A study on Kannan type contractive mappings

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.