Expansive and degenerate behaviour of multiplicative Kannan-type contractions for \( β> 1 \)

This paper analyzes multiplicative Kannan-type contractions (MKC) pp , q rpp, qs rpp, qs 1´ , 0 ă 1, outside the contractive range, with emphasis on the case ą 1.We show that when ą 1 the MKC condition induces an expansive mapping, so that distances between successive iterates increase, rendering classical Cauchy-based fixed point techniques ineffective.A classification is obtained according to the behaviour of the distances between points and their images under the mapping.If these distances are unbounded or admit arbitrarily small values, then any map satisfying (MKC) on subsets of R is necessarily degenerate, in the sense that the image of all non-fixed points collapses to a singleton.Non-degenerate maps arise only when these distances remain bounded and bounded away from zero.These results refine and correct existing assertions on (MKC) for ą 1.