The main aim of this paper is to prove the existence of the fixed point of the sum of two operators in setting of the cone-normed spaces with the values of cone-norm belonging to an ordered locally convex space. We apply this result to prove the existence of global solution of the Cauchy problem with perturbation of the form x (t) = f [t, x (t)] + [t, x (t)] , t ∈ [0, ∞), x (0) = x 0 ∈ F 1 , in a scale of Banach spaces {(F s ,. s) : s ∈ (0, 1]}