Let be a cone metric space over a normal cone in a real Banach space in the sense of L.-G. Huang and X. Zhang [J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)]. Let satisfy the following properties: (q1) for all ; (q2) for all ; (q3) if is a sequence in converging to and for some and , for each , then ; (q4) for each with , there exists with , such that and imply . The function is called a -distance in (this notion is a cone metric version of the notion of -distance of O. Kada, T. Suzuki and W. Takahashi [Math. Japon. 44, No. 2, 381–391 (1996; Zbl 0897.54029)]).
The authors prove the following common fixed point result in terms of a -distance. Let , be constants with and let be two mappings satisfying the condition for all . Suppose that and is a complete subset of . If and satisfy for all with or , then and have a common fixed point in . No compatibility assumptions have to be used.