The article deals with new generalization of coincidence point theorem. It is assumed that are self-mappings of a metric space satisfying, for all , the inequality
where is a function with properties: (F) for all ; (F) for all ; (F) for all . Furthermore, it is assumed that there exist two sequences such that
and that and are closed. Under these conditions, it is stated that both pairs and have a coincidence point and, moreover, have a unique common fixed point, provided that both pairs and are weakly compatible ( is called weakly compatible if for some implies , and, similarly, for ). Some modifications of this statement are also given. A number of examples of functions are presented. Comparisons between old and new results are gathered at the end of the article.