We consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X , i.e. ‖ T I x − I T x ‖ ≤ ‖ I x − T x ‖       for       any       x       in       X , and satisfy the inequality ‖ T x − T y ‖ ≤ a ‖ I x − I y ‖ + ( 1 − a ) max { ‖ T x − I x ‖ , ‖ T y − I y ‖ } for all x , y in C , where 0 < a < 1 . It is proved that if I is linear and non-expansive in C and such that I C contains T C , then T and I have a unique common fixed point in C .