Let be a complete partially ordered metric space and be two maps with bounded values, fulfilling
for all , where satisfies some mild conditions,
for each , there exists with such that , where ,
if fulfills for all and as , then for all .
In addition, assume that there exists a continuous strictly increasing function with for all , such that
for some with ,
Then there exists in a common fixed point for and .
Reviewer’s remark: The seminal 2004 fixed point result of A. C. M. Ran and M. C. B. Reurings [Proc. Am. Math. Soc. 132, No. 5, 1435–1443 (2004; Zbl 1060.47056)] was obtained in 1986 by the reviewer [J. Math. Anal. Appl. 117, 100–127 (1986; Zbl 0613.47037)].