banach operator pair and common fixed points for nonexpansive maps

let x be a normed space and t and i be two nonexpansive self-maps on the closed convex subset c⊂x. it is proved that if (i,t) is a banach operator pair t(c) is compact then f(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.