Let be the set of all continuous functions satisfying the following conditions: (i) is non-increasing in and ; (ii) there exists such that if each of the relations or imply ; (iii) for all .
The following theorem is the main result of the paper. Let be a metric space and four mappings satisfying the following conditions: (a) , and one of the sets and is complete; (b) for all and
Then each of the pairs of mappings and has a coincidence point. Moreover, if each of the pairs of mappings and commute at their coincidence points, then and have a unique common fixed point.
The previous theorem improves under many aspects an earlier result of V. Popa [Demonstr. Math. 32, 157–163 (1999; Zbl 0926.54030]. Next the author establishes related results and give illustrative examples which demonstrate the utility of the proved results.