Nonlinear Analysis 65 (2006) 918–932 www.elsevier.com/locate/na Equilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges K. Wlodarczyk ∗ , D. Klim Department of Nonlinear Analysis, Faculty of Mathematics, University of L´ od´ z, Banacha 22, 90-238 L´ od´ z, Poland Received 25 January 2005; accepted 5 October 2005 Abstract Let C be a nonempty subset of a topological vector space E . We state and prove new various fixed point theorems of Fan–Browder type for set-valued maps F : C → 2 E such that C ⊂ F (C ) (called expansive), without assuming that the sets C and F (C ) are convex or compact or equal, and E is Hausdorff. Let K be a convex subset of E and let C be a nonempty subset of K . Our proofs use a technique based on the investigations of the images of maps and restated for maps f : C × K → R ∪ {-∞, +∞} of G.X.-Z. Yuan’s results concerning the existence of equilibrium points and minimax inequalities for maps f : K × K → R ∪ {-∞, +∞}. Examples are provided. c  2005 Elsevier Ltd. All rights reserved. Keywords: Fixed point; Expansive set-valued map; Nonconvex and noncompact domain and range; Existence of equilibrium point; Topological vector space 1. Introduction In minimax theory the famous result of Knaster–Kuratowski–Mazurkiewicz [22] is the main ingredient and its infinite dimensional extension by Ky Fan [15] (called FKKM principle) has proved itself to be an elegant and powerful tool. Lemma 1.1 (FKKM Principle). Let X be an arbitrary set in a Hausdorff topological vector space E. To each x ∈ X let a closed set F (x ) in E be given such that F (x ) is compact for at least one x ∈ X . If, for each finite subset A of X , conv( A) ⊂  x ∈ A F (x ), then  x ∈ X F (x ) = ∅. ∗ Corresponding author. E-mail addresses: wlkzxa@math.uni.lodz.pl (K. Wlodarczyk), klimdr@math.uni.lodz.pl (D. Klim). 0362-546X/$ - see front matter c  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.10.006