Fixed point theorem for generalized Kannan type mappings
Evgeniy Petrov, Ravindra K. Bisht
TL;DR
This paper introduces generalized Kannan-type mappings on metric spaces, defined by $d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz) \le \lambda(d(x,Tx)+d(y,Ty)+d(z,Tz))$ for pairwise distinct $x,y,z$ with $\lambda \in [0,2/3)$. It proves a fixed-point theorem on complete spaces: if $|X|\ge 3$, $T$ has no periodic points of prime period $2$, and $T$ is a generalized Kannan-type mapping, then $T$ has a fixed point and at most two fixed points; the method uses the triplet inequality to show the orbit is Cauchy. The work further extends these results under asymptotic regularity and continuity to generalized $F$-Kannan-type and generalized $\mathcal{B}$-Kannan-type mappings, allowing $\lambda$ to range up to $\infty$ under suitable conditions. It also establishes analogous fixed-point results in incomplete metric spaces and discusses approximate fixed-point sequences, emphasizing the role of continuity and accumulation points in ensuring convergence to fixed points.
Abstract
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.