Cyclical coincidence of multivalued maps
We prove the following result using Brouwer's fixed-point theorem for a simplex. \(\forall\) \(i\in {\mathbb{Z}}_ m\), let \(X_ i\) be a nonempty convex subset of a topological vector space and \(T_ i: X_ i\to 2^{X_{i+1}}\) have nonempty convex values. If each \(T_ i\) is either ''Browder-Fan'' or of ''Kakutani type'' then \(\exists (x_ 0,...,x_{m-1})\subset X_ 0\times...\times X_{m-1}\) such that, \(\forall\) \(i\in {\mathbb{Z}}_ m\), \(x_{i+1}\in T_ iX_ i.\) The two kind of maps can be mixed in any order. The definitions will require that some (but possibly not all) of the sets \(X_ i\) are compact and some (but possibly not all) of the underlying topological vector spaces are locally convex. This result extends fixed-point and coincidence theorems of Browder, Fan and Takahashi.
Brouwer's fixed-point theorem for a simplex
scientific article; zbMATH DE number 3998598