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An extension of Mehta theorem with applications. (English) Zbl 1251.54059
In this paper, the authors extend the notion of measure of precompactness, introduced by C. J. Himmelberg, J. R. Porter and F. S. Van Vleck [Proc. Am. Math. Soc. 23, 635–641 (1969; Zbl 0195.14902)], to $l·c·$-spaces with precompact polytope and they obtain a generalization of Mehta’s theorem, see [G. Mehta, Appl. Math. Lett. 3, No. 2, 69–71 (1990; Zbl 0717.47020)]. Based on these results, the authors obtain a new fixed point result for a class of condensing mappings. This result is a generalization of a similar result from the paper E. Tarafdar and P. J. Watson, [Coincidence and the Fan-Glicksberg fixed point theorem in locally H-convex spaces. Research report, The University of Queensland, (1997)].
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 55M20 Fixed points and coincidences (algebraic topology)
##### References:
 [1] Ding, X. P.; Kim, W. K.; Tan, K. K.: Equilibria of non-compact generalized games with L$*$-majorized preference correspondences, J. math. Anal. appl. 164, No. 2, 508-517 (1992) · Zbl 0765.90092 · doi:10.1016/0022-247X(92)90130-6 [2] Himmelberg, C. J.; Porter, J. R.; Van Vleck, F. S.: Fixed point theorems for condensing multifunctions, Proc. amer. Math. soc. 23, 635-641 (1969) · Zbl 0195.14902 · doi:10.2307/2036602 [3] Huang, Y. Y.; Kuo, T. Y.; Jeng, J. C.: Fixed point theorems for condensing multimaps on locally G-convex spaces, Nonlinear anal. 67, 1522-1531 (2007) · Zbl 1122.47046 · doi:10.1016/j.na.2006.07.034 [4] Kelley, J. L.: General topology, (1975) [5] Kim, W. K.: A maximal element of condensing multimaps, J. chung. Math. soc. 6, 59-64 (1993) [6] Lin, L. J.; Ansari, Q. H.: Collective fixed points and maximal elements with applications to abstract economies, J. math. Anal. appl. 296, 455-472 (2004) · Zbl 1051.54028 · doi:10.1016/j.jmaa.2004.03.067 [7] Mehta, G.: Maximal elements of condensing preference maps, Appl. math. Lett. 3, No. 2, 69-71 (1990) · Zbl 0717.47020 · doi:10.1016/0893-9659(90)90017-6 [8] Tarafdar, E.: Fixed point theorems in H-spaces and equilibrium points of abstract economies, J. austral. Math. soc. Ser. A 53, 252-260 (1992) · Zbl 0761.47041 [9] Tarafdar, E.: A fixed point theorems in H-spaces and related results, Bull. aust. Math. soc. 42, 133-140 (1990) · Zbl 0714.47039 · doi:10.1017/S0004972700028239 [10] E. Tarafdar, P.J. Watson, Coincidence and the Fan-Glicksberg fixed point theorem in locally H-convex spaces, Research report, The University of Queensland, 1997. [11] Watson, P. J.: Coincidence and fixed points in locally G-convex spaces, Bull. aust. Math. soc. 59, 297-304 (1999) · Zbl 0943.47044 · doi:10.1017/S0004972700032901 [12] Wu, X.; Shen, Z. F.: Equilibrium of abstract economy and generalized quasi-variational inequality in H-spaces, Topology appl. 153, 123-132 (2005) · Zbl 1084.91036 · doi:10.1016/j.topol.2003.08.037