W. A. Kirk [8] introduced the notion of asymptotic contractions and proved fixed point theorem for this class of mappings. In note [1] we present a new short and simple proof of Kirk’s theorem. Further results on this class of mappings was obtained by: J. Jachymski, I. Joźwik [7], Y.-Z. Chen [4], P. Gerhardy [5], [6], T. Suzuki [9], H. K. Xu [12], M. Arav, F. E. C. Santos, S. Reich, A. Zaslavski [2] and K. W lodarczyk, D. Klim, R. Plebaniak [10], [11]. The papers [10] and [11] presents some ideas for application of the theory of asymptotic contractions in the analysis of set-valued dynamic systems. In this paper we present one fixed point theorem of Kirk’s type unifying and generalizing recent results of W. A. Kirk [8], J. Jachymski, I. Joźwik [7] and Y.-Z. Chen [4]. Let X be a nonempty set and f : X → X arbitrary mapping. x ∈ X is a fixed point for f if x = f(x). If x0 ∈ X , we say that a sequence (xn) defined by xn = f (x0) is a sequence of Picard iterates of f at point x0 or that (xn) is the orbit of f at point x0. In [2] M. Arav, F. E. C. Santos, S. Reich and A. Zaslavski proved the following result: