Common Fixed Points for Finite Families of Nonexpansive Mappings

Markov ([320])(see also Kakutani [270]) showed that if a commuting family of bounded linear transformations T_α, α ϵ ▵ (▵ an arbitrary index set) of a normed linear space E into itself leaves some nonempty compact convex subset K of E invariant, then the family has at least one common fixed point. (The actual result of Markov is more general than this but this version is adequate for our purposes).

Motivated by this result, De Marr studied the problem of the existence of a common fixed point for a family of nonlinear maps, and proved the following theorem.