We show that the range of contraction constant in a result of Karapınar et al. can not be extended to $[0,1)$. However, by using some additional conditions, we prove that the range can be extended to $[0,1)$. The first result gives a negative answer to the open question on a Ćirić type theorem in $b$-metric spaces, and the next results are improvements of Ćirić type theorems in $b$-metric spaces in the literature.