From the text: “We derive a very general fixed point theorem for nonexpansive mappings defined in product spaces from a theorem due to R. E. Bruck. Our result is inspired by, and analogous to, one recently obtained by T. Kuczumow [Proc. Am. Math. Soc. 108, No. 3, 727-729 (1990; Zbl 0696.47051)].”
One of the results is Theorem 3: Let \(K\) be a separable closed convex subset of a Banach space \(X\) and let \(M\) be a separable complete metric space. Suppose \(T\) is a nonexpansive mapping on \(K\times M\) with values in \(K\). Then there exists a nonexpansive mapping \(r\) on \(K\times M\) with values in \(K\) such that for all \((x,u)\) in \(K\times M\), (a) \(T(r(x,u),u)= r(x,u)\), und (b) if \(T(x,u)=x\) then \(r(x,u)=x\).