Almost-fixed-point and fixed-point theorems for discrete-valued maps (Q1186120)

The authors prove three theorems which in a certain sense are discrete analogues of fixed-point theorems in analysis. In fact, in the proofs, generalizations of Brouwer's and Kakutani's fixed-point theorems are used. The results are of the following type: Let \(f\) be a map from \(I\) into \(\mathbb{Z}^ d\), where \(I\) is an interval in \(\mathbb{Z}^ d\). If \(f\) satisfies certain boundary conditions then \(f\) has an ``almost-fixed- point'', meaning that, in a certain sense, \(f(x)\) is not too far from \(x\).