Fixed point theorem for generalized Kannan type mappings

TL;DR

This paper introduces generalized Kannan type mappings in metric spaces, proves a fixed-point theorem for them, and explores conditions under which these mappings are continuous at fixed points, extending fixed-point results beyond complete spaces.

Contribution

It defines a new class of mappings called generalized Kannan type mappings and establishes fixed-point theorems for them, including cases with asymptotic regularity and continuity.

Findings

Generalized Kannan mappings can be discontinuous but are continuous at fixed points.

Fixed-point theorems hold for these mappings even in non-complete metric spaces.

Additional conditions extend the applicability of fixed-point results.

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.