A study on Kannan type contractive mappings

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In this article, we consider Kannan type contractive self-map T on a metric space (X,d) such that d(Tx,Ty)<1/2{d(x,Tx)+d(y,Ty)} for all x ≠ y ∈ 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. Górnicki [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.