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Fixed points, coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory. (English) Zbl 1157.54017

In 1961, Ky Fan [Math. Ann. 142, 305–310 (1961; Zbl 0093.36701)] gave a generalization of the classical Knaster-Kuratowski-Mazurkievicz theorem and established an elementary but very basic “geometry” lemma for multivalued maps. Afterwards, in the year 1968, F. E. Browder [ibid. 177, 283–301 (1968; Zbl 0176.45204)] obtained a fixed point theorem which is the more convenient form of Fan’s lemma. Since then this result has been known as the Fan-Browder fixed point theorem.
The authors establish the existence of fixed point theorems for multivalued maps in generalized convex spaces. To establish this result, they use the Fan-Browder fixed point theorem [Theorem 4] of Z. Yu and L. Lin [Nonlinear Anal., Theory Methods Appl. 52A, No. 2, 445–455 (2003; Zbl 1033.54011)]. This new established result is further used to derive several coincidence theorems and existence theorems for maximal elements. In the next section of the paper, the authors also give applications of these results in the area of equilibrium problems and minimax theory.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
91B50 General equilibrium theory
49J35 Existence of solutions for minimax problems
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