  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.
