In a recent paper [Pata, V., A fixed point theorem in metric spaces, J. Fixed Point Theory Appl.,10 (2011), No. 2, 299–305], the author stated and proved a fixed point theorem that is intended to generalize the well known Banach’s contraction mapping principle. In this note we show that the main result in the above paper does not hold at least in two extremal cases for the parameter involved in the contraction condition used there. We also present some illustrative examples and related results.