We show that the range of contraction constant in a result of Karap?nar 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 Ciric type theorem in b-metric spaces, and the next results are improvements of Ciric type theorems in b-metric spaces in the literature.