zbl 0830.47052 · doi:10.1016/0165-0114(94)90178-3 |
| [21] | Croitoru A, Văideanu C: On pointwise measurability of multifunctions.An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2009, 17:69-76. · Zbl 1249.26050 |
| [22] | Croitoru A: Fuzzy integral of measurable multifunctions.Iran. J. Fuzzy Syst. 2012,9(4):133-140. · Zbl 1260.28014 |
| [23] | Hu, S.; Papageorgiu, NS, I (1997), Dordrecht · doi:10.1007/978-1-4615-6359-4 |
| [24] | Klein E, Thompson C: Theory of Correspondences. Wiley-Interscience, New York; 1984. · Zbl 0556.28012 |
| [25] | Uea, NO; Kumam, P., A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces, No. 2010 (2010) · Zbl 1210.47102 |
| [26] | Patriche, M., Equilibrium of Bayesian fuzzy economies and quasi-variational inequalities with random fuzzy mappings, No. 2013 (2013) · Zbl 1285.91081 |
| [27] | Patriche, M., Bayesian abstract fuzzy economies, random quasi-variational inequalities with random fuzzy mappings and random fixed point theorems (2013) · Zbl 1285.91081 |
| [28] | Patriche, M., Bayesian abstract economy with a measure space of agents, No. 2009 (2009) · Zbl 1176.91079 |
| [29] | Patriche M: Bayesian abstract economies.Optimization 2011,60(4):451-461. · Zbl 1237.91155 · doi:10.1080/02331930903358830 |
| [30] | Nash JF: Equilibrium points inn-person games.Proc. Natl. Acad. Sci. USA 1950,36(1):48-49. · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48 |
| [31] | Yu H, Zhang Z: Pure strategy equilibria in games with countable actions.J. Math. Econ. 2007, 43:192-200. · Zbl 1280.91037 · doi:10.1016/j.jmateco.2006.09.008 |
| [32] | Aliprantis CD, Border KC: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin; 1994. · Zbl 0839.46001 · doi:10.1007/978-3-662-03004-2 |
| [33] | Fan K: Fixed-point and minimax theorems in locally convex topological linear spaces.Proc. Natl. Acad. Sci. USA 1952, 38:121-126. · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121 |
| [34] | Yuan XZ: Non-compact random generalized games and random quasi-variational inequalities.J. Appl. Math. Stoch. Anal. 1994,7(4):467-486. · Zbl 0821.47049 · doi:10.1155/S1048953394000377 |
| [35] | Yuan XZ: Remarks on quasi-variational inequalities and fixed points in locally convex topological vector spaces.Appl. Math. Lett. 1997,10(6):55-61. · Zbl 0893.49006 · doi:10.1016/S0893-9659(97)00105-5 |
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