A note on a paper of I.D. Arand̄elović on asymptotic contractions

W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirkʹs result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand̄elovićʹs theorem is false. We also explain how to correct this result.