Given a Banach space and a closed set , let be a nonlinear operator which satisfies for some and the contraction-type condition
(, ). It was shown by B. Lou [Proc. Am. Math. Soc. 127, No. 8, 2259-2264 (1999; Zbl 0918.47046)] that has then a unique fixed point. In the present paper, the authors prove that this is equivalent to, and even weaker than, the Banach contraction principle in the setting of -normed spaces. Moreover, they show by means of a counterexample that the term above cannot be replaced by .