Another perspective on Kannan contraction

Inspired by the well-known result stating that if any iterate of a mapping is a Banach contraction on a complete metric space, then the mapping itself possesses a unique fixed point, we investigate that claim for a Kannan contraction but by retaining the left-hand side of the inequality as per the mapping itself. With an additional assumption of $k$-continuity, the existence and uniqueness of a fixed point is obtained for a new class of contractions, $m$-Kannan contraction, on a complete metric space. Several examples are given in order to substantiate many theoretical claims such as discontinuity at the unique limit point of the iterative sequence or the ones testifying that this class is wider than the class of Kannan mappings.