A study on Kannan type contractive mappings

TL;DR

This paper investigates fixed point results for Kannan type contractive mappings in metric spaces without assuming compactness or continuity, and addresses an open question in the field.

Contribution

It establishes new fixed point theorems for Kannan type maps without compactness or continuity assumptions and confirms an open problem posed by Górnicki.

Findings

Proved fixed point theorems for Kannan type maps without compactness.

Connected fixed point properties to space completeness.

Provided examples illustrating the theoretical results.

Abstract

In this article, we consider Kannan type contractive self-map $T$ on a metric space $(X,d)$ such that \[d(Tx,Ty)<\frac{1}{2}\{d(x,Tx)+d(y,Ty)\} \mbox{ for all } x \neq y \in X, \] and establish some new fixed point results without taking the compactness of $X$ and also without assuming continuity of $T$. Further, we anticipate a result ensuring the completeness of the space $X$ via FPP of this map. Finally, we are able to give an affirmative answer to the open question posed by J. G\'{o}rnicki [\textit{Fixed point theorems for Kannan type mappings}, J. Fixed Point Theory Appl. 2017]. Apart from these, our manuscript consists of several non-trivial examples which signify the motivation of our investigations.