Fixed point theorem for generalized Kannan type mappings

We introduce a new type of mappings in metric spaces which are three-point analogue of the well-known Kannan type mappings and call them generalized Kannan type mappings. It is shown that in general case such mappings are discontinuous but continuous at fixed points as well as Kannan type mappings and that these two classes of mappings are independent. The fixed-point theorem for generalized Kannan type mappings is proved. Additional conditions of asymptotic regularity and continuity allow us to extent the class of mappings for which the fixed-point theorems hold. Following Kannan, we also obtain two other fixed-point theorems for generalized Kannan type mappings in metric spaces which are not obligatory complete.