UDC 517.9 Let ~'be a Bmlach space over a field K E {R,C}, and let B(a,p) be the open ball in ~'with center a E ~" and radius p > 0. By Lipk(~ a, p), k E NU {0}, we denote the Banach space of continuous mappings (operators) defined on the closed ball B(a, p) = B(a, p), ranging in ~'. and k times differentiable on B(a, p) with the kth derivative satisfying the Lipschitz condition. We set Lip(~: a, p) = Lip~ a, p). Let d: D(~/) C ~'--* ~" be a closed linear operator with dense domain D(~/) (it will play the role of the nonperturbed operator). In the present paper, we consider a perturbed mapping of the form t= ~/- e-~: D(~) M B(0, p) C ~'--* ~, where -~ E Lipl(~, 0, p), p > 0, and e E K is a small parameter. Under certain conditions imposed on ~ and on the perturbation ~, the mapping .~e can be reduced to a mapping of a "simpler" structure. The transformation used to perform the reduction is an abstract analog of the Krylov-Bogolyubov transformation [1-3], which is applied in the justification of the averaging method (on an infinite interval) for ordinary differential equations (see the example below). If ~ belongs to the Banach algebra End ~V of linear bounded operators in ~, then we arrive at the method of similar operators [4, 5]. Definition 1. Let ~1, ~2 E Lip(~,a,p). We say that a mapping .~- ~1: D(.~) N-B(a,p) C ~f--~ J(f is equivalent to a mapping .~/- ~: D(~') A B(a, p) C ~'--* ~ on the ball B(a, P0), P0 ~ p, if there exists a mapping ~ Lip 1 (.~ a, Po), such that the linear operators ~ E End ~ x E B(a, Po), are invertible, supxe:~(a,po ) [[~]['(x)-~H < oo, ~ ~ D(~/) MB(a, p0) for any x ~ D(.~/) N-B(a, po), and (oe/- -~)(o?z'(x)) = o~"(x)(~/- ~2)(x), x e B(a, p0) n D(~t).