BULLETIN OF THE POLISH ACADEMY OF SCIENCES MATHEMATICS Vol. 56, No. 1, 2008 OPERATOR THEORY Two Results on Jachymski–Schr¨oder–Stein Contractions by Simeon REICH and Alexander J. ZASLAVSKI Presented by Bogdan BOJARSKI Summary. We establish two fixed point theorems for certain mappings of contractive type. 1. Introduction. Throughout this paper, (X, d) is a complete metric space, N 0 a natural number, and φ : [0, ∞) → [0, ∞) a function which is upper semicontinuous from the right and satisfies φ(t) <t for all t> 0. We call a mapping T : X → X for which (1.1) min{d(T i x, T i y): i ∈{1,...,N 0 }} ≤ φ(d(x, y)) for all x, y ∈ X a Jachymski–Schr¨oder–Steincontraction (with respect to φ). Such mappings with φ(t)= γt for some γ ∈ (0, 1) have recently been of considerable interest [1, 7–11]. In the present paper we study general Jachymski–Schr¨oder–Stein contractions and prove two fixed point theorems for them (Theorems 2.1 and 3.1 below). In our first result we establish con- vergence of iterates to a fixed point, and in the second this conclusion is strengthened to obtain uniform convergence on bounded subsets of X . This last type of convergence is useful in the study of inexact orbits [4]. Our the- orems contain the (by now classical) results of [2, 3] as well as Theorem 2 of [8], where condition (1.1) was first introduced. In contrast with our The- orem 2.1, it was assumed in [8, Theorem 2] that the function φ was upper semicontinuous and that lim inf t→∞ (t - φ(t)) > 0. Moreover, our argument is completely different from the one presented in [8], where the Cantor Inter- section Theorem was employed. We remark in passing that Cantor’s theorem was also used for a linear φ in [5, p. 22] (cf. also [6, p. 2]). 2000 Mathematics Subject Classification : 47H10, 54E50, 54H25. Key words and phrases : complete metric space, contractive mapping, fixed point, iteration. [53] c  Instytut Matematyczny PAN, 2008