Coincidence Points and Common Fixed Points of Cyclic Maps by Implicit Contractive Conditions

The study of common fixed points and coincidence points has occupied a large part of the priorities of researchers, especially in the metric space and its generalizations. This done by weakening the assumption of commutativity by using: compatible mappings, weakly compatible mappings, R-sub weakly commuting mapping, Cq-commuting maps, occasionally weakly compatible maps, etc. The notion of occasionally weakly compatible and occasionally weakly biased mappings reduces to weak compatibility due to the unique coincidence points of the involved maps. It was pointed out that some results did not lead to effective generalizations for occasionally weakly compatible maps and occasionally weakly biased maps, which were used under contractive conditions. Given these facts, the concept of occasionally weak compatibility has been presented by defining the idea of conditionally commuting mappings, which construct a proper status in the context of studying common fixed points for a pair of mappings. In this consideration, the concept of cyclic form is used to prove fixed point theorems for mappings satisfying an implicit relation, including an altering distance function in a complete b-metric space. And prove some common fixed point in a complete b-metric space for a pair and two pairs of weakly compatible, discontinuous mappings having an implicit relation with an altering distance function employed for this purpose. Also, obtaining common fixed point theorems for pairs operators (f,S) which are occasionally weakly S-biased (or f-biased) of type (A). As well as, there are many examples that have been presented to illustrate the results of the research.