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A further generalization of Ky Fan’s minimax inequality and its applications. (English) Zbl 0554.49005

This paper generalizes the well-known Ky Fan minimax inequality to non- compact convex sets in a Hausdorff topological vector space and to a pair of functions. The proof is based on the Brézis-Nirenberg-Stampacchia lemma about the non-empty intersection of all values of some KKM multi- valued map. By using this generalization of Ky Fan’s inequality a Dugundji-Granas variational inequality is generalized to set-valued maps and to non-compact sets in Hausdorff space. As a consequence two new fixed point theorems for pseudo-contractive and for nonexpansive maps in Hilbert space are proved.
Reviewer: Z.Wyderka

MSC:

49J35 Existence of solutions for minimax problems
47H10 Fixed-point theorems
49J40 Variational inequalities
54C60 Set-valued maps in general topology
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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