Principles for the existence of common fixed points for two functions were established by Sehie Park. Such principles cannot be generalized for more than two functions, but the authors make some progress in improving fixed point theorems involving four maps by weakening the condition of compatibility of two of the maps. Two selfmaps and of a metric space are said to be compatible if for every sequence in such that , . Two maps and are said to be weakly compatible if they commute at coincidence points. The following theorem is typical for the results of the paper.
Theorem 1. Let be selfmaps of a complete metric space satisfying
(i) and .
(ii) for each in , where
and is upper semicontinuous and such that for each . If either
(iii) are compatible, or is continuous and are weakly compatible, or
(iv) are compatible, or is continuous and are weakly compatible
then have a unique common fixed point .
Moreover, the theorem contains an assertion about sequences arising from the maps and converging to the common fixed point. The authors also point out that some fixed point theorems in the literature for three or four functions have incorrect or incomplete proofs.