Quasi-Contractions on a Nonnormal Cone Metric Space

Ilić and Rakočević [Appl. Math. Lett., 22:5 (2009), 728–731] proved a fixed point theorem for quasi-contractive mappings on cone metric spaces when the underlying cone is normal. Recently, Z. Kadelburg, S. Radenović, and V. Rakočević obtained a similar result without using the normality condition but only for a contractive constant $\lambda\in(0,1/2)$ [Appl. Math. Lett., 22:11 (2009), 1674–1679]. In this note, using a new method of proof, we prove this theorem for any contractive constant $\lambda \in (0,1)$.