Let S be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in R . Any pair of points in S of the form x,−x is a pair of antipodes in S. Let ∆ be the d-simplex formed by the convex hull of the standard unit vectors in R. Equivalently, ∆ = {(x1, ..., xd+1) : ∑ i xi = 1, xi ≥ 0}. The following are two classical results about closed covers of these topological spaces:

Economic Models and Generalized Games 229 4.

A new generalized nonlinear variational-like inequality is introduced and studied. By applying the auxiliary principle technique and KKM theory, we construct a new iterative algorithm for solving the generalized nonlinear variational-like inequality. By means of the Banach fixed-point theorem, we establish the existence and uniqueness of solution for the generalized nonlinear variational-like i...

We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lustern...

A d-partite hypergraph is called fractionally balanced if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using topological version of Hall’s theorem we prove lower bounds the matching number such hypergraphs. These yield rainbow versions KKM for products simplices, which turn are used to obtain some results multiple-cak...