Banach operator pair and common fixed points for nonexpansive maps

Let XX be a normed space and TT and II be two nonexpansive self-maps on the closed convex subset C⊂XC⊂X. It is proved that if (I,T)(I,T) is a Banach operator pair and T(C)¯ is compact then F(I,T)≠0̸F(I,T)≠0̸. A family of nonexpansive maps is also investigated; the well-known De Marr’s fixed point theorem is extended to the noncommuting case by introducing a notion of Banach operator family. As an application of the theorem, the problem for the common fixed point in invariant approximation for convex set is solved directly in a quite different way from the others before.