  This paper investigates whether some fixed point theorems for
quasi-contractions on metric spaces introduced by \`Cir\`ic in [1] and
generalised by Kumam et al. in [2] can be improved further. It turns out that
the answer is negative. We provide two examples of complete metric spaces and
two operators without fixed points. We prove that for any possible
straightforward relaxation of generalised quasi-contractive conditions, one of
these operators satisfies the condition.
