Let be a cone metric space in the sense of [L-G. Huang and X. Zhang, J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)] over a regular cone , such that for .
Let be a mapping satisfying for all , where are continuous and increasing, satisfying: (a) iff ; (b) , for ; (c) for and (d) either or for and . Under these assumptions, the authors prove that has a unique fixed point in . A similar result is obtained for the existence of a common fixed point for two self-mappings.