Some extensions of the Brouwer fixed point theorem

We study the existence of fixed points for continuous maps f from an n-ball X in Rⁿ to Rⁿ with n≥ 1. We show that f has a fixed point if, for some absolute retract Y⊂∂ X, f(Y)⊂ X and ∂ X-Y is an $(f, X)$-blockading set. For n≥ 2, let D be an n-ball in X and Y be an $(n-1)$-ball in ∂ X. Relying on the result just mentioned, we show the existence of a fixed point of f, if D and Y are well placed and behave well under f, and deg(fD)=- deg(f∂ Y), where fD=f|D: D → Rⁿ and f∂ Y=f|∂ Y: ∂ Y → ∂ Y. The degree deg(fD) of fD is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem.