# Common Fixed Points for Finite Families of Nonexpansive Mappings

Chapter

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.

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