On Kannan-Geraghty maps as an extension of Kannan maps

Abstract

Extending the concept of weakly Kannan maps on metric spaces, we study the maps as $f:X\rightarrow X$ on a metric space $(X, d)$ satisfying condition $d(f(x), f(y)) \leq (1/2)\beta(d(x, y))[d(x ,f(x)) + d(y, f(y))]$ for every $x, y\in X$ and a function $\beta: [0, \infty)\rightarrow [0,1)$ where for every sequence $t=\{t_n\}$ of non-negative real numbers satisfying $\beta(t_n)\rightarrow 1,$ while $t_n\rightarrow 0$. Such a map is named the Kannan-Geraghty map because of its relation to weakly Kannan map and Geraghty contraction. Firstly, we show that our new condition is different from weakly Kannan condition. Having proven the fixed point theorem, we present two useful results on Kannan-Geraghty maps. Also, we illustrate some examples of Kannan-Graghty map having interesting properties.