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Collective fixed points and maximal elements with applications to abstract economies. (English) Zbl 1051.54028
Summary: We first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is $\Phi$-condensing. As an application of our collective fixed points theorem, we derive a coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of $L_S$-majorized multivalued maps whose product is $\Phi$-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not $L_S$-majorized but their product is $\Phi$-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54C60 Set-valued maps (general topology) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 91B50 General equilibrium theory in economics 47N10 Applications of operator theory in optimization, convex analysis, programming, economics
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##### References:
 [1] Ansari, Q. H.; Yao, J. C.: A fixed point theorem and its applications to the system of variational inequalities. Bull. austral. Math. soc. 54, 433-442 (1999) · Zbl 0944.47037 [2] Ansari, Q. H.; Idzik, A.; Yao, J. C.: Coincidence and fixed point theorems with applications. Topol. methods nonlinear anal. 15, 191-202 (2000) · Zbl 1029.54047 [3] Borglin, A.; Keiding, H.: Existence of equilibrium actions and of equilibrium: a note on the ’new’ existence theorem. J. math. Econom. 3, 313-316 (1976) · Zbl 0349.90157 [4] Chebli, S.; Florenzano, M.: Maximal elements and equilibria for condensing correspondences. Nonlinear anal. 38, 995-1002 (1999) · Zbl 0946.47038 [5] Deguire, P.; Lassonde, M.: Families selections. Topol. methods nonlinear anal. 5, 261-269 (1995) · Zbl 0877.54017 [6] Deguire, P.; Tan, K. K.; Yuan, G. X. Z: The study of maximal element, fixed point for ls-majorized mappings and their applications to minimax and variational inequalities in product spaces. Nonlinear anal. 37, 933-951 (1999) · Zbl 0930.47024 [7] Ding, X. P.: Best approximation and coincidence theorems. J. sichuan normal univ. Nat. sci. 18, 103-113 (1995) [8] Djafari-Rouhani, B.; Tarafdar, E.; Watson, P. J.: Fixed point theorems, coincidence theorems and variational inequalities. Lecture notes in econom. And math. Systems 502, 183-188 (2001) · Zbl 0994.47056 [9] Fitzpatrick, P. M.; Petryshyn, P. M.: Fixed point theorems for multivalued noncompact acyclic mappings. Pacific J. Math. 54, 17-23 (1974) · Zbl 0312.47047 [10] Lan, K.; Webb, J.: New fixed point theorems for a family of mappings and applications to problems on sets with convex sections. Proc. amer. Math. soc. 126, 1127-1132 (1998) · Zbl 0891.46004 [11] L.J. Lin, Z.T. Yu, Q.H. Ansari, L.-P. Lai, Fixed point and maximal element theorem for a family of multivalued maps and their applications to generalized abstract economics and minimax inequalities, J. Math. Anal. Appl., in press [12] Lin, L. J.; Park, S.; Yu, Z. T.: Remarks on fixed points, maximal elements and equilibria of generalized games. J. math. Anal. appl. 233, 581-596 (1999) · Zbl 0949.91004 [13] Lin, L. J.: Applications of a fixed point theorem in G-convex space. Nonlinear anal. 46, 601-608 (2001) · Zbl 1001.47041 [14] Mehta, G.: Maximal elements of condensing preference maps. Appl. math. Lett. 3, 69-71 (1990) · Zbl 0717.47020 [15] Tarafdar, E.: On nonlinear variational inequalities. Proc. amer. Math. soc. 671, 95-98 (1997) · Zbl 0369.47029 [16] Tarafdar, E.: A fixed point theorem equivalent to the Fan--knaster--Kuratowski--mazurkiewirz theorem. J. math. Anal. appl. 128, 475-479 (1987) · Zbl 0644.47050 [17] Tarafdar, E.: A fixed point theorem and equilibrium point of an abstract economy. Math. econom. 20, 211-218 (1991) · Zbl 0718.90014 [18] Yannelis, N. C.; Prabhakar, N. D.: Existence of maximal elements and equilibria in linear topological spaces. J. math. Econom. 12, 233-245 (1983) · Zbl 0536.90019 [19] Yuan, G. X. Z: KKM theory and applications in nonlinear analysis. (1999) · Zbl 0936.47034