common fixed point theorems for sequences of mappings with some weaker conditions

in this paper, we prove a common fixed point theorem for six mappings (two set valued and four single valued mappings) without assuming compatibility and continuity of any mapping on non complete metric spaces. to prove the theorem, we use a non compatible condition, that is, weak commutativity of type (kb). we show that completeness of the whole space is not necessary for the existence and uniqueness of common fixed point, and give an example to support our theorem. also, we prove a common fixed point theorem for two self mappings and two sequences set-valued mappings by the same weaker conditions. our results improve, extend and generalizes the corresponding results given by many authors.