A fixed point theorem for mappings with a contractive iterate

THEOREM. Let (X, d) be a complete metric space, and f: X->X a continuous mapping satisfying the condition: there exists a k u for each xoE0X. LEMMA. If f: X)X be any mapping satisfying the condition of the above theorem then for each xEX, r(x) =sup,n d(fp(x), x) is finite. PROOF. Let xEX and let