Cyclic quasi-contractions of Ciric type in b-metric spaces

In this paper, we give a negative answer to the open question raised by Radenovic et al. [S. Radenovic, T. Dosenovic, T. A. Lampert, Z. Golubovıc, Appl. Math. Comput., 273 (2016), 155–164]. Namely, we give two examples which show that the set of fixed points for cyclic quasi-contractive mappings of Ciric type may be empty. Then, by using a new lemma, we give some sufficient conditions for the existence of fixed point for cyclic and non-cyclic quasi-contractive mappings of Ciric type in b-metric spaces. In particular, we show that the condition of Fatou property in the result of Amini-Harandi [A. Amini-Harandi, Fixed Point Theory, 15 (2014), 351–358] may be omitted.