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We prove that main result of asymptotic contractions by I. A r a n đ e l o v i ć [Appl. Anal. Disrete Math. 1 (2007), 211-216, Theorem 1, p. 212] has been for the first time proved 21 years ago in T a s k o v i ć [Fundamental elements of the fixed point theory, ZUNS1986, Theorem 4, p. 170]. But, the author (and next other authors) this historical fact is to neglect and to ignore. 1. Main results and facts Let X be a topological space, T : X → X, and let A : X × X → R+. In 1986 Tasković [6] investigated the concept of TCS-convergence in a space X, i.e., a topological space X := (X,A) satisfies the condition of TCSconvergence iff x ∈ X and if A(Tnx, Tn+1x) → 0 (n → ∞) implies that {T(x)}n∈N has a convergent subsequence. The following results, given in the next two theorems are given in 1986 by M. R. Tasković [6] as a natural extension of characterization statements of asymptotically conditions of fixed point theorem given in 1985 by Tasković [7]. These results are according to topological spaces. Theorem 1 (Tasković [6]). Let T be a mapping of topological space X := (X,A) into itself, where X satisfies the condition of TCS-convergence. Suppose that there exist a sequence of nonnegative real functions {αn(x, y)}n∈N such that αn(x, y) → 0 (n→∞) and positive integer m(x, y) such that (B) A ( T(x), T(y) ) ≤ αn(x, y) for all n ≥ m(x, y), and for all x, y ∈ X, where A : X × X → R+. If x 7→ A(x, T (x)) is a T-orbitally lower semicontinuous function and A(a, b) = 0 implies a = b, 2000 Mathematics Subject Classification. Primary: 54E15, 47H10, 05A15; Secondary: 54E35, 54H25.

On a statement by I. Arandjelovic for asymptotic contractions in Appl. Anal. Discrete Math.